Solutions for College Algebra 12th Us Edition by Lial by 6alsm

Chapter 2

GRAPHS AND FUNCTIONS

Section 2.1 Rectangular Coordinates and Graphs

1. The point (–1, 3) lies in quadrant II in the rectangular coordinate system.

2. The point (4, 6) lies on the graph of the equation y = 3x – 6. Find the y-value by letting x = 4 and solving for y

3461266 y

3. Any point that lies on the x-axis has y-coordinate equal to 0.

4. The y-intercept of the graph of y = –2x + 6 is (0, 6).

5. The x-intercept of the graph of 2x + 5y = 10 is (5, 0). Find the x-intercept by letting y = 0 and solving for x

250102105

6. The distance from the origin to the point (–3, 4) is 5. Using the distance formula, we have

9. False. The midpoint of the segment joining (0, 0) and (4, 4) is

404044 , 2222,2,2.

10. False. The distance between the point (0, 0) and (4, 4) is 2222 (,)(40)(40)44 16163242

11. Any three of the following:

2,5,1,7,3,9,5,17,6,21

12. Any three of the following:

3,3,5,21,8,18,4,6,0,6

13. Any three of the following: (1999, 35), (2001, 29), (2003, 22), (2005, 23), (2007, 20), (2009, 20)

14. Any three of the following:

2002,86.8,2004,89.8,2006,90.7, 2008,97.4,2010,106.5,2012,111.4, 2014,111.5

7. True

15. P(–5, –6), Q(7, –1)

(a) 22 22 (,)[7–(–5)][1–(–6)] 12516913 dPQ  

(b) The midpoint M of the segment joining points P and Q has coordinates –576(1)27 , , 2222 7 1,. 2

8. True

16. P(–4, 3), Q(2, –5)

(a) 22 22 (,)[2–(–4)](–5–3) 6(–8)10010 dPQ 

(b) The midpoint M of the segment joining points P and Q has coordinates  – 423(–5)22 , , 2222 1,1.

17. (8,2), (3,5) PQ

(a)  22 2 2 (,)(3–8)(5–2) 53 25934 dPQ 

(b) The midpoint M of the segment joining points P and Q has coordinates 8325117 , , 2222

18. P ( 8, 4), Q (3, 5)

(a)  2 2 22 (,)3–854 11(–9)12181 202 dPQ

(b) The midpoint M of the segment joining points P and Q has coordinates –834(–5)51 , 2222,.

19. P(–6, –5), Q(6, 10)

(a) 22 22 (,)[6–(–6)][10–(–5)] 1215144225 369341

(b) The midpoint M of the segment joining points P and Q has coordinates – 66–510055 ,,0,. 22222

20. P(6, –2), Q(4, 6)

(a) 22 22 (,)(4–6)[6–(–2)] (–2)8 46468217

(b) The midpoint M of the segment joining points P and Q has coordinates

21.   32, 45, P   2,–5 Q (a)

22 22 (,) 2–32–5–45 –22–55 8125133

(b) The midpoint M of the segment joining points P and Q has coordinates 32245(–5) , 22 423535 222,22,.

22.     – 7, 83, 57, –3 PQ (a) 22 22 (,) [57–(–7)](–3–83) (67)(–93)252243 495355 dPQ

(b) The midpoint M of the segment joining points P and Q has coordinates –75783(–3) , 22 477373 , 27, . 222 

23. Label the points A(–6, –4), B(0, –2), and C(–10, 8). Use the distance formula to find the length of each side of the triangle.

22 22 (,)[0–(–6)][–2–(–4)] 6236440 dAB   22 22 (,)(–10–0)[8–(–2)] (10)10100100

200 dBC 

  22 22 (,)[–10–(–6)][8–(–4)] (–4)1216144160 dAC 

 Because       222 40160200,  triangle ABC is a right triangle.

24. Label the points A(–2, –8), B(0, –4), and C(–4, –7). Use the distance formula to find the length of each side of the triangle.

22 22 (,)[0–(–2)][–4–(–8)] 2441620 dAB 



dBC 

22 22 (,)(–4–0)[–7–(–4)] (–4)(–3)169 255





dAC 

22 22 (,)[–4–(–2)][–7–(–8)] (–2)1415



Because 222 (5)(20)520255,  triangle ABC is a right triangle.

25. Label the points A(–4, 1), B(1, 4), and C(–6, –1).

22 22 (,)[1–(–4)](4–1) 5325934 dAB 



dBC 

22 22 (,)(–6–1)(–1–4) (–7)(–5)492574



dAC 

22 22 (,)[–6–(–4)](–1–1) (–2)(–2)448



Because 222 (8)(34)(74)  because 8344274,  triangle ABC is not a right triangle.

26. Label the points A(–2, –5), B(1, 7), and C(3, 15).

dAB 

22 22 (,)[1(2)][7(5)] 3129144153



dBC  

22 22 (,)(31)(157) 2846468

dAC 

22 22 (,)[3(2)][15(5)] 52025400425



Because  2 22 (68)(153)425  because 68153221425  , triangle ABC is not a right triangle.

27. Label the points A(–4, 3), B(2, 5), and C(–1, –6).

dAB  

 2 2 22 (,)2––453 6236440

dBC 





dAC

  2 2 2 2 (,)–1––463 3998190

   

Because       222 4090130,  triangle

ABC is a right triangle.

dAB

28. Label the points A(–7, 4), B(6, –2), and C(0, –15).   2 2 2 2 (,)6––724 136 16936205

dBC

2 2 22 (,)06–15––2 –6–13 36169205

dAC

2 2 2 2 (,)0––7154 71949361410

Because

  

 222 205205410,  triangle ABC is a right triangle.

29. Label the given points A(0, –7), B(–3, 5), and C(2, –15). Find the distance between each pair of points.



  2 2 2 2 (,)305––7 –3129144 153317 dAB   

dBC   

   2 2 2 2 (,)2–3–15–5 5–2025400 425517

 

  2 2 2 2 (,)2015–7 2–868217 dAC  



  22 22 (,)1265 (–3)(–11) 9121130

Because (,)(,)(,) dABdACdBC  or 317217517,  the points are collinear.

30. Label the points A(–1, 4), B(–2, –1), and C(1, 14). Apply the distance formula to each pair of points.

  2 2 22 (,)–2––1–1–4 –1–526

dBC  

dAB     22 22 (,)1––214––1 315234326

dAC    

  2 2 22 (,)1––114–4 210104226

Because 26226326  , the points are collinear.

31. Label the points A(0, 9), B(–3, –7), and C(2, 19).

dAB  

22 22 (,)(–3–0)(–7–9) (–3)(–16)9256 26516.279



dBC  

22 22 (,)2––319––7 52625676 70126.476

 

dAC 

22 22 (,)2–019–9 2104100 10410.198





Because (,)(,)(,) dABdACdBC  or 265104701 16.27910.19826.476, 26.47726.476,    the three given points are not collinear. (Note, however, that these points are very close to lying on a straight line and may appear to lie on a straight line when graphed.)

32. Label the points A(–1, –3), B(–5, 12), and C(1, –11).

22 2 2 (,)–5––112––3 – 41516225 24115.5242

Because d(A,B) + d(A,C)  d(B,C) or 24168565 15.52428.246223.7697 23.770423.7697,    the three given points are not collinear. (Note, however, that these points are very close to lying on a straight line and may appear to lie on a straight line when graphed.)

33. Label the points A(–7, 4), B(6,–2), and C(–1,1).

dAB

2 2 2 2 (,)6––724 13616936 20514.3178

2 2 2 2 (,)161–2 73499 587.6158 dBC

2 2 2 2 (,)1––714 6–3369 456.7082

Because d(B,C) + d(A,C)  d(A,B) or 5845205

7.61586.708214.3178 14.324014.3178,    the three given points are not collinear. (Note, however, that these points are very close to lying on a straight line and may appear to lie on a straight line when graphed.)

34. Label the given points A(–4, 3), B(2, 5), and C(–1, 4). Find the distance between each pair of points.

 2 2 (,)2––4536222 36440210 dAB   

dBC 

  22 2 2 (,)(12)(45) 3(–1)9110



 2 2 22 (,)1–443 319110 dAC   



Because (,)(,)(,) dBCdACdAB  or 1010210,  the points are collinear.

35. Midpoint (5, 8), endpoint (13, 10) 1310 5and8 22 1310and1016 –3and6. xy xy xy

The other endpoint has coordinates (–3, 6).

36. Midpoint (–7, 6), endpoint (–9, 9) –99 –7and6 22 –9–14and912 –5and3. xy

The other endpoint has coordinates (–5, 3).

37. Midpoint (12, 6), endpoint (19, 16) 1916 12and6 22 1924and1612 5and–4. xy xy xy

The other endpoint has coordinates (5, –4).

38. Midpoint (–9, 8), endpoint (–16, 9) –169 –9and8 22 –16–18and916 –2and7 xy xy xy

The other endpoint has coordinates (–2, 7).

39. Midpoint (a, b), endpoint (p, q) and 22 2and2 2and2 pxqyab pxaqyb xapybq

The other endpoint has coordinates (2,2) apbq

40. Midpoint 

6,6ab , endpoint 

3,5ab 35 6and6 22 312and512 9and7 axbyab axabyb xayb

The other endpoint has coordinates (9a, 7b

41. The endpoints of the segment are (1990, 21.3) and (2012, 30.1).

1990201221.330.9 , 22 2001,26.1 M

The estimate is 26.1%. This is very close to the actual figure of 26.2%.

42. The endpoints are (2006, 7505) and (2012, 3335)

2006201275053335 , 22 2009,5420

According to the model, the average national advertising revenue in 2009 was $5420 million. This is higher than the actual value of $4424 million.

43. The points to use are (2011, 23021) and (2013, 23834). Their midpoint is 2011201323,02123,834 , 22 (2012,23427.5).

In 2012, the poverty level cutoff was approximately $23,428.

44. (a) To estimate the enrollment for 2003, use the points (2000, 11,753) and (2006, 13,180)

2000200611,75313,180 , 22 2003,12466.5

The enrollment for 2003 was about 12,466.5 thousand.

(b) To estimate the enrollment for 2009, use the points (2006, 13,180) and (2012, 14,880)

2006201213,18014,880 , 22 2009,14030 M

The enrollment for 2009 was about 14,030 thousand.

45. The midpoint M has coordinates 1212 , .

(continued)

this shows (,)(,)(,) dPMdMQdPQ  and (,)(,). dPMdMQ 

46. The distance formula, 22 2121 (–)(–),dxxyy  can be written as 221/2 2121 [(–)(–)].dxxyy 

In exercises 47 58, other ordered pairs are possible.

47. (a) x y 0 2 y-intercept:  1 2 0 022 x y

4 0 x-intercept: 1 2 1 2 0 02 24 y x xx   

2 1 additional point

(b)

48. (a) x y 0 2 y-intercept:  0 1 022 2 x y  

4 0 x-intercept: 0 1 02 2 1 24 2 y x xx   

2 1 additional point

(b)

49. (a) x y 0 5 3 y-intercept:  5 3 0 2035 35 x y yy    5 2 0 x-intercept:  5 2 0 2305 25 y x xx    4 1 additional point

(b)

50. (a) x y

0 3 y-intercept:  0 3026 263 x y yy   

2 0 x-intercept:  0 3206 362 y x xx 

4 3 additional point

(b)

51. (a) x y 0 0 x- and y-intercept: 002 

1 1 additional point

2 4 additional point (b)

52. (a) x y 0 2 y-intercept: 2 0 02 022 x y yy   

1 3 additional point

2 6 additional point no x-intercept: 2 2 002 22 yx xx  

53. (a) x y 3 0 x-intercept: 0 03 033 y x xx   

4 1 additional point

7 2 additional point no y-intercept: 0033xyy 

54. (a) x y 0 3 y-intercept: 0 03 033 x y yy   

4 1 additional point

9 0 x-intercept: 0 03 39 y x xx    (b)

55. (a) x y

0 2 y-intercept: 0 02 22 x y yy  

2 0 x-intercept: 0 02 022 y x xx  

2 4 additional point

4 2 additional point

(b)

56. (a) x y 2 2 additional point

4 0 x-intercept: 0 04 04 044 y x x xx 

0 4 y-intercept: 0 04 44 x y yy  

(b)

57. (a) x y

0 0 x- and y-intercept: 003 

1 1 additional point

2 8 additional point

(b)

58. (a) x y 0 0 x- and y-intercept: 003  1 1additional point 2 8additional point

(b)

59. Points on the x-axis have y-coordinates equal to 0. The point on the x-axis will have the same x-coordinate as point (4, 3). Therefore, the line will intersect the x-axis at (4, 0).

60. Points on the y-axis have x-coordinates equal to 0. The point on the y-axis will have the same y-coordinate as point (4, 3). Therefore, the line will intersect the y-axis at (0, 3).

61. Because (a, b) is in the second quadrant, a is negative and b is positive. Therefore, (a, – b) will have a negative x–coordinate and a negative y-coordinate and will lie in quadrant III.

(–a, b) will have a positive x-coordinate and a positive y-coordinateand will lie in quadrant I. (–a, – b) will have a positive x-coordinate and a negative y-coordinate and will lie in quadrant IV.

(b, a) will have a positive x-coordinate and a negative y-coordinate and will lie in quadrant IV.

62. Label the points (2,2),A (13,10), B (21,5), C and (6,13). D To determine which points form sides of the quadrilateral (as opposed to diagonals), plot the points.

Use the distance formula to find the length of each side.

dBC





dDA  

2 2 2 2 (,)26213 815 6422528917





Because all sides have equal length, the four points form a rhombus.

63. To determine which points form sides of the quadrilateral (as opposed to diagonals), plot the points.

Use the distance formula to find the length of each side.

 22 22 (,)5121 4116117

 22 2 2 (,)3542 22448 dBC

dCD

22 22 (,)1334 41 16117

2 2 2 2 (,)1113 22448

Because d(A, B) = d(C, D) and d(B, C) = d(D, A), the points are the vertices of a parallelogram. Because d(A, B) ≠ d(B, C), the points are not the vertices of a rhombus.

64. For the points A(4, 5) and D(10, 14), the difference of the x-coordinates is 10 – 4 = 6 and the difference of the y-coordinates is 14 – 5 = 9. Dividing these differences by 3, we obtain 2 and 3, respectively. Adding 2 and 3 to the x and y coordinates of point A, respectively, we obtain B(4 + 2, 5 + 3) or B(6, 8).

Adding 2 and 3 to the x- and y- coordinates of point B, respectively, we obtain C(6 + 2, 8 + 3) or C(8, 11). The desired points are B(6, 8) and C(8, 11).

We check these by showing that d(A, B) = d(B, C) = d(C, D) and that d(A, D) = d(A, B) + d(B, C) + d(C, D). 

dAB





dBC



 22 22 22 22 (,)6485 234913 (,)86118 234913

dCD dAD



   22 22 22 22 (,)1081411 234913 (,)104145 693681 1179(13)313









d(A, B), d(B, C), and d(C, D) all have the same measure and d(A, D) = d(A, B) + d(B, C) + d(C, D) Because 313131313. 

Section 2.2 Circles

1. The circle with equation 22 49 xy has center with coordinates (0, 0) and radius equal to 7.

2. The circle with center (3, 6) and radius 4 has equation  2 3616.xy

3. The graph of  22 479xy has center with coordinates (4, –7).

4. The graph of 2 2 59 xy has center with coordinates (0, 5).

5. This circle has center (3, 2) and radius 5. This is graph B.

6. This circle has center (3, –2) and radius 5. This is graph C.

7. This circle has center (–3, 2) and radius 5. This is graph D.

8. This circle has center (–3, –2) and radius 5. This is graph A.

9. The graph of 22 0 xy has center (0, 0) and radius 0. This is the point (0, 0). Therefore, there is one point on the graph.

10. 100 is not a real number, so there are no points on the graph of 22 100. xy

11. (a) Center (0, 0), radius 6

22 22 222 006 00636 xy xyxy   (b)

12. (a) Center (0, 0), radius 9

22 22 222 009 00981 xy

(b)

13. (a) Center (2, 0), radius 6

22 22 2 22 206 206 (–2)36 xy xy xy

14. (a) Center (3, 0), radius 3

 22 2 2 303 39 xy xy   (b)

15. (a) Center (0, 4), radius 4   22 2 2 044 416 xy xy   (b)

16. (a) Center (0, –3), radius 7

2 2 2 2 2 22 037 037 (3)49 xy xy xy

17. (a) Center (–2, 5), radius 4  2 2 222 22 254 [–(–2)](–5)4 (2)(–5)16 xy xy xy

18. (a) Center (4, 3), radius 5

22 22 2 22 435 435 4325 xy xy xy

19. (a) Center (5, –4), radius 7

2 2 222 22 547 (–5)[–(–4)]7 (–5)(4)49 xy xy xy

20. (a) Center (–3, –2), radius 6

22 22 2 22 326 326 (3)(2)36 xy xy xy

21. (a) Center   2,2 , radius 2

22. (a) Center   3,3 , radius 3

(b)

23. (a) The center of the circle is located at the midpoint of the diameter determined by the points (1, 1) and (5, 1). Using the midpoint formula, we have

1511 22,3,1 C

. The radius is one-half the length of the diameter:

The equation of the circle is  22 314xy

(b) Expand  22 314xy to find the equation of the circle in general form:

24. (a) The center of the circle is located at the midpoint of the diameter determined by the points ( 1, 1) and ( 1, 5). Using the midpoint formula, we have

1(1)1(5),1,2. 22 C

The radius is one-half the length of the diameter:

The equation of the circle is

22 129xy

(b) Expand

22 129xy to find the equation of the circle in general form:

25. (a) The center of the circle is located at the midpoint of the diameter determined by the points ( 2, 4) and ( 2, 0). Using the midpoint formula, we have  2(2)40,2,2. 22 C     

The radius is one-half the length of the diameter:

2 2 1 22402 2 r  

The equation of the circle is  22 224xy

(b) Expand  22 224xy to find the equation of the circle in general form:

22 22 22 224 44444 4440 xy xxyy xyxy

26. (a) The center of the circle is located at the midpoint of the diameter determined by the points (0, 3) and (6, 3). Using the midpoint formula, we have

063(3),3,3 22 C

The radius is one-half the length of the diameter:

2 2 1 60333 2 r

The equation of the circle is  22 339xy

(b) Expand  22 339xy to find the equation of the circle in general form:  22 22 22 339 69699 6690 xy xxyy xyxy   

27. 22 6890xyxy

Complete the square on x and y separately.  

22 22 22 68–9 69816–9916 3416 xxyy xxyy xy 

 

Yes, it is a circle. The circle has its center at (–3, –4) and radius 4.

28. 22 8–6160xyxy

Complete the square on x and y separately.

22 22 22 8–616 816–69–16169 4–39 xxyy xxyy xy

Yes, it is a circle. The circle has its center at (–4, 3) and radius 3.

29. 22 4124xyxy

Complete the square on x and y separately.

22 22 22 –412–4 – 441236–4436 –2636 xxyy xxyy xy

Yes, it is a circle. The circle has its center at (2, –6) and radius 6.

30. 22 – 1210–25xyxy

Complete the square on x and y separately.

22 22 22 – 1210–25 –12361025 – 253625 –6536

Yes, it is a circle. The circle has its center at (6, –5) and radius 6.

31. 22 444–16–190 xyxy

Complete the square on x and y separately.

Yes, it is a circle with center

32. 22 9912–18–230

Complete the square on x and y separately.



Yes, it is a circle with center   2 3 ,1 and radius 2.

33. 22 2–6140xyxy

Complete the square on x and y separately.

22 22 22 2–6–14 21–69–1419 1–3–4 xxyy xxyy xy   

The graph is nonexistent.

34. 22 4–8320xyxy

Complete the square on x and y separately.

22 22 22 4–8–32 44–816 – 32416 2–4–12 xxyy xxyy xy 

  

The graph is nonexistent.

35. 22 66180xyxy

Complete the square on x and y separately.

 22 22 22 6618 69691899 330 xxyy xxyy xy 





The graph is the point (3, 3).

36. 22 4480xyxy

Complete the square on x and y separately.

22 22 22 448 4444844 220 xxyy xxyy xy 





The graph is the point ( 2, 2).

37. 22 9966230 xyxy

Complete the square on x and y separately.

  22 2222 33 2 22 23 212111 3939999 222 255 11 3393 969623 9923 xxyy xxyy xxyy xy 









Yes, it is a circle with center   11 33 , and radius 5 3

38. 22 444470 xyxy

Complete the square on x and y separately.

Yes, it is a circle with center   11 22 , and radius 3 2

39. The equations of the three circles are 22 (7)(4)25 xy , 22 (9)(4)169 xy , and 22 (3)(9)100 xy . From the graph of the three circles, it appears that the epicenter is located at (3, 1).

Check algebraically: 22 22 22 (7)(4)25 (37)(14)25 43252525

xy



 22 22 22 (9)(4)169 (39)(14)169 125169169169

xy



 22 22 22 (3)(9)100 (33)(19)100 6(8)100100100

xy



 (3, 1) satisfies all three equations, so the epicenter is at (3, 1).

40. The three equations are 22 (3)(1)5 xy , 22 (5)(4)36 xy , and 22 (1)(4)40 xy . From the graph of the three circles, it appears that the epicenter is located at (5, 2).

Check algebraically: 22 22 22 (3)(1)5 (53)(21)5 21555

xy   22 22 2 (5)(4)36 (55)(24)36 6363636

xy 

 22 22 22 (1)(4)40 (51)(24)40 6(2)404040 xy  

(5, 2) satisfies all three equations, so the epicenter is at (5, 2).

41. From the graph of the three circles, it appears that the epicenter is located at ( 2, 2).

Check algebraically: 22 22 22 (2)(1)25 (22)(21)25 (4)(3)25 2525

xy    22 22 22 (2)(2)16 (22)(22)16 0(4)16 1616

xy   

xy    22 22 22 (1)(2)9 (21)(22)9 (3)09 99

( 2, 2) satisfies all three equations, so the epicenter is at ( 2, 2).

42. From the graph of the three circles, it appears that the epicenter is located at (5, 0).

Check algebraically: 22

xy

22 22 (2)(4)25 (52)(04)25 3(4)25 2525

  





 22 22 2 2 2 1316 129616 281016 2860 430 xx xxxx xx xx xx





To solve this equation, we can use the quadratic formula with a = 1, b = –4, and c = –3.

  2 44413 21 41612428 22 427 27 2 x  

xy   

22 22 22 (1)(3)25 (51)(03)25 4325 2525

xy

22 22 22 (3)(6)100 (53)(06)100 86100 100100



  (5, 0) satisfies all three equations, so the epicenter is at (5, 0).

43. The radius of this circle is the distance from the center C(3, 2) to the x-axis. This distance is 2, so r = 2.

222 22 (–3)(–2)2 (–3)(–2)4 xy xy

 

44. The radius is the distance from the center C(–4, 3) to the point P(5, 8).

22 22 [5–(–4)](8–3) 95106 r 



The equation of the circle is 222 23 [–(–4)](–3)(106) (4)(–3)106 xy xy 



45. Label the points P(x, y) and Q(1, 3).

If (, )4 dPQ  ,  22 134 xy 

 22 1316. xy 

If x = y, then we can either substitute x for y or y for x. Substituting x for y we solve the following:

Because x = y, the points are

27, 27 and 2–7, 27.

46. Let P(–2, 3) be a point which is 8 units from Q(x, y). We have

 22 (,)238 dPQxy  22 2364. xy 

Because x + y = 0, x = –y. We can either substitute x for y or y for x. Substituting x for y we solve the following: 

  

 2 2 22 22 2 2 2364 2364 449664 2101364 210510 xx xx xxxx xx xx

 





To solve this equation, use the quadratic formula with a = 2, b = 10, and c = –51.  

 

 2 10104251 22 10100408 4 104127 10508 44 1021275127 42 x

 

  

 

Because yx  the points are 51275127 , 22      and –51275127 , . 22 

   

47. Let P(x, y) be a point whose distance from A(1, 0) is 10 and whose distance from B(5, 4) is 10 . d(P, A) = 10 , so

22 (1)(0)10 xy 

22 (1)10. xy  (,)10, so dPB 

22 (5)(4)10 xy 

22 (5)(4)10. xy  Thus, 2222 22 22 (1)(5)(4) 12 2510168 1241108 8408 5 xyxy xxy xxyy xxy yx yx

Substitute 5 – x for y in the equation 22 (1)10 xy  and solve for x 22 22 22 2 (1)(5)10 12251010 2122610212160

680(2)(4)0 20or40 2or4 xx xxxx xxxx xxxx xx xx 

To find the corresponding values of y use the equation y = 5 – x. If x = 2, then y = 5 – 2 = 3. If x = 4, then y = 5 – 4 = 1. The points satisfying the conditions are (2, 3) and (4, 1).

48. The circle of smallest radius that contains the points A(1, 4) and B(–3, 2) within or on its boundary will be the circle having points A and B as endpoints of a diameter. The center will be M, the midpoint:

13 4226 , , 2222

( 1, 3).

The radius will be the distance from M to either A or B:

The equation of the circle is

49. Label the points A(3, y) and B(–2, 9).

If d(A, B) = 12, then

22 22 22 2 2 2 23912 5912 5912 258118144 18380 y y y yy yy

Solve this equation by using the quadratic formula with a = 1, b = –18, and c = –38:

 2 18184138 21 1832415218476 212 y

184119 182119 9119 22  

The values of y are 9119  and 9119

50. Because the center is in the third quadrant, the radius is 2 , and the circle is tangent to both axes, the center must be at (2, 2).

Using the center-radius of the equation of a circle, we have

51. Let P(x, y) be the point on the circle whose distance from the origin is the shortest. Complete the square on x and y separately to write the equation in center-radius form: 22 22 22 1614880 16641449 886449 (8)(7)25 xxyy xxyy xy

So, the center is (8, 7) and the radius is 5.

22 (,)87113dCO  . Because the length of the radius is 5, (,)1135dPO 

52. Using compasses, draw circles centered at Wickenburg, Kingman, Phoenix, and Las Vegas with scaled radii of 50, 75, 105, and 180 miles respectively. The four circles should intersect at the location of Nothing.

53. The midpoint M has coordinates 

3–9 –1+546 , , (2, –3).

54. Use points C(2, –3) and P(–1, 3).

2 2 2 2 (,)–1–23––3 –36936 4535

The radius is 35.

55. Use points C(2, –3) and Q(5, –9).

2 2 2 2 (,)5–2–9––3 3–6936 4535

The radius is 35.

56. Use the points P(–1, 3) and Q(5, –9).

Because  2 2 (,)5––1–9–3dPQ   

2 2 6–1236144180

65  ,the radius is 1 (,). 2 dPQ Thus

1 6535. 2 r 

57. The center-radius form for this circle is 222 (–2)(3)(35) xy 22 (–2)(3)45. xy

58. Label the endpoints of the diameter P(3, –5) and Q(–7, 3). The midpoint M of the segment joining P and Q has coordinates 3(–7)–534–2,,(–2, –1). 2222

   

The center is C(–2, –1). To find the radius, we can use points C(–2, –1) and P(3, –5)

dCP  

  22 2 2 (,)3––2–5––1 5–4251641



We could also use points C(–2, –1).and Q(–7, 3).   22 2 2 (,)7––23––1 54251641 dCQ

We could also use points P(3, –5) and Q(–7, 3) to find the length of the diameter. The length of the radius is one-half the length of the diameter.

 2 2 2 2 (,)7–33––5 10810064 164241 dPQ

11 22(,)24141 dPQ 

The center-radius form of the equation of the circle is 222 22 [–(–2)][–(–1)](41) (2)(1)41 xy xy 



59. Label the endpoints of the diameter P(–1, 2) and Q(11, 7). The midpoint M of the segment joining P and Q has coordinates 111279 222,5,.      

The center is   9 2 5,. C To find the radius, we can use points   9 2 5, C and P( 1, 2).

  2 2 9 2 2 2 516913 242 (,)5––12 6 dCP   



We could also use points   9 2 5, C and Q(11, 7).

   2 2 9 2 2 2 516913 242 (,)511–7 6 dCQ   (continuedonnextpage)

(continued)

Using the points P and Q to find the length of the diameter, we have

22 22 ,11127 125 16913

 1113 ,13 222

The center-radius form of the equation of the circle is

22 2 913 22 2 2 9169 24 5 5 xy

60. Label the endpoints of the diameter P(5, 4) and Q( 3, 2). The midpoint M of the segment joining P and Q has coordinates

5(3)4(2),1,1. 22

The center is C(1, 1). To find the radius, we can use points C(1, 1) and P(5, 4).

43255 dCP  

We could also use points C(1, 1) and Q( 3, 2).

22 22 (,)131–2 43255

Using the points P and Q to find the length of the diameter, we have

22 22 ,5342 8610010 dPQ 

11 22(,)105 dPQ 

The center-radius form of the equation of the circle is

22 2 22 115 1125 xy xy

61. Label the endpoints of the diameter P(1, 4) and Q(5, 1). The midpoint M of the segment joining P and Q has coordinates

5 2 1541 22,3,.  

The center is   5 2 3,. C

The length of the diameter PQ is

222 2 154143255. 

The length of the radius is   5 1 225. 

The center-radius form of the equation of the circle is

62. Label the endpoints of the diameter P( 3, 10) and Q(5, 5). The midpoint M of the segment joining P and Q has coordinates

5 2 3510(5),1,. 22



The center is   5 2 1,. C

The length of the diameter PQ is 

2 22 351058152 28917.

The length of the radius is   17 1 2217. 

The center-radius form of the equation of the circle is

22 2 517 22 2 2 5289 24 1 1 xy xy 

Section 2.3 Functions

1. The domain of the relation        3,5,4,9,10,13 is  3,4,10.

2. The range of the relation in Exercise 1 is  5,9,13.

3. The equation y = 4x – 6 defines a function with independent variable x and dependent variable y

4. The function in Exercise 3 includes the ordered pair (6, 18).

5. For the function   42,fxx     24228210. f 

6. For the function   , gxx    993. g 

7. The function in Exercise 6,   , gxx  has domain   0,. 

8. The function in Exercise 6,   , gxx  has range   0,. 

For exercises 9 and 10, use this graph.

9. The largest open interval over which the function graphed here increases is   ,3.

10. The largest open interval over which the function graphed here decreases is   3,. 

11. The relation is a function because for each different x-value there is exactly one y-value. This correspondence can be shown as follows.

12. The relation is a function because for each different x-value there is exactly one y-value. This correspondence can be shown as follows.

13. Two ordered pairs, namely (2, 4) and (2, 6), have the same x-value paired with different y-values, so the relation is not a function.

14. Two ordered pairs, namely (9, 2) and (9, 1), have the same x-value paired with different y-values, so the relation is not a function.

15. The relation is a function because for each different x-value there is exactly one y-value. This correspondence can be shown as follows.

16. The relation is a function because for each different x-value there is exactly one y-value. This correspondence can be shown as follows.

17. The relation is a function because for each different x-value there is exactly one y-value. This correspondence can be shown as follows.

18. The relation is a function because for each different x-value there is exactly one y-value. This correspondence can be shown as follows.

19. Two sets of ordered pairs, namely (1, 1) and (1, 1) as well as (2, 4) and (2, 4), have the same x-value paired with different y-values, so the relation is not a function.

domain: {0, 1, 2}; range: { 4, 1, 0, 1, 4}

20. The relation is not a function because the x-value 3 corresponds to two y-values, 7 and 9. This correspondence can be shown as follows.

domain: {2, 3, 5}; range: {5, 7, 9, 11}

21. The relation is a function because for each different x-value there is exactly one y-value.

domain: {2, 3, 5, 11, 17}; range: {1, 7, 20}

22. The relation is a function because for each different x-value there is exactly one y-value.

domain: {1, 2, 3, 5}; range: {10, 15, 19, 27}

23. The relation is a function because for each different x-value there is exactly one y-value. This correspondence can be shown as follows.

Domain: {0, 1, 2}; range: {0, 1, 2}

24. The relation is a function because for each different x-value there is exactly one y-value. This correspondence can be shown as follows.

Domain: {0, 1, 2}; range: {0, 1, 2}

25. The relation is a function because for each different year, there is exactly one number for visitors.

domain: {2010, 2011, 2012, 2013} range: {64.9, 63.0, 65.1, 63.5}

26. The relation is a function because for each basketball season, there is only one number for attendance.

domain: {2011, 2012, 2013, 2014} range: {11,159,999, 11,210,832, 11,339,285, 11,181,735}

27. This graph represents a function. If you pass a vertical line through the graph, one x-value corresponds to only one y-value.

domain:   ,; range:   , 

28. This graph represents a function. If you pass a vertical line through the graph, one x-value corresponds to only one y-value.

domain: 

,; range:   ,4

29. This graph does not represent a function. If you pass a vertical line through the graph, there are places where one value of x corresponds to two values of y domain:   3,;  range:   , 

30. This graph does not represent a function. If you pass a vertical line through the graph, there are places where one value of x corresponds to two values of y domain: [ 4, 4]; range: [ 3, 3]

31. This graph represents a function. If you pass a vertical line through the graph, one x-value corresponds to only one y-value.

domain:   ,; range:   , 

32. This graph represents a function. If you pass a vertical line through the graph, one x-value corresponds to only one y-value. domain: [ 2, 2]; range: [0, 4]

33. 2 yx  represents a function because y is always found by squaring x. Thus, each value of x corresponds to just one value of yx can be any real number. Because the square of any real number is not negative, the range would be zero or greater.

domain:   ,; range:   0, 

34. 3 yx  represents a function because y is always found by cubing x. Thus, each value of x corresponds to just one value of y. x can be any real number. Because the cube of any real number could be negative, positive, or zero, the range would be any real number.

domain:   ,; range:   , 

35. The ordered pairs (1, 1) and (1, 1) both satisfy 6 . xy  This equation does not represent a function. Because x is equal to the sixth power of y, the values of x are nonnegative. Any real number can be raised to the sixth power, so the range of the relation is all real numbers.

domain:   0,  range:   , 

36. The ordered pairs (1, 1) and (1, 1) both satisfy 4 . xy  This equation does not represent a function. Because x is equal to the fourth power of y, the values of x are nonnegative. Any real number can be raised to the fourth power, so the range of the relation is all real numbers.

domain:

37. 25yx represents a function because y is found by multiplying x by 2 and subtracting 5. Each value of x corresponds to just one value of yx can be any real number, so the domain is all real numbers. Because y is twice x, less 5, y also may be any real number, and so the range is also all real numbers.

domain:

38. 64yx represents a function because y is found by multiplying x by 6 and adding 4. Each value of x corresponds to just one value of y. x can be any real number, so the domain is all real numbers. Because y is 6 times x, plus 4, y also may be any real number, and so the range is also all real numbers.

domain:   ,; range:  

39. By definition, y is a function of x if every value of x leads to exactly one value of y. Substituting a particular value of x, say 1, into x + y < 3 corresponds to many values of y. The ordered pairs (1, –2), (1, 1), (1, 0), (1, 1), and so on, all satisfy the inequality. Note that the points on the graphed line do not satisfy the inequality and only indicate the boundary of the solution set. This does not represent a function. Any number can be used for x or for y, so the domain and range of this relation are both all real numbers.

domain: 

,; range:

40. By definition, y is a function of x if every value of x leads to exactly one value of y.

Substituting a particular value of x, say 1, into xy < 4 corresponds to many values of y. The ordered pairs (1, 1), (1, 0), (1, 1), (1, 2), and so on, all satisfy the inequality. Note that the points on the graphed line do not satisfy the inequality and only indicate the boundary of the solution set. This does not represent a function. Any number can be used for x or for y, so the domain and range of this relation are both all real numbers.

domain: 

range:

41. For any choice of x in the domain of , yx  there is exactly one corresponding value of y, so this equation defines a function. Because the quantity under the square root cannot be negative, we have 0. x  Because the radical is nonnegative, the range is also zero or greater. domain:

42. For any choice of x in the domain of , yx  there is exactly one corresponding value of y, so this equation defines a function. Because the quantity under the square root cannot be negative, we have 0. x  The outcome of the radical is nonnegative, when you change the sign (by multiplying by 1), the range becomes nonpositive. Thus the range is zero or less. domain:

43. Because 2 xy  can be rewritten as 2 , yx  we can see that y can be found by dividing x into 2. This process produces one value of y for each value of x in the domain, so this equation is a function. The domain includes all real numbers except those that make the denominator equal to zero, namely x = 0. Values of y can be negative or positive, but never zero. Therefore, the range will be all real numbers except zero.

domain:     ,00,; 

range:     ,00,  

44. Because xy = 6 can be rewritten as 6 , yx  we can see that y can be found by dividing x into 6. This process produces one value of y for each value of x in the domain, so this equation is a function. The domain includes all real numbers except those that make the denominator equal to zero, namely x = 0. Values of y can be negative or positive, but never zero. Therefore, the range will be all real numbers except zero.

domain: 

range: 

  ,00,; 

,00,

45. For any choice of x in the domain of 41yx there is exactly one corresponding value of y, so this equation defines a function. Because the quantity under the square root cannot be negative, we have 41041 xx 1 4 . x  Because the radical is nonnegative, the range is also zero or greater.

domain:  1 4 ,;    range: 

0, 

46. For any choice of x in the domain of 72 yx  there is exactly one corresponding value of y, so this equation defines a function. Because the quantity under the square root cannot be negative, we have 72027 xx  77 22 or . xx Because the radical is nonnegative, the range is also zero or greater.

domain:  7 2 ,;

range:

47. Given any value in the domain of 2 3 , yx  we find y by subtracting 3, then dividing into 2. This process produces one value of y for each value of x in the domain, so this equation is a function. The domain includes all real numbers except those that make the denominator equal to zero, namely x = 3. Values of y can be negative or positive, but never zero. Therefore, the range will be all real numbers except zero.

domain: 

,33,;  range:  

,00,  

48. Given any value in the domain of 7 5 , yx  we find y by subtracting 5, then dividing into 7. This process produces one value of y for each value of x in the domain, so this equation is a function. The domain includes all real numbers except those that make the denominator equal to zero, namely x = 5. Values of y can be negative or positive, but never zero. Therefore, the range will be all real numbers except zero.

domain:     ,55,; 

range:     ,00,  

49. B. The notation f(3) means the value of the dependent variable when the independent variable is 3.

50. Answers will vary. An example is: The cost of gasoline depends on the number of gallons used; so cost is a function of number of gallons.

51.    34 0304044 fxx f

52.  

34 33349413 fxx f

53.

54.

2 2 41 22421 48111

2 2 41 10104101 10040159 gxxx g

55.    11 33 34 34143 fxx f  

56.

57.

58.



77 33 34 347411 fxx f  

 2 2 111 222 111 44 41 41 21 gxxx g   

 2 2 111 444 11 1616 41 41 11 gxxx g   

59.    34 34 fxx fpp

60.

61.

62.

2 2 41 41 gxxx gkkk

34 3434 fxx fxxx

2 2 2 41 41 41 gxxx gxxx xx

63.

34 2324 36432 fxx fxx xx

64.

65.

66.

34 4344 312438 fxx faa aa

34 233234 694613 fxx fmm mm

34 323324 964910 fxx ftt tt

67. (a)

68. (a)

 22 f  (b)   13 f 

25 f  (b)

111 f

69. (a)   215 f  (b)   110 f 

70. (a)

21 f  (b)

17 f 

71. (a)   23 f  (b) 

72. (a)

 23 f  (b)

13 f 

12 f 

73. (a)   20 f  (b) 

77. (a)  11 33 312 312 12 3 44 xy yx x y yxfxx

(b)     1 3 334143 f 

78. (a)  11 44 48 84 84 8 4 22 xy xy xy x y yxfxx    

(b)     3385 1 332244444 f 

79. (a)

2 2 2 23 23 23 yxx yxx fxxx  

(b) 2 32333 293318 f  

80. (a)

2 2 2 32 32 32 yxx yxx fxxx

(b) 2 33332 393232 f  

81. (a)  4488 3333 438 438 483 48 3 xy xy xy x y yxfxx     

(b)     88 4124 3333333 f 

12 f  (d)

 44 f 

04 f  (c)

74. (a)   25 f  (b) 

00 f  (c)

12 f  (d)

75. (a)   23 f

10 f

76. (a) 

23 f

13 f

(b)

(d)

(b)

44 f 

02 f

42 f

03 f

43 f

82. (a)  2299 5555 259 529 29 5 xy yx x y yxfxx     

(b)     96915 2 55555 333 f 

83.   34 f 

84. Because     2 0.20.230.21 f 

= 0.04 + 0.6 + 1 = 1.64, the height of the rectangle is 1.64 units. The base measures 0.3 0.2 = 0.1 unit. Because the area of a rectangle is base times height, the area of this rectangle is 0.1(1.64) = 0.164 square unit.

85.  3 f is the y-component of the coordinate, which is 4.

86.   2 f is the y-component of the coordinate, which is 3.

87. (a)   2,0 (b)   ,2

0, 

88. (a)  3,1 (b)   1, 

(c)

,3

89. (a)     ,2;2, 

(b) 

90. (a) 

2,2 (c) none

3,3 (b) 

91. (a) 

(b)

 ,3;3, 

1,0;1, 

,1;0,1

92. (a) 

,2;0,2

(b)   

2,0;2, 

93. (a) Yes, it is the graph of a function.

(b) [0, 24]

(d) The most electricity was used at 17 hr or 5 P.M. The least electricity was used at 4 A.M.

(e)   121900 f  At 12 noon, electricity use is about 1900 megawatts.

(f) increasing from 4 A.M. to 5 P.M.; decreasing from midnight to 4 A.M. and from 5 P.M. to midnight

94. (a) At t = 2, y = 240 from the graph. Therefore, at 2 seconds, the ball is 240 feet high.

(b) At y = 192, x = 1 and x = 5 from the graph. Therefore, the height will be 192 feet at 1 second and at 5 seconds.

(d) The coordinate of the highest point is (3, 256). Therefore, it reaches a maximum height of 256 feet at 3 seconds.

(e) At x = 7, y = 0. Therefore, at 7 seconds, the ball hits the ground.

95. (a) At t = 12 and t = 20, y = 55 from the graph. Therefore, after about 12 noon until about 8 P.M. the temperature was over 55º.

(b) At t = 6 and t = 22, y = 40 from the graph. Therefore, until about 6 A.M. and after 10 P.M. the temperature was below 40º.

(c) The temperature at noon in Bratenahl, Ohio was 55º. Because the temperature in Greenville is 7º higher, we are looking for the time at which Bratenahl, Ohio was at or above 48º. This occurred at approximately 10 A.M and 8:30 P.M.

(d) The temperature is just below 40° from midnight to 6 A.M., when it begins to rise until it reaches a maximum of just below 65° at 4 P.M. It then begins to fall util it reaches just under 40° again at midnight.

96. (a) At t = 8, y = 24 from the graph. Therefore, there are 24 units of the drug in the bloodstream at 8 hours.

(b) The level increases between 0 and 2 hours after the drug is taken and decreases between 2 and 12 hours after the drug is taken.

(c) The coordinates of the highest point are (2, 64). Therefore, at 2 hours, the level of the drug in the bloodstream reaches its greatest value of 64 units.

(d) After the peak, y = 16 at t = 10. 10 hours – 2 hours = 8 hours after the peak. 8 additional hours are required for the level to drop to 16 units.

(e) When the drug is administered, the level is 0 units. The level begins to rise quickly for 2 hours until it reaches a maximum of 64 units. The level then begins to decrease gradually until it reaches a level of 12 units, 12 hours after it was administered.

Section 2.4 Linear Functions

1. B; ()fx = 3x + 6 is a linear function with y-intercept (0, 6).

2. H; x = 9 is a vertical line.

3. C; ()fx = 8 is a constant function.

4. G; 2x – y = –4 or y = 2x + 4 is a linear equation with x-intercept (–2, 0) and y-intercept (0, 4).

5. A; ()fx = 5x is a linear function whose graph passes through the origin, (0, 0). f(0) = 5(0) = 0.

6. D; 2 () fxx  is a function that is not linear.

7. –3 m  matches graph C because the line falls rapidly as x increases.

8. m = 0 matches graph A because horizontal lines have slopes of 0.

9. m = 3 matches graph D because the line rises rapidly as x increases.

10. m is undefined for graph B because vertical lines have undefined slopes.

11. ()–4fxx 

Use the intercepts.

(0)0–4–4: f  y-intercept

0–44: xx  x-intercept

Graph the line through (0, –4) and (4, 0).

The domain and range are both

Section 2.4 Linear Functions

12. ()–4fxx

Use the intercepts.

(0)–044: f  y-intercept

0–44: xx  x-intercept

Graph the line through (0, 4) and (4, 0).

The domain and range are both   ,.

13. 1 2 ()–6fxx 

Use the intercepts.

  1 2 (0)0–6–6: f  y-intercept 11 22 0–6612: xxx  x-intercept

Graph the line through (0, –6) and (12, 0).

The domain and range are both

14. 2 3 ()2fxx

Use the intercepts.

  2 3 (0)022: f  y-intercept 22 33 02–23:-intercept xxxx

Graph the line through (0, 2) and (–3, 0).

The domain and range are both   ,. 

15. ()3 fxx 

The x-intercept and the y-intercept are both zero. This gives us only one point, (0, 0). If x = 1, y = 3(1) = 3. Another point is (1, 3). Graph the line through (0, 0) and (1, 3).

The domain and range are both

16.   –2 fxx 

The x-intercept and the y-intercept are both zero. This gives us only one point, (0, 0). If x = 3, y = –2(3) = –6, so another point is (3, –6). Graph the line through (0, 0) and (3, –6).

The domain and range are both 

,. 

17. ()–4fx  is a constant function.

The graph of ()4fx  is a horizontal line with a y-intercept of –4.

domain:   ,; range: {–4}

18. ()3fx  is a constant function whose graph is a horizontal line with y-intercept of 3.

domain:   ,; range: {3}

19. ()0fx  is a constant function whose graph is the x-axis.

domain:   ,; range: {0}

20.   9 fxx 

The domain and range are both   ,. 

21. 4312 xy  Use the intercepts.

  40312312 4:-intercept yy yy 

  43012412 3:-intercept xx xx  

Graph the line through (0, 4) and ( 3, 0).

The domain and range are both   ,. 

22. 2510; xy Use the intercepts.

  20510510 2:-intercept yy yy     25010210 5:-intercept xx xx  

Graph the line through (0, 2) and (5, 0):

The domain and range are both   ,. 

23. 340 yx

Use the intercepts.

3400300:-intercept yyyy 

 3040400:-intercept xxxx

The graph has just one intercept. Choose an additional value, say 3, for x

34303120 3124 yy yy

Graph the line through (0, 0) and (3, 4):

The domain and range are both   ,. 

24. 320 xy

Use the intercepts.

3020200:-intercept yyyy

3200300: xxx  x-intercept

The graph has just one intercept. Choose an additional value, say 2, for x

 3220620 263 yy yy 



Graph the line through (0, 0) and (2, 3):

The domain and range are both

25. x = 3 is a vertical line, intersecting the x-axis at (3, 0).

domain: {3}; range:   , 

26. x = –4 is a vertical line intersecting the x-axis at (–4, 0).

domain: { 4}; range:   , 

27. 2x + 4 = 0 242 xx  is a vertical line intersecting the x-axis at (–2, 0).

domain: { 2}; range:   , 

28. 360–362 xxx  is a vertical line intersecting the x-axis at (2, 0).

domain: {2}; range:   , 

29. 505xx  is a vertical line intersecting the x-axis at (5, 0).

domain: {5}; range:   , 

30. 303 xx  is a vertical line intersecting the x-axis at   3,0.

domain: { 3}; range:   , 

31. y = 5 is a horizontal line with y-intercept 5. Choice A resembles this.

32. y = –5 is a horizontal line with y-intercept –5. Choice C resembles this.

33. x = 5 is a vertical line with x-intercept 5. Choice D resembles this.

34. x = –5 is a vertical line with x-intercept –5. Choice B resembles this. 35.

39. The rise is 2.5 feet while the run is 10 feet so the slope is 2.5 1 104 0.2525%  . So A = 0.25, 2.5 10 C  , D = 25%, and 1 4 E  are all expressions of the slope.

40. The pitch or slope is 1 4 . If the rise is 4 feet then 1rise4 4run x  or x = 16 feet. So 16 feet in the horizontal direction corresponds to a rise of 4 feet.

41. Through (2, –1) and (–3, –3)

Let 112 2, –1, –3, and xyx 2 –3. y 

Then rise–3 – (–1)–2 y  and run–3 – 2–5. x 

The slope is rise–22 run–55 y m x   

42. Through ( 3, 4) and (2, 8)

Let 1122 3, 4, 2, and –8. xyxy 

Then rise–8 – 4–12 y  and run2 – (3)5. x 

The slope is rise–1212 . run55 y m x   

43. Through (5, 8) and (3, 12)

Let 1122 5, 8, 3, and 12. xyxy 

Then rise1284 y  and run352. x 

The slope is rise4 2. run2 y m x   

44. Through (5, –3) and (1, –7) 1122 Let 5, –3,1,and –7. xyxy 

Then rise–7 – (–3)–4 y  and run1 – 5–4. x 

The slope is 4 1. 4 y m x   

45. Through (5, 9) and (–2, 9) 21 21 9 – 90 0 –2 5–7 yyy m xxx   

46. Through (–2, 4) and (6, 4) 21 21 4 40 0 6 (–2)8 yyy m xxx   

47. Horizontal, through (5, 1)

The slope of every horizontal line is zero, so m = 0.

48. Horizontal, through (3, 5)

The slope of every horizontal line is zero, so m = 0.

49. Vertical, through (4, –7)

The slope of every vertical line is undefined; m is undefined.

50. Vertical, through (–8, 5)

The slope of every vertical line is undefined; m is undefined.

51. (a) 35yx

Find two ordered pairs that are solutions to the equation.

If x = 0, then   3055.yy

If x = 1, then   315352.yyy 

Thus two ordered pairs are (0, 5) and ( 1, 2)

21 21 rise 253 3. run 101 yy m xx 

(b)

52. 24yx

Find two ordered pairs that are solutions to the equation. If 0, x  then   204 y 

4. y  If 1, x  then   214 y 

242.yy Thus two ordered pairs are   0,4 and   1,2.

21 21 24 rise2 2. run 101 yy m xx 

(b)

53. 23yx 

Find two ordered pairs that are solutions to the equation. If 0, x  then 200. yy If 3, y  then   23363 xx 

2. x  Thus two ordered pairs are   0,0 and

 2,3.

21 21 rise 303 . run 202 yy m xx 

54. 45yx

Find two ordered pairs that are solutions to the equation. If 0, x  then 400. yy 

If 4, x  then   454420 yy 

5. y  Thus two ordered pairs are   0,0 and   4,5.

21 21 rise 505 run 404 yy m xx  (b)

55. 5210 xy

Find two ordered pairs that are solutions to the equation. If 0, x  then   50210 y 

5. y  If 0, y  then   52010 x  510 x  2. x 

Thus two ordered pairs are  0,5 and   2,0.   21 21 05 rise5 . run202 yy m xx  (b)

56. 4312 xy

Find two ordered pairs that are solutions to the equation. If 0, x  then   40312 y 

312 y  4. y  If 0, y  then   43012412 xx  3. x  Thus two ordered pairs are   0,4 and   3,0.

21 21 rise044 . run303 yy m xx

(b)

57. Through (–1, 3), 3 2 m 

First locate the point (–1, 3). Because the slope is 3 2 , a change of 2 units horizontally (2 units to the right) produces a change of 3 units vertically (3 units up). This gives a second point, (1, 6), which can be used to complete the graph.

58. Through (–2, 8), 2 5 m  . Because the slope is 2 5 , a change of 5 units horizontally (to the right) produces a change of 2 units vertically (2 units up). This gives a second point (3, 10), which can be used to complete the graph. Alternatively, a change of 5 units to the left produces a change of 2 units down. This gives the point ( 7, 6).

59. Through (3, –4), 1 3 – m  . First locate the point (3, –4). Because the slope is 1 3 –, a change of 3 units horizontally (3 units to the right) produces a change of –1 unit vertically (1 unit down). This gives a second point, (6, –5), which can be used to complete the graph.

60. Through (–2, –3), 3 4 – m  . Because the slope is 3–3 44 –,  a change of 4 units horizontally (4 units to the right) produces a change of –3 units vertically (3 units down). This gives a second point (2, –6), which can be used to complete the graph.

61. Through   1 2 ,4 , m = 0.

The graph is the horizontal line through   1 2 ,4.

Exercise 61

Exercise 62

62. Through   3 2 ,2, m = 0.

The graph is the horizontal line through   3 2 ,2.

63. Through   5 2 , 3 , undefined slope. The slope is undefined, so the line is vertical, intersecting the x-axis at   5 2 , 0.

64. Through   9 4 , 2 , undefined slope. The slope is undefined, so the line is vertical, intersecting the x-axis at

9 4 , 0.

65. The average rate of change is

fbfa m ba

20416

$4 044  (thousand) per year. The value of the machine is decreasing $4000 each year during these years.

66. The average rate of change is

fbfa m ba  2000200

$50 404  per month. The amount saved is increasing $50 each month during these months.

67. The graph is a horizontal line, so the average rate of change (slope) is 0. The percent of pay raise is not changing—it is 3% each year.

68. The graph is a horizontal line, so the average rate of change (slope) is 0. That means that the number of named hurricanes remained the same, 10, for the four consecutive years shown.

69.

256250852523 2012198032 78.8 thousand per year fbfa m ba 

The number of high school dropouts decreased by an average of 78.8 thousand per year from 1980 to 2012.

70.

17095302 20132006 3593 $513.29 7 fbfa m

Sales of plasma flat-panel TVs decreased by an average of $513.29 million per year from 2006 to 2013.

71. (a) The slope of –0.0167 indicates that the average rate of change of the winning time for the 5000 m run is 0.0167 min less. It is negative because the times are generally decreasing as time progresses.

(b) The Olympics were not held during World Wars I (1914 1919) and II (1939–1945).

The model predicts a winning time of 13.05 minutes. The times differ by 13.35 13.05 = 0.30 min.

72. (a) From the equation, the slope is 200.02. This means that the number of radio stations increased by an average of 200.02 per year.

(b) The year 2018 is represented by x = 68.   200.02682727.716,329.06 y 

According to the model, there will be about 16,329 radio stations in 2018.

73.

 20132008 335,652270,334 2013200820132008 65,318 13,063.6 5 ff  

The average annual rate of change from 2008 through 2013 is about 13,064 thousand.

74.     20142006 3.744.53 2014200620142006 0.79 0.099 8 ff  

The average annual rate of change from 2006 through 2014 is about –0.099.

75. (a)

The average rate of change was 8.17 thousand mobile homes per year.

(b) The negative slope means that the number of mobile homes decreased by an average of 8.17 thousand each year from 2003 to 2013.

76.

There was an average decrease of 1.6 births per thousand per year from 1991 through 2013.

77. (a)

10500Cxx

35 Rxx

(d)

1050035 50025 20 CxRx xx x x

20 units; do not produce

78. (a)   1502700Cxx

(b)

280 Rxx  (c)

PxRxCx xx xx x

2801502700 2801502700 1302700

(d)

79. (a)

 21 units; produce

CxRx xx x x 

1502700280 2700130 20.77 or 21 units

4001650Cxx

(b)   305 Rxx  (c)

PxRxCx xx xx x

3054001650 3054001650 951650

(d) 

4001650305 9516500 951650 17.37units CxRx xx x x x     

This result indicates a negative “breakeven point,” but the number of units produced must be a positive number. A calculator graph of the lines   1 4001650yCxx and   2 305 yRxx  in the window [0, 70] × [0, 20000] or solving the inequality 3054001650 xx will show that

RxCx  for all positive values of x (in fact whenever x is greater than 17.4). Do not produce the product because it is impossible to make a profit.

80. (a)   11180Cxx (b)   20 Rxx 

(c)

2011180 20111809180

1118020 1809 20 CxRx xx x x 

  20 units; produce

  2001000240 10004025 CxRxxx xx   The break-even point is 25 units.   (25)200251000$6000 C  which is the same as   (25)24025$6000 R 

82.     2201000240 10002050 CxRxxx xx 



The break-even point is 50 units instead of 25 units. The manager is not better off because twice as many units must be sold before beginning to show a profit.

83. The first two points are A(0, –6) and B(1, –3).

3 – (–6)3 3 1 – 01 m 

84. The second and third points are B(1, –3) and C(2, 0).

0 – (–3)3 3 2 – 11 m 

85. If we use any two points on a line to find its slope, we find that the slope is the same in all cases.

86. The first two points are A(0, –6) and B(1, –3).

22 22 (,)(1 – 0)[–3 – (–6)] 131910 dAB  

87. The second and fourth points are B(1, –3) and D(3, 3).

22 22 (,)(3 – 1)[3 – (–3)] 26436 40210 dBD 

88. The first and fourth points are A(0, –6) and D(3, 3).

22 22 (,)(3 – 0)[3 – (–6)] 39981 90310 dAD  

89. 10210310;  The sum is 310, which is equal to the answer in Exercise 88.

90. If points A, B, and C lie on a line in that order, then the distance between A and B added to the distance between B and C is equal to the distance between A and C

91. The midpoint of the segment joining A(0, –6) and G(6, 12) has coordinates

  06612 22 ,    66 22 ,    3,3.  The midpoint is M(3, 3), which is the same as the middle entry in the table.

92. The midpoint of the segment joining E(4, 6) and F(5, 9) has coordinates

    4569915 2222 ,,   = (4.5, 7.5). If the x-value 4.5 were in the table, the corresponding y-value would be 7.5.

Chapter 2 Quiz (Sections 2.1 2.4)

1.   22 2121 22 22 (,) 8(4)32 (4)(5)162541

2. To find an estimate for 2006, find the midpoint of (2004, 6.55) and (2008, 6.97:  200420086.556.97 , 22 2006,6.76 M 

The estimated enrollment for 2006 was 6.76 million.

To find an estimate for 2010, find the midpoint of (2008, 6.97) and (2012, 7.50):

 200820126.977.50 , 22 2010,7.235 M 

The estimated enrollment for 2006 was about 7.24 million.

3. 4.

5. 22 4830xyxy

Complete the square on x and y separately. 22 22 (44)(816)3416 (2)(4)17 xxyy xy  

The radius is 17 and the midpoint of the circle is (2, 4).

6. From the graph, f( 1) is 2.

7. Domain: (,);  range: [0,) 

8. (a) The largest open interval over which f is decreasing is (,3) 

(b) The largest open interval over which f is increasing is   3, 

9. (a) 11563 5142 m 

(b) 440 0 1(7)6 m 

10. The points to use are (2009, 10,602) and (2013, 15,884). The average rate of change is 15,88410,6025282 1320.5 201320094 

The average rate of change was 1320.5 thousand cars per year. This means that the number of new motor vehicles sold in the United States increased by an average of 1320.5 thousand per year from 2009 to 2013.

Section 2.5 Equations of Lines and Linear Models

1. The graph of the line   348yx has slope 4 and passes through the point (8, 3).

2. The graph of the line 27yx has slope –2 and y-intercept (0, 7).

3. The vertical line through the point (–4, 8) has equation x = –4.

4. The horizontal line through the point (–4, 8) has equation y = 8. For exercises 5 and 6, 6 7 67076 xyyxyx

5. Any line parallel to the graph of 670 xy must have slope 6 7

6. Any line perpendicular to the graph of 670 xy must have slope 7 6

7. 1 4 2 yx is graphed in D. The slope is 1 4 and the y-intercept is (0, 2).

8. 4x + 3y = 12 or 3y = –4x + 12 4 3 or –4yx is graphed in B. The slope is 4 3 – and the y-intercept is (0, 4).

9.     3 2 – 11yx is graphed in C. The slope is 3 2 and a point on the graph is (1, –1).

10. y = 4 is graphed in A y = 4 is a horizontal line with y-intercept (0, 4).

11. Through (1, 3), m = –2.

Write the equation in point-slope form.

    11 – –3–2–1yymxxyx 

Then, change to standard form. 3–2225yxxy 

12. Through (2, 4), m = –1

Write the equation in point-slope form.

   11 – –4–1–2yymxxyx 

Then, change to standard form. – 4–26yxxy

13. Through (–5, 4), 3 2 m 

Write the equation in point-slope form.

3 2 – 45yx

Change to standard form.

2–4–35 2–8–3–15 32–7 yx yx xy   

14. Through 3 4 (–4, 3), m 

Write the equation in point-slope form.

3 4 – 34yx

Change to standard form.

4334 412312 3424or 3424 yx yx xyxy

15. Through (–8, 4), undefined slope

Because undefined slope indicates a vertical line, the equation will have the form x = a

The equation of the line is x = –8.

16. Through (5, 1), undefined slope

This is a vertical line through (5, 1), so the equation is x = 5.

17. Through (5, 8), m = 0

This is a horizontal line through (5, 8), so the equation is y = 8.

18. Through ( 3, 12), m = 0

This is a horizontal line through ( 3, 12), so the equation is y = 12.

19. Through (–1, 3) and (3, 4)

First find m. 4–31

3–(–1)4 m 

Use either point and the point-slope form.

 1 4 – 4–3 4163 413 413 yx yx xy xy

20. Through (2, 3) and ( 1, 2)

First find m 2311 1233

Use either point and the point-slope form.

1 3 3–2 3–92 37 37 yx yx xy xy

21. x-intercept (3, 0), y-intercept (0, –2)

The line passes through (3, 0) and (0, –2). Use these points to find m. –2 –02 0–33 m 

Using slope-intercept form we have 2 3 – 2. yx 

22. x-intercept (–4, 0), y-intercept (0, 3)

m 

The line passes through the points (–4, 0) and (0, 3). Use these points to find m 3–03 0–(–4)4

Using slope-intercept form we have 3 4 3. yx

23. Vertical, through (–6, 4)

The equation of a vertical line has an equation of the form x = a. Because the line passes through (–6, 4), the equation is x = –6. (Because the slope of a vertical line is undefined, this equation cannot be written in slope-intercept form.)

24. Vertical, through (2, 7)

The equation of a vertical line has an equation of the form x = a. Because the line passes through (2, 7), the equation is x = 2. (Because the slope of a vertical line is undefined, this equation cannot be written in slope-intercept form.)

25. Horizontal, through ( 7, 4)

The equation of a horizontal line has an equation of the form y = b. Because the line passes through ( 7, 4), the equation is y = 4.

26. Horizontal, through ( 8, 2)

The equation of a horizontal line has an equation of the form y = b. Because the line passes through ( 8, 2), the equation is y = 2.

27. m = 5, b = 15

Using slope-intercept form, we have 515.yx

28. m = 2, b = 12

Using slope-intercept form, we have 212.yx

29. Through ( 2, 5), slope = 4

542 542 548 43 yx yx yx yx

30. Through (4, 7), slope = 2

 724 728 21 yx yx yx

31. slope 0, y-intercept   3 2 0,

These represent 3 2 0 and . mb

Using slope-intercept form, we have   33 22 0.yxy

32. slope 0, y-intercept   5 4 0,

These represent 5 4 0 and . mb

Using slope-intercept form, we have   55 44 0.yxy

33. The line x + 2 = 0 has x-intercept (–2, 0). It does not have a y-intercept. The slope of his line is undefined. The line 4y = 2 has y-intercept   1 2 0,. It does not have an x-intercept. The slope of this line is 0.

34. (a) The graph of y = 3x + 2 has a positive slope and a positive y-intercept. These conditions match graph D.

(b) The graph of y = –3x + 2 has a negative slope and a positive y-intercept. These conditions match graph B.

(d) The graph of y = –3x – 2 has a negative slope and a negative y-intercept. These conditions match graph C.

35. y = 3x – 1

This equation is in the slope-intercept form, y = mx + b slope: 3; y-intercept: (0, –1)

36. y = –2x + 7 slope: –2; y-intercept: (0, 7)

37. 4x – y = 7

Solve for y to write the equation in slopeintercept form. –

–474–7yxyx  slope: 4; y-intercept: (0, –7)

38. 2x + 3y = 16

Solve the equation for y to write the equation in slope-intercept form.

16 2 33 3–216–yxyx  slope: 2 3 – ; y-intercept:   16 3 0,

39. 33 44 4–3– or 0 yxyxyx 

slope: 3 4 ; y-intercept (0, 0)

40. 11 2or22 0 yxyxyx 

slope is 1 2 ; y-intercept: (0, 0)

41. 24xy

Solve the equation for y to write the equation in slope-intercept form.

1 2 2–4–2 yxyx 

slope: 1 2 – ; y-intercept: (0, 2)

42. 39xy

Solve the equation for y to write the equation in slope-intercept form. 1 3

3–9–3 yxyx 

slope: 1 3 – ; y-intercept: (0, 3)

43. 3 2 10 yx

Solve the equation for y to write the equation in slope-intercept form.

33 22101yxyx 

slope: 3 2 ; y-intercept: (0, 1)

44. (a) Use the first two points in the table, A(–2, –11) and B(–1, –8). –8 – (–11)3 3 –1 – (–2)1 m 

(b) When x = 0, y = 5. The y-intercept is (0, –5).

3–5ymxbyx 

45. (a) The line falls 2 units each time the x value increases by 1 unit. Therefore the slope is 2. The graph intersects the y-axis at the point (0, 1) and intersects the x-axis at   1 2 ,0 , so the y-intercept is (0, 1) and the x-intercept is   1 2 ,0.

(b) An equation defining f is y = 2x + 1.

46. (a) The line rises 2 units each time the x value increases by 1 unit. Therefore the slope is 2. The graph intersects the y-axis at the point (0, 1) and intersects the x-axis at   1 2 ,0 , so the y-intercept is (0, 1) and the x-intercept is   1 2 ,0.

(b) An equation defining f is y = 2x 1.

47. (a) The line falls 1 unit each time the x value increases by 3 units. Therefore the slope is 1 3 . The graph intersects the y-axis at the point (0, 2), so the y-intercept is (0, 2). The graph passes through (3, 1) and will fall 1 unit when the x value increases by 3, so the x-intercept is (6, 0).

(b) An equation defining f is 1 3 2. yx

48. (a) The line rises 3 units each time the x value increases by 4 units. Therefore the slope is 3 4 . The graph intersects the y-axis at the point (0, 3) and intersects the x-axis at (4, 0), so the y-intercept is (0, 3) and the x-intercept is 4.

(b) An equation defining f is 3 4 3. yx

49. (a) The line falls 200 units each time the x value increases by 1 unit. Therefore the slope is 200. The graph intersects the y-axis at the point (0, 300) and intersects the x-axis at   3 2 ,0, so the y-intercept is (0, 300) and the x-intercept is   3 2 ,0.

(b) An equation defining f is y = 200x + 300.

50. (a) The line rises 100 units each time the x value increases by 5 units. Therefore the slope is 20. The graph intersects the y-axis at the point (0, 50) and intersects the x-axis at   5 2 ,0, so the y-intercept is (0, 50) and the x-intercept is   5 2 ,0.

(b) An equation defining f is y = 20x 50.

51. (a) through (–1, 4), parallel to x + 3y = 5

Find the slope of the line x + 3y = 5 by writing this equation in slope-intercept form.

5 1 33 353–5 –xyyx yx  

The slope is 1 3

Because the lines are parallel, 1 3 is also the slope of the line whose equation is to be found. Substitute 1 3 m  , 1 –1, x  and 1 4 y  into the point-slope form.

11 1 3 1 3 – –4––1 –41 3–12––1311 yymxx yx yx yxxy 

(b) Solve for y 111 33 3–11–yxyx 

52. (a) through (3, –2), parallel to 2x – y = 5

Find the slope of the line 2x – y = 5 by writing this equation in slope-intercept form.

2–5––25 2–5 xyyx yx  

The slope is 2. Because the lines are parallel, the slope of the line whose equation is to be found is also 2. Substitute m = 2, 1 3 x  , and 1 2 y  into the point-slope form.

11 – –22–322–6 –2–8 or 2–8 yymxx yxyx xyxy   

(b) Solve for yy = 2x – 8

53. (a) through (1, 6), perpendicular to 3x + 5y = 1

Find the slope of the line 3x + 5y = 1 by writing this equation in slope-intercept form.

3515–31 –xyyx yx  

3 1 55

This line has a slope of 3 5 . The slope of any line perpendicular to this line is 5 3 , because   35 53 1. Substitute 5 3 m  , 1 1, x  and 1 6 y  into the point-slope form.

5 3 –6(–1) 3(–6)5(–1) 3–185–5 –135–3or 5–3–13 yx yx yx xyxy    

(b) Solve for y. 513 33 3513yxyx 

54. (a) through (–2, 0), perpendicular to 8x – 3y = 7

Find the slope of the line 8x – 3y = 7 by writing the equation in slope-intercept form.

87 33 8–37–3–87 –xyyx yx  

This line has a slope of 8 3 . The slope of any line perpendicular to this line is 3 8 , because   83 38 1.

Substitute 3 8 m  , 1 2, x  and 1 0 y  into the point-slope form.

3 8 –0–(2) 8–3(2) 8–3–638–6 yx yx yxxy   

(b) Solve for y. 36 88 33 84 8–3–6––yxyx yx 

55. (a) through (4, 1), parallel to y = 5

Because y = 5 is a horizontal line, any line parallel to this line will be horizontal and have an equation of the form y = b Because the line passes through (4, 1), the equation is y = 1.

(b) The slope-intercept form is y = 1.

56. (a) through   2,2, parallel to y = 3.

Because y = 3 is a horizontal line, any line parallel to this line will be horizontal and have an equation of the form y = b. Because the line passes through   2,2, the equation is y = –2.

(b) The slope-intercept form is y = 2

57. (a) through (–5, 6), perpendicular to x = –2.

Because x = –2 is a vertical line, any line perpendicular to this line will be horizontal and have an equation of the form y = b. Because the line passes through (–5, 6), the equation is y = 6.

(b) The slope-intercept form is y = 6.

58. (a) Through (4, –4), perpendicular to x = 4

Because x = 4 is a vertical line, any line perpendicular to this line will be horizontal and have an equation of the form y = b. Because the line passes through (4, –4), the equation is y = –4.

(b) The slope-intercept form is y = –4.

59. (a) Find the slope of the line 3y + 2x = 6. 2 3 3263–26 –2 yxyx yx 

Thus, 2 3 –. m  A line parallel to 3y + 2x = 6 also has slope 2 3 –. Solve for k using the slope formula.

(b) Find the slope of the line 2y – 5x = 1. 5 1 22 251251 yxyx yx

Thus, 5 2 . m  A line perpendicular to 2y – 5x = 1 will have slope 2 5 –, because

5 2 25 – –1.  Solve this equation for k

60. (a) Find the slope of the line 2–34. xy  24 33 2–34–3–24 –xyyx yx 

Thus, 2 3 m  A line parallel to 2x – 3y = 4 also has slope 2 3 . Solve for r using the slope formula.

–62–62 –4–23–63 –62 66 –63 642 rr r rr

(b) Find the slope of the line x + 2y = 1. 11 22 212–1 –xyyx yx  

Thus, 1 2 m  A line perpendicular to the line x + 2y = 1 has slope 2, because 1 2 – (2)–1.  Solve for r using the slope formula.

–6–6 22 –4–2–6 –6–12–6 rr rr  

61. (a) First find the slope using the points (0, 6312) and (3, 7703).

770363121391 463.67 303 m 

The y-intercept is (0, 6312), so the equation of the line is 463.676312.yx

(b) The value x = 4 corresponds to the year 2013.   463.67463128166.68 y 

The model predicts that average tuition and fees were $8166.68 in 2013. This is $96.68 more than the actual amount.

62. (a) First find the slope using the points (0, 6312) and (2, 7136).

71366312824 412 202 m 

The y-intercept is (0, 6312), so the equation of the line is 4126312.yx

(b) The value x = 4 corresponds to the year 2013.

  412463127960 y 

The model predicts that average tuition and fees were $7960 in 2013. This is $110 less than the actual amount.

63. (a) First find the slope using the points (0, 22036) and (4, 24525). 24525220362489 622.25 404 m 

The y-intercept is (0, 22036), so the equation of the line is 622.2522,036.yx

The slope of the line indicates that the average tuition increase is about $622 per year from 2009 through 2013.

(b) The year 2012 corresponds to x = 3.

 622.25322,03623,902.75 y 

According to the model, average tuition and fees were $23,903 in 2012. This is $443 more than the actual amount $23,460.

64. (a) See the graph in the answer to part (b).There appears to be a linear relationship between the data. The farther the galaxy is from Earth, the faster it is receding.

(b) Using the points (520, 40,000) and (0, 0), we obtain 40,000–040,000 76.9. 520–0520 m 

The equation of the line through these two points is y = 76.9x.

According to the model, the galaxy Hydra is approximately 780 megaparsecs away.

Using m = 76.9, we estimate that the age of the universe is approximately 12.35 billion years.

(e) 11 109 11 9 9.510 1.910 or 1910 50 9.510 9.510 100 A A

The range for the age of the universe is between 9.5 billion and 19 billion years.

65. (a) The ordered pairs are (0, 32) and (100, 212).

The slope is 212–321809 100–01005 m 

Use 9 11 5 (, )(0,32) and xym in the point-slope form. 11 9 5 9 5 99 55 –(–) –32(–0) –32 32 32 yymxx yx yx yxFC    

(b)  9 5 5 9 32 5932 5916095–160 95(–32)(–32) FC FC FCCF CFCF 

66. (a) The ordered pairs are (0, 1) and (100, 3.92).

The slope is 3.92–12.92 0.0292 100–0100 m  and 1. b 

Using slope-intercept form we have 0.02921yx or()0.02921. pxx

(b) Let x = 60. P(60) = 0.0292(60) + 1 = 2.752

The pressure at 60 feet is approximately 2.75 atmospheres.

67. (a) Because we want to find C as a function of I, use the points (12026, 10089) and (14167, 11484), where the first component represents the independent variable, I. First find the slope of the line. 11484100891395 0.6516 14167120262141 m 

Now use either point, say (12026, 10089), and the point-slope form to find the equation.

100890.6516(–12026)

100890.6516–7836 0.65162253

(b) Because the slope is 0.6516, the marginal propensity to consume is 0.6516.

68. D is the only possible answer, because the x-intercept occurs when y = 0. We can see from the graph that the value of the x-intercept exceeds 10.

69. Write the equation as an equivalent equation with 0 on one side: 2742 xxx 27420 xxx . Now graph 2742yxxx  in the window [–5, 5] × [–5, 5] to find the x-intercept:

Solution set: {3}

70. Write the equation as an equivalent equation with 0 on one side: 724531 xxx  7245310 xxx  . Now graph 724531yxxx  in the window [–5, 5] × [–5, 5] to find the x-intercept:

Solution set: {1}

71. Write the equation as an equivalent equation with 0 on one side:     321225 xx

 3212250 xx . Now graph

321225yxx  in the window [–5, 5] × [–5, 5] to find the x-intercept:

Solution set:   1 2 or 0.5

72. Write the equation as an equivalent equation with 0 on one side:

 43422362 xxxx 

  434223620 xxxx  Now graph 

  43422362yxxxx  in the window [–2, 8] × [–5, 5] to find the x-intercept:

Solution set: {4}

73. (a)   252 2102 102 12 xx xx x x 

Solution set: {12}

(b) Answers will vary. Sample answer: The solution does not appear in the standard viewing window x-interval [10, –10]. The minimum and maximum values must include 12.

74. Rewrite the equation as an equivalent equation with 0 on one side.

326482 6184820 xxx xxx

Now graph y = 6x 18 ( 4x + 8 2x) in the window [–10, 10] × [–30, 10].

The graph is a horizontal line that does not intersect the x-axis. Therefore, the solution set is  . We can verify this algebraically.

 326482 61868026 xxx xx

Because this is a false statement, the solution set is 

75. A(–1, 4), B(–2, –1), C(1, 14)

For A and B, 145 5 2(1)1 m 

For B and C, 14(1)15 5 1(2)3 m 

For A and C, 14410 5 1(1)2 m 

Since all three slopes are the same, the points are collinear.

76. A(0, 7), B(–3, 5), C(2, 15)

For A and B, 5(7)12 4 303 m 

For B and C, 15520 4 2(3)5 m 

For A and C, 15(7)8 4 202 m 

Since all three slopes are the same, the points are collinear.

77. A( 1, 3), B( 5, 12), C(1, 11)

For A and B, 12(3)15 5(1)4 m 

For B and C, 111223 1(5)6 m 

For A and C, 11(3)8 4 1(1)2 m 

Since all three slopes are not the same, the points are not collinear.

78. A(0, 9), B(–3, –7), C(2, 19)

For A and B, 791616 3033 m 

For B and C, 19(7)26 2(3)5 m 

For A and C, 19910 5 202 m 

Because all three slopes are not the same, the points are not collinear.

79. 22 111 222 111 (,)(–0)(–0) dOPxmx xmx  

80. 22 222 222 222 (,)(–0)(–0) dOQxmx xmx  

81. 22 212211 (,)(–)(–) dPQxxmxmx 

    

82. 

[(,)][(,)][(,)] dOPdOQdPQ xmxxmx xxmxmx xmxxmx xxmxmx

222 22 222222 111222 2 22 212211 222222 111222 22 212211

  222222 111222 2222 221122 22 121211 211212 12121 2 2 022 220 xmxxmx xxxxmx mmxxmx xxmmxx mmxxxx     

83. 121212 1212 –2–20 –2(1)0 mmxxxx xxmm  

84. 1212 –2(1)0 xxmm  Because 12 0 and 0, xx we have 1212 10 implying that –1. mmmm

85. If two nonvertical lines are perpendicular, then the product of the slopes of these lines is –1.

Summary Exercises on Graphs, Circles, Functions, and Equations

1. (3,5), (2,3) PQ

(a)

22 22 (,)(23)(35) 18 16465

(b) The midpoint M of the segment joining points P and Q has coordinates

53 32525 , 22222,,1

Use either point and the point-slope form.

– 58–3yx 

Change to slope-intercept form. 5824819yxyx 

2. P(–1, 0), Q(4, –2)

(a) 22 22 (,)[4–(–1)](–2–0) 5(–2) 25429 dPQ 

(b) The midpoint M of the segment joining points P and Q has coordinates – 140(–2)32 , , 2222 3 ,1. 2

Use either point and the point-slope form.

2 5

Change to slope-intercept form.

22 55 521 522

3. P(–2, 2), Q(3, 2)

(b) The midpoint M of the segment joining points P and Q has coordinates – 2322 , 22

All lines that have a slope of 0 are horizontal lines. The equation of a horizontal line has an equation of the form y = b. Because the line passes through (3, 2), the equation is y = 2.

22,2, P

2,32 Q (a)

22 22 (,)2–2232–2 –222 2810 dPQ

(b) The midpoint M of the segment joining points P and Q has coordinates 222232 , 22  

324232 222,,22.

Use either point and the point-slope form.

 – 22–22yx 

Change to slope-intercept form. – 2242252yxyx 

5. P(5, –1), Q(5, 1) (a) 22 22 (,)(5–5)[1–(–1)] 020442 dPQ  

(b) The midpoint M of the segment joining points P and Q has coordinates 55–11 , 22 

100 22,5,0. 

All lines that have an undefined slope are vertical lines. The equation of a vertical line has an equation of the form x = a The line passes through (5, 1), so the equation is x = 5. (Because the slope of a vertical line is undefined, this equation cannot be written in slope-intercept form.)

6. (1,1), (3,3) PQ

(a)

22 22 (,)(3–1)(3–1) 44 16163242 dPQ

(b) The midpoint M of the segment joining points P and Q has coordinates

1313 , 22

22 , 22

Use either point and the point-slope form.

Change to slope-intercept form. 11 yxyx 

23,35,63,35PQ

(a)

(b) The midpoint M of the segment joining points P and Q has coordinates

23633535 , 22 8365 22,43,35.

All lines that have a slope of 0 are horizontal lines. The equation of a horizontal line has an equation of the form y = b. Because the line passes through   23,35, the equation is 35. y 

8. P(0, –4), Q(3, 1) (a) 22 22 (,)(3–0)[1–(–4)] 3592534 dPQ 

(b) The midpoint M of the segment joining points P and Q has coordinates 03–413333 , 222222,,.

Using slope-intercept form we have 5 3 – 4. yx 

9. Through  2,1 and  4,1

First find m: 1–121 4–(–2)63 m 

Use either point and the point-slope form.     1 3 – 1–4yx 

Change to slope-intercept form.

11 33 314334 31 yxyx yxyx

10. the horizontal line through (2, 3)

The equation of a horizontal line has an equation of the form y = b. Because the line passes through (2, 3), the equation is y = 3.

11. the circle with center (2, –1) and radius 3 2 22 22 (–2)–(–1)3 (–2)(1)9 xy xy 

12. the circle with center (0, 2) and tangent to the x-axis

The distance from the center of the circle to the x-axis is 2, so r = 2. 22222 (–0)(–2)2(–2)4 xyxy

13. the line through (3, 5) with slope 5 6 Write the equation in point-slope form.

Change to standard form.

14. the vertical line through ( 4, 3)

The equation of a vertical line has an equation of the form x = a. Because the line passes through ( 4, 3), the equation is x = 4.

15. a line through (–3, 2) and parallel to the line 2x + 3y = 6

First, find the slope of the line 2x + 3y = 6 by writing this equation in slope-intercept form. 2 3 2363262 xyyxyx 

The slope is 2 3 . Because the lines are parallel, 2 3 is also the slope of the line whose equation is to be found. Substitute 2 3 m  , 1 3 x  , and 1 2 y  into the pointslope form.

2 11 3 2 3 – – 2 3 3223362–6 3–2 yymxxyx yxyx yxyx

16. a line through the origin and perpendicular to the line 342 xy

First, find the slope of the line 342 xy by writing this equation in slope-intercept form. 3321 4442 3424–32 xyyx yxyx 

This line has a slope of 3 4 The slope of any line perpendicular to this line is 4 3 , because   3 4 34 1. Using slope-intercept form we have 44 33 0 or .yxyx  (continuedonnextpage)

17. 22424 xxyy

Complete the square on x and y separately.

22 22 22 424 4421441 219 xxyy xxyy

Yes, it is a circle. The circle has its center at (2, 1) and radius 3.

18. 22610360 xxyy

Complete the square on x and y separately.

22 22 22 61036 691025–36925 352 xxyy xxyy

No, it is not a circle.

19. 2212200xxy

Complete the square on x and y separately.

22 22 2 2 1220 1236–2036 616 xxy xxy xy





Yes, it is a circle. The circle has its center at (6, 0) and radius 4.

20. 2221661 xxyy

Complete the square on x and y separately.



22 22 22 21661 211664–61164 184 xxyy xxyy xy

Yes, it is a circle. The circle has its center at ( 1, 8) and radius 2.

21. 222100xxy

Complete the square on x and y separately.

 22 22 2 2 210 21–101 19 xxy xxy xy  



No, it is not a circle.

22. 22 – 890xyy

Complete the square on x and y separately.  

22 22 2 2 –89 – 816916 –425 xyy xyy xy

Yes, it is a circle. The circle has its center at (0, 4) and radius 5.

23. The equation of the circle is 222 (4)(5)4. xy

Let y = 2 and solve for x: 222 (4)(25)4 x  2222 (4)(3)4(4)7 xx  4747xx 

The points of intersection are   47,2  and   47,2

24. Write the equation in center-radius form by completing the square on x and y separately:





 22 22 22 22 10241440 10241440 (1025)(24144)25 (5)(12)25 xyxy xxyy xxyy xy





The center of the circle is (5, 12) and the radius is 5.

Now use the distance formula to find the distance from the center (5, 12) to the origin:

 22 2121 22 (50)(120)2514413 dxxyy 



The radius is 5, so the shortest distance from the origin to the graph of the circle is 13 5 = 8.

(continuedonnextpage)

25. (a) The equation can be rewritten as 1163 4442 46 . yxyxyx  x can be any real number, so the domain is all real numbers and the range is also all real numbers.

domain:   ,; range:   , 

(b) Each value of x corresponds to just one value of y. 46xy represents a function.

1133 4242yxfxx 

  33 112 2242222 1 f 

26. (a) The equation can be rewritten as 2 5.yx  y can be any real number. Because the square of any real number is not negative, 2 y is never negative. Taking the constant term into consideration, domain would be   5,. domain:   5,; range:   , 

(b) Because ( 4, 1) and ( 4, 1) both satisfy the relation, 2 5 yx does not represent a function.

27. (a) 2 2 225xy is a circle centered at ( 2, 0) with a radius of 5. The domain will start 5 units to the left of –2 and end 5 units to the right of –2. The domain will be [ 2 5, 2 + 5] = [ 7, 3]. The range will start 5 units below 0 and end 5 units above 0. The range will be [0 5, 0 + 5] = [ 5, 5].

(b) Because ( 2, 5) and ( 2, 5) both satisfy the relation, 2 2 225xy does not represent a function.

28. (a) The equation can be rewritten as 22 3 1 22 23. yxyx  x can be any real number. Because the square of any real number is not negative, 2 1 2 x is never negative. Taking the constant term into consideration, range would be

3 2 ,.

domain:   ,; range:  3 2 ,   

(b) Each value of x corresponds to just one value of y. 2 23xy represents a function.

112233 2222yxfxx 

Section 2.6 Graphs of Basic Functions

1. The equation   2 fxx  matches graph E. The domain is   ,. 

2. The equation of   fxx  matches graph G.

The function is increasing on   0,. 

3. The equation   3 fxx  matches graph A.

The range is   ,. 

4. Graph C is not the graph of a function. Its equation is 2 xy  .

5. Graph F is the graph of the identity function. Its equation is   fxx 

6. The equation     fxx  matches graph B.   1.51 f 

7. The equation   3 fxx  matches graph H.

No, there is no interval over which the function is decreasing.

8. The equation of   fxx  matches graph D.

The domain is   0,. 

9. The graph in B is discontinuous at many points. Assuming the graph continues, the range would be {…, 3, 2, 1, 0, 1, 2, 3, …}.

10. The graphs in E and G decrease over part of the domain and increase over part of the domain. They both increase over   0,  and decrease over   ,0.

11. The function is continuous over the entire domain of real numbers   ,. 

12. The function is continuous over the entire domain of real numbers   ,. 

13. The function is continuous over the interval

0,. 

14. The function is continuous over the interval

,0.

15. The function has a point of discontinuity at (3, 1). It is continuous over the interval

,3 and the interval 

3,. 

16. The function has a point of discontinuity at x = 1. It is continuous over the interval 

,1 and the interval

1,. 

17. 2 if –1 () – 1 if –1 fxxxxx   

(a) (–5)2(–5)–10 f 

(b) (–1)2(–1)–2 f 

(d) f(3) = 3 – 1 = 2

18.  – 2 if 3 () 5– if 3 fxxx xx   

(a) f(–5) = –5 – 2 = –7

(b) f(–1) = –1 – 2 = –3

(d) f(3) = 5 – 3 = 2

19. 2 if –4 ()– if –42 3 if 2 xx fxxx xx

(a) f(–5) = 2 + (–5) = –3

(b) f(–1) = – (–1) = 1

(d) (3)339 f 

20.  –2 if –3 3–1 if –32 –4 if 2 xx fxxx xx 

(a) f(–5) = –2(–5) = 10

(b) f(–1) = 3(–1) – 1 = –3 – 1 = –4

(d) f(3) = –4(3) = –12

21.  – 1 if 3 () 2 if 3 fxxx x   

Draw the graph of y = x – 1 to the left of x = 3, including the endpoint at x = 3. Draw the graph of y = 2 to the right of x = 3, and note that the endpoint at x = 3 coincides with the endpoint of the other ray.

22. 6– if 3 () 3 if 3 fxxxx   

Graph the line y = 6 – x to the left of x = 3, including the endpoint. Draw y = 3 to the right of x = 3. Note that the endpoint at x = 3 coincides with the endpoint of the other ray.

23. 4– if 2 () 12 if 2 fxxx xx   

Draw the graph of y = 4 x to the left of x = 2, but do not include the endpoint. Draw the graph of y = 1 + 2x to the right of x = 2, including the endpoint.

24. 21 if0 () if 0 fxxx xx   

Graph the line y = 2x + 1 to the right of x = 0, including the endpoint. Draw y = x to the left of x = 0, but do not include the endpoint.

25.  3 if 1 () 1if 1 fxx x   

Graph the line y = 3 to the left of x = 1, including the endpoint. Draw y = 1 to the right of x = 1, but do not include the endpoint.

26.  2 if 1 () 2 if 1 fxx x   

Graph the line y = – 2 to the left of x = 1, including the endpoint. Draw y = 2 to the right of x = 1, but do not include the endpoint.

Exercise 26

Exercise 27

27. 2 if –4 ()– if –45 3 if 5 xx fxxx xx

Draw the graph of y = 2 + x to the left of –4, but do not include the endpoint at x = 4. Draw the graph of y = –x between –4 and 5, including both endpoints. Draw the graph of y = 3x to the right of 5, but do not include the endpoint at x = 5.

28. –2 if –3 ()3–1 if –32 –4 if 2 xx fxxx xx

Graph the line y = –2x to the left of x = –3, but do not include the endpoint. Draw y = 3x – 1 between x = –3 and x = 2, and include both endpoints. Draw y = –4x to the right of x = 2, but do not include the endpoint. Notice that the endpoints of the pieces do not coincide.

2 1 2 1 2 – +2 if 2 () if 2 xx fx xx

Graph the curve 2 1 2 2 yx to the left of x = 2, including the endpoint at (2, 0). Graph the line 1 2 yx  to the right of x = 2, but do not include the endpoint at (2, 1). Notice that the endpoints of the pieces do not coincide.

3 2 5if 0 () if 0 fxxx xx

Graph the curve 3 5 yx to the left of x = 0, including the endpoint at (0, 5). Graph the line 2 yx  to the right of x = 0, but do not include the endpoint at (0, 0). Notice that the endpoints of the pieces do not coincide.

31. 2 2if 51 ()2if 10 2if 02 xx fxx xx

Graph the line 2 yx  between x = 5 and x = 1, including the left endpoint at ( 5, 10), but not including the right endpoint at ( 1, 2). Graph the line y = 2 between x = 1 and x = 0, including the left endpoint at ( 1, 2) and not including the right endpoint at (0, 2). Note that ( 1, 2) coincides with the first two sections, so it is included. Graph the curve 2 2 yx from x = 0 to x = 2, including the endpoints at (0, 2) and (2, 2). Note that (0, 2) coincides with the second two sections, so it is included. The graph ends at x = 5 and x = 2.

32. 2 2 0.5if 42 ()if 22 4if 24 xx fxxx xx

Graph the curve 2 0.5 yx  between x = 4 and x = 2, including the endpoints at ( 4, 8) and ( 2, 2). Graph the line yx  between x = 2 and x = 2, but do not include the endpoints at ( 2, 2) and (2, 2). Graph the curve 2 4 yx from x = 2 to x = 4, including the endpoints at (2, 0) and (4, 12). The graph ends at x = 4 and x = 4.

33. 3 2 3if 20 ()3if 01 4if 13 xx fxxx xxx

Graph the curve 3 3 yx between x = 2 and x = 0, including the endpoints at ( 2, 5) and (0, 3). Graph the line y = x + 3 between x = 0 and x = 1, but do not include the endpoints at (0, 3) and (1, 4). Graph the curve 2 4 yxx  from x = 1 to x = 3, including the endpoints at (1, 4) and (3, 2). The graph ends at x = 2 and x = 3.

34. 2 3 1 2 2if 31 ()1 if 12 1if 23 xx fxxx xx

Graph the curve y = 2x to from x = 3 to x = 1, including the endpoint ( 3, 6), but not including the endpoint ( 1, 2). Graph the curve 2 1 yx from x = 1 to x = 2, including the endpoints ( 1, 2) and (2, 5).

Graph the curve 3 1 2 1 yx from x = 2 to x = 3, including the endpoint (3, 14.5) but not including the endpoint (2, 5). Because the endpoints that are not included coincide with endpoints that are included, we use closed dots on the graph.

35. The solid circle on the graph shows that the endpoint (0, –1) is part of the graph, while the open circle shows that the endpoint (0, 1) is not part of the graph. The graph is made up of parts of two horizontal lines. The function which fits this graph is

 –1 if 0 () 1 if 0. fxx x   

domain:   ,; range: {–1, 1}

36. We see that y = 1 for every value of x except x = 0, and that when x = 0, y = 0. We can write the function as

1 if 0 () 0 if 0. fxx x   

domain:   ,; range: {0, 1}

37. The graph is made up of parts of two horizontal lines. The solid circle shows that the endpoint (0, 2) of the one on the left belongs to the graph, while the open circle shows that the endpoint (0, –1) of the one on the right does not belong to the graph. The function that fits this graph is  2 if 0 () –1 if 1. fxx x   

domain: (, 0](1, );   range: {–1, 2}

38. We see that y = 1 when 1 x  and that y = –1 when x > 2. We can write the function as

 1 if –1 () –1 if 2. fxx x   

domain: (–, –1](2, );   range: {–1, 1}

39. For 0 x  , that piece of the graph goes through the points ( 1, 1) and (0, 0). The slope is 1, so the equation of this piece is y = x. For x > 0, that piece of the graph is a horizontal line passing through (2, 2), so its equation is y = 2. We can write the function as  if 0 (). 2 if 0 fxxx x   

domain: (–, )  range: (,0]{2}  

40. For x < 0, that piece of the graph is a horizontal line passing though ( 3, 3), so the equation of this piece is y = 3. For 0 x  , the curve passes through (1, 1) and (4, 2), so the equation of this piece is yx  . We can write the function as 3 if 0 (). if 0 x fx xx     

domain: (–, )  range: {3}[0,)  

41. For x < 1, that piece of the graph is a curve passes through ( 8, 2), ( 1, 1) and (1, 1), so the equation of this piece is 3 yx  . The right piece of the graph passes through (1, 2) and (2, 3). 23 1 12 m  , and the equation of the line is 211yxyx  . We can write the function as 3 if 1 () 1if 1 fxxx xx       domain: (–, )  range: (,1)[2,)  

42. For all values except x = 2, the graph is a line. It passes through (0, 3) and (1, 1). The slope is 2, so the equation is y = 2x 3. At x = 2, the graph is the point (2, 3). We can write the function as  3if 2 (). 23if 2 fxx xx   

domain: (–, )  range: (,1)(1,)  

43. f (x) =   x Plot points.

x –xf (x) =   x –2 2 2

–1.5 1.5 1 –1 1 1

–0.5 0.5 0 0 0 0

0.5 –0.5 –1 1 –1 –1 1.5 –1.5 –2 2 –2 –2

More generally, to get y = 0, we need 0–10110. xxx 

To get y = 1, we need 1–2 x  12 –21.xx 

Follow this pattern to graph the step function.

domain:   ,; range: {...,–2,–1,0,1,2,...}

44. f (x) =   x Plot points.

Follow this pattern to graph the step function.

domain:   ,; range: {...,–2,–1,0,1,2,...}

45. f (x) =   2 x

To get y = 0, we need 1 2 0210. xx 

To get y = 1, we need 1 2 1221. xx 

To get y = 2, we need 3 2 2231. xx 

Follow this pattern to graph the step function.

domain:   ,; range: {...,–2,–1,0,1,2,...}

46. g (x) =  21 x

To get y = 0, we need 1 2 02111221. xxx 

To get y = 1, we need 3 2 12–122231. xxx 

Follow this pattern to graph the step function.

domain:   ,; range: {..., 2,–1,0,1,2,...}

47. The cost of mailing a letter that weighs more than 1 ounce and less than 2 ounces is the same as the cost of a 2-ounce letter, and the cost of mailing a letter that weighs more than 2 ounces and less than 3 ounces is the same as the cost of a 3-ounce letter, etc.

48. The cost is the same for all cars parking between 1 2 hour and 1-hour, between 1 hour and 1 2 1 hours, etc.

51. (a) For 08, x 

49.834.2 1.95 80 m  , so 1.9534.2.yx For 813 x  , 52.249.8 0.48 138 m  , so the equation is 52.20.48(13)yx  0.4845.96yx

(b) 1.9534.2if 08 () 0.4845.96if 813fxxx xx   

52. When 03 x  , the slope is 5, which means that the inlet pipe is open, and the outlet pipe is closed. When 35 x  , the slope is 2, which means that both pipes are open. When 58 x  , the slope is 0, which means that both pipes are closed. When 810 x  , the slope is 3, which means that the inlet pipe is closed, and the outlet pipe is open.

53. (a) The initial amount is 50,000 gallons. The final amount is 30,000 gallons.

(b) The amount of water in the pool remained constant during the first and fourth days.

(d) The slope of the segment between (1, 50000) and (3, 40000) is 5000, so the water was being drained at 5000 gallons per day.

54. (a) There were 20 gallons of gas in the tank at x = 3.

(b) The slope is steepest between t = 1 and t ≈ 2.9, so that is when the car burned gasoline at the fastest rate.

55. (a) There is no charge for additional length, so we use the greatest integer function. The cost is based on multiples of two feet, so 2 ()0.8 fxx

if 618 x

(b)

56. (a) 6.5 if 04 ()5.548 if 46 –30195 if 66.5 xx fxxx xx

Draw a graph of y = 6.5x between 0 and 4, including the endpoints. Draw the graph of y = –5.5x + 48 between 4 and 6, including the endpoint at 6 but not the one at 4. Draw the graph of y = –30x + 195, including the endpoint at 6.5 but not the one at 6. Notice that the endpoints of the three pieces coincide.

(b) From the graph, observe that the snow depth, y, reaches its deepest level (26 in.) when x = 4, x = 4 represents 4 months after the beginning of October, which is the beginning of February.

(c) From the graph, the snow depth y is nonzero when x is between 0 and 6.5. Snow begins at the beginning of October and ends 6.5 months later, in the middle of April.

Section 2.7 Graphing Techniques

1. To graph the function   2 3, fxx shift the graph of 2 yx  down 3 units.

2. To graph the function   2 5, fxx shift the graph of 2 yx  up 5 units.

3. The graph of  2 4 fxx is obtained by shifting the graph of 2 yx  to the left 4 units.

4. The graph of  2 7 fxx is obtained by shifting the graph of 2 yx  to the right 7 units.

5. The graph of   fxx  is a reflection of the graph of   fxx  across the x-axis.

6. The graph of   fxx  is a reflection of the graph of   fxx  across the y-axis.

7. To obtain the graph of 3 23, fxx shift the graph of 3 yx  2 units to the left and 3 units down.

8. To obtain the graph of 3 36, fxx shift the graph of 3 yx  3 units to the right and 6 units up.

9. The graph of   fxx  is the same as the graph of yx  because reflecting it across the y-axis yields the same ordered pairs.

10. The graph of 2 xy  is the same as the graph of  2 xy  because reflecting it across the x-axis yields the same ordered pairs.

11. (a) B; 2 (7)yx is a shift of 2 , yx  7 units to the right.

(b) D; 2 7 yx is a shift of 2 , yx  7 units downward.

(d) A; 2 (7)yx is a shift of 2 , yx  7 units to the left.

(e) C; 2 7 yx is a shift of 2 , yx  7 units upward.

12. (a) E; 3 4 yx  is a vertical stretch of 3 , yx  by a factor of 4.

(b) C; 3 yx  is a reflection of 3 , yx  over the x-axis.

(d) A; 3 4 yx is a shift of 3 , yx  4 units to the right.

(e) B; 3 4 yx is a shift of 3 , yx  4 units down.

13. (a) B; 2 2 yx is a shift of 2 , yx  2 units upward.

(b) A; 2 2 yx is a shift of 2 , yx  2 units downward.

(d) C; 2 (2)yx is a shift of 2 , yx  2 units to the right.

(e) F; 2 2 yx  is a vertical stretch of 2 , yx  by a factor of 2.

(f) D; 2 yx  is a reflection of 2 , yx  across the x-axis.

(g) H; 2 (2)1yx is a shift of 2 , yx  2 units to the right and 1 unit upward.

(h) E; 2 (2)1yx is a shift of 2 , yx  2 units to the left and 1 unit upward.

(i) I; 2 (2)1yx is a shift of 2 , yx  2 units to the left and 1 unit down.

14. (a) G; 3 yx is a shift of , yx  3 units to the left.

(b) D; 3 yx is a shift of , yx  3 units downward.

(d) B; 3 yx  is a vertical stretch of , yx  by a factor of 3.

(e) C; yx  is a reflection of yx  across the x-axis.

(f) A; 3 yx is a shift of , yx  3 units to the right.

(g) H; 32 yx is a shift of , yx  3 units to the right and 2 units upward.

(h) F; 32 yx is a shift of , yx  3 units to the left and 2 units upward.

(i) I; 32 yx is a shift of , yx  3 units to the right and 2 units downward.

15. (a) F; 2 yx is a shift of yx  2 units to the right.

(b) C; 2 yx is a shift of yx  2 units downward.

(d) D; 2 yx  is a vertical stretch of yx  by a factor of 2.

(e) G; yx  is a reflection of yx  across the x-axis.

(f) A; yx  is a reflection of yx  across the y-axis.

(g) E; 2 yx  is a reflection of 2 yx  across the x-axis. 2 yx  is a vertical stretch of yx  by a factor of 2.

(h) I; 22 yx is a shift of yx  2 units to the right and 2 units upward.

(i) B; 22 yx is a shift of yx  2 units to the left and 2 units downward.

16. The graph of 3 216fxx is the graph of   3 fxx  stretched vertically by a factor of 2, shifted left 1 unit and down 6 units. 17.

continuedonnextpage)

continued)

35. (a)   4 yfx is a horizontal translation of f, 4 units to the left. The point that corresponds to (8, 12) on this translated function would be 

84,124,12.

(b)   4 yfx is a vertical translation of f, 4 units up. The point that corresponds to (8, 12) on this translated function would be     8,1248,16. 

36. (a)   1 4 yfx  is a vertical shrinking of f, by a factor of 1 4 The point that corresponds to (8, 12) on this translated function would be

 1 4 8,128,3. 

(b)   4 yfx  is a vertical stretching of f, by a factor of 4. The point that corresponds to (8, 12) on this translated function would be 

 8,4128,48. 

37. (a) (4)yfx  is a horizontal shrinking of f, by a factor of 4. The point that corresponds to (8, 12) on this translated function is

  1 4 8,122,12. 

(b)   1 4 yfx  is a horizontal stretching of f, by a factor of 4. The point that corresponds to (8, 12) on this translated function is     84,1232,12. 

38. (a) The point that corresponds to (8, 12) when reflected across the x-axis would be (8, 12).

(b) The point that corresponds to (8, 12) when reflected across the y-axis would be ( 8, 12).

39. (a) The point that is symmetric to (5, –3) with respect to the x-axis is (5, 3).

(b) The point that is symmetric to (5, –3) with respect to the y-axis is (–5, –3).

40. (a) The point that is symmetric to (–6, 1) with respect to the x-axis is (–6, –1).

(b) The point that is symmetric to (–6, 1) with respect to the y-axis is (6, 1).

41. (a) The point that is symmetric to (–4, –2) with respect to the x-axis is (–4, 2).

(b) The point that is symmetric to (–4, –2) with respect to the y-axis is (4, –2).

42. (a) The point that is symmetric to (–8, 0) with respect to the x-axis is (–8, 0) because this point lies on the x-axis.

(b) The point that is symmetric to the point (–8, 0) with respect to the y-axis is (8, 0).

43. The graph of y = |x 2| is symmetric with respect to the line x = 2.

44. The graph of y = |x + 1| is symmetric with respect to the line x = 1.

45. 2 5 yx

Replace x with –x to obtain 22 ()55yxx . The result is the same as the original equation, so the graph is symmetric with respect to the y-axis. Because y is a function of x, the graph cannot be symmetric with respect to the x-axis. Replace x with –x and y with –y to obtain 222 ()222. yxyxyx 

The result is not the same as the original equation, so the graph is not symmetric with respect to the origin. Therefore, the graph is symmetric with respect to the y-axis only.

46. 4 23yx

Replace x with –x to obtain 44 2()323yxx 

The result is the same as the original equation, so the graph is symmetric with respect to the y-axis. Because y is a function of x, the graph cannot be symmetric with respect to the x-axis. Replace x with –x and y with –y to obtain 44 – 2()323yxyx 

4 23yx . The result is not the same as the original equation, so the graph is not symmetric with respect to the origin. Therefore, the graph is symmetric with respect to the y-axis only.

47. 22 12 xy

Replace x with –x to obtain 2222 ()1212 xyxy 

The result is the same as the original equation, so the graph is symmetric with respect to the y-axis. Replace y with –y to obtain 2222()1212 xyxy

The result is the same as the original equation, so the graph is symmetric with respect to the x-axis. Because the graph is symmetric with respect to the x-axis and y-axis, it is also symmetric with respect to the origin.

48. 22 6 yx

Replace x with –x to obtain 2 22266 yxyx

The result is the same as the original equation, so the graph is symmetric with respect to the y-axis. Replace y with –y to obtain 2222 ()66 yxyx 

49. 3 4 yxx 

Replace x with –x to obtain 33 3 4()()4() 4. yxxyxx yxx  

The result is not the same as the original equation, so the graph is not symmetric with respect to the y-axis. Replace y with –y to obtain 33 44. yxxyxx 

The result is not the same as the original equation, so the graph is not symmetric with respect to the x-axis. Replace x with –x and y with –y to obtain 33 33 4()()4() 44. yxxyxx yxxyxx

 

The result is the same as the original equation, so the graph is symmetric with respect to the origin. Therefore, the graph is symmetric with respect to the origin only.

50. 3 yxx 

Replace x with –x to obtain 33 ()(). yxxyxx 

The result is not the same as the original equation, so the graph is not symmetric with respect to the y-axis. Replace y with –y to obtain 33 .yxxyxx  The result is not the same as the original equation, so the graph is not symmetric with respect to the x-axis. Replace x with –x and y with –y to obtain ()()33 yxxyxx 

3 yxx  The result is the same as the original equation, so the graph is symmetric with respect to the origin. Therefore, the graph is symmetric with respect to the origin only.

51. 2 8 yxx

Replace x with –x to obtain ()()822 8. yxxyxx 

2288.yxxyxx 

The result is not the same as the original equation, so the graph is not symmetric with respect to the origin. Therefore, the graph has none of the listed symmetries.

52. y = x + 15

Replace x with –x to obtain ()15 15. yxyx 

The result is not the same as the original equation, so the graph is not symmetric with respect to the y-axis. Because y is a function of x, the graph cannot be symmetric with respect to the x-axis. Replace x with –x and y with –y to obtain ()1515.yxyx  The result is not the same as the original equation, so the graph is not symmetric with respect to the origin. Therefore, the graph has none of the listed symmetries.

53.   3 2 fxxx 

3 33 2 22

The function is odd.

54.   53 2 fxxx     53 5353 2 22 fxxx xxxxfx





The function is odd.

55.   42 0.526fxxx   42 42 0.526 0.526 fxxx xxfx  

The function is even.

56.   2 0.754fxxx



2 2 0.75 4 0.754 fxxx xxfx  

The function is even.

57.   3 9 fxxx

3 33 9 99 fxxx xxxxfx 



The function is neither.

58.

  4 58fxxx   4 4 58 58 fxxx xxfx 



The function is neither.

59.   2 1 fxx

This graph may be obtained by translating the graph of 2 yx  1 unit downward.

60.

  2 2 fxx

This graph may be obtained by translating the graph of 2 yx  2 units downward.

61.   2 2 fxx

This graph may be obtained by translating the graph of 2 yx  2 units upward.

62.   2 3 fxx

This graph may be obtained by translating the graph of 2 yx  3 units upward.

63. 2 4 gxx

This graph may be obtained by translating the graph of 2 yx  4 units to the right.

64. 2 2 gxx

This graph may be obtained by translating the graph of 2 yx  2 units to the right.

65. 2 2 gxx

This graph may be obtained by translating the graph of 2 yx  2 units to the left.

66.

2 3 gxx

This graph may be obtained by translating the graph of 2 yx  3 units to the left.

67.   1 gxx

The graph is obtained by translating the graph of yx  1 unit downward.

68.   32 gxx

This graph may be obtained by translating the graph of yx  3 units to the left and 2 units upward.

69.

(1)3hxx

This graph may be obtained by translating the graph of 3 yx  1 unit to the left. It is then reflected across the x-axis.

70.

(1)3hxx

This graph can be obtained by translating the graph of 3 yx  1 unit to the right. It is then reflected across the x-axis. (We may also reflect the graph about the x-axis first and then translate it 1 unit to the right.) 71.

2 21hxx

This graph may be obtained by translating the graph of 2 yx  1 unit down. It is then stretched vertically by a factor of 2. 72.

2 32hxx

This graph may be obtained by stretching the graph of 2 yx  vertically by a factor of 3, then shifting the resulting graph down 2 units.

2 2(2)4fxx

This graph may be obtained by translating the graph of 2 yx  2 units to the right and 4 units down. It is then stretched vertically by a factor of 2. 74.

2 3(2)1fxx

This graph may be obtained by translating the graph of 2 yx  2 units to the right and 1 unit up. It is then stretched vertically by a factor of 3 and reflected over the x-axis. 75.

2 fxx

This graph may be obtained by translating the graph of yx  two units to the left.

76. 

3 fxx

This graph may be obtained by translating the graph of yx  three units to the right.

77. 

fxx 

This graph may be obtained by reflecting the graph of yx  across the x-axis.

78.   2 fxx

This graph may be obtained by translating the graph of yx  two units down.

79.

21fxx

This graph may be obtained by stretching the graph of yx  vertically by a factor of two and then translating the resulting graph one unit up.

80.   32fxx

This graph may be obtained by stretching the graph of yx  vertically by a factor of three and then translating the resulting graph two units down.

81.

3 1 2 4 gxx

This graph may be obtained by stretching the graph of 3 yx  vertically by a factor of 1 2 , then shifting the resulting graph down four units.

82.   3 1 2 2 gxx

This graph may be obtained by stretching the graph of 3 yx  vertically by a factor of 1 2 , then shifting the resulting graph up two units.

83. 3 3 gxx

This graph may be obtained by shifting the graph of 3 yx  three units left.

84. 3 2 fxx

This graph may be obtained by shifting the graph of 3 yx  two units right.

85.  2 2 3 2 fxx

This graph may be obtained by translating the graph of 2 yx  two units to the right, then stretching the resulting graph vertically by a factor of 2 3 .

86. Because () (),gxxxfx  the graphs are the same.

87. (a) y = g(–x)

The graph of g(x) is reflected across the y-axis.

(b) y = g(x – 2)

The graph of g(x) is translated to the right 2 units.

The graph of g(x) is reflected across the x-axis.

(d) y = –g(x) + 2

The graph of g(x) is reflected across the x-axis and translated 2 units up.

88. (a)   yfx 

The graph of f(x) is reflected across the x-axis.

(b)   2 yfx 

The graph of f(x) is stretched vertically by a factor of 2.

The graph of f(x) is reflected across the y-axis.

(d)   1 2 yfx 

The graph of f(x) is compressed vertically by a factor of 1 2

89. It is the graph of   fxx  translated 1 unit to the left, reflected across the x-axis, and translated 3 units up. The equation is 13. yx

90. It is the graph of   gxx  translated 4 units to the left, reflected across the x-axis, and translated two units up. The equation is 42. yx

91. It is the graph of   fxx  translated one unit right and then three units down. The equation is 13. yx

92. It is the graph of   fxx  translated 2 units to the right, shrunken vertically by a factor of 1 2 , and translated one unit down. The equation is 1 2 21. yx

93. It is the graph of   gxx  translated 4 units to the left, stretched vertically by a factor of 2, and translated four units down. The equation is 244.yx

94. It is the graph of   fxx  reflected across the x-axis and then shifted two units down. The equation is 2 yx .

95. Because f (3) = 6, the point (3, 6) is on the graph. Because the graph is symmetric with respect to the origin, the point (–3, –6) is on the graph. Therefore, f (–3) = –6.

96. Because f (3) = 6, (3, 6) is a point on the graph. The graph is symmetric with respect to the y-axis, so (–3, 6) is on the graph. Therefore, f (–3) = 6.

97. Because f (3) = 6, the point (3, 6) is on the graph. The graph is symmetric with respect to the line x = 6 and the point (3, 6) is 3 units to the left of the line x = 6, so the image point of (3, 6), 3 units to the right of the line x = 6 is (9, 6). Therefore, f (9) = 6.

98. Because f (3) = 6 and f (–x) = f (x), f (–3) = f (3). Therefore, f (–3) = 6.

99. Because (3, 6) is on the graph, (–3, –6) must also be on the graph. Therefore, f (–3) = –6.

100. If f is an odd function, f (–x) = –f (x). Because f (3) = 6 and f (–x) = –f (x), f (–3) = –f (3). Therefore, f (–3) = –6.

101. f (x) = 2x + 5

Translate the graph of ()fx up 2 units to obtain the graph of ()(25)227. txxx 

Now translate the graph of t(x) = 2x + 7 left 3 units to obtain the graph of ()2(3)7267213. gxxxx 

(Note that if the original graph is first translated to the left 3 units and then up 2 units, the final result will be the same.)

102. f (x) = 3 – x

Translate the graph of ()fx down 2 units to obtain the graph of ()(3)21. txxx 

Now translate the graph of   1 txx right 3 units to obtain the graph of ()(3)1314. gxxxx 

(Note that if the original graph is first translated to the right 3 units and then down 2 units, the final result will be the same.)

103. (a) Because f(–x) = f(x), the graph is symmetric with respect to the y-axis.

(b) Because f(–x) = –f(x), the graph is symmetric with respect to the origin.

104. (a) f(x) is odd. An odd function has a graph symmetric with respect to the origin. Reflect the left half of the graph in the origin.

(b) f(x) is even. An even function has a graph symmetric with respect to the y-axis. Reflect the left half of the graph in the y-axis.

Chapter 2 Quiz (Sections 2.5 2.7)

1. (a) First, find the slope: 95 2 1(3) m 

Choose either point, say, ( 3, 5), to find the equation of the line: 52((3))2(3)5yxyx  211yx

(b) To find the x-intercept, let y = 0 and solve for x: 11 2 0211xx  . The x-intercept is   11 2 ,0.

2. Write 3x 2y = 6 in slope-intercept form to find its slope: 3 2 3263. xyyx 

Then, the slope of the line perpendicular to this graph is 2 3 2 3 4((6))yx 22 33 (6))4 yxyx 

3. (a) 8 x  (b) 5 y 

4. (a) Cubing function; domain: (,)  ; range: (,)  ; increasing over (,) 

(b) Absolute value function; domain: (,)  ; range: [0,)  ; decreasing over (,0)  ; increasing over (0,) 

A 5.5-minute call costs $2.75.

6. if 0 () 23if 0 fxxx xx      

For values of x < 0, the graph is the line y = 2x + 3. Do not include the right endpoint (0, 3). Graph the line yx  for values of x ≥ 0, including the left endpoint (0, 0).

7. 3 ()1fxx

Reflect the graph of 3 () fxx  across the x-axis, and then translate the resulting graph one unit up.

8. ()213fxx

Shift the graph of () fxx  one unit right, stretch the resulting graph vertically by a factor of 2, then shift this graph three units up.

9. This is the graph of () gxx  , translated four units to the left, reflected across the x-axis, and then translated two units down. The equation is 42 yx

10. (a)   2 7 fxx

Replace x with –x to obtain

2 2 ()7 7 fxx fxxfx

 

The result is the same as the original function, so the function is even.

(b)   3 1 fxxx

Replace x with –x to obtain

3 3 1 1 fxxx xxfx 



The result is not the same as the original equation, so the function is not even.

Because     , fxfx  the function is not odd. Therefore, the function is neither even nor odd.

Replace x with –x to obtain

10199 10199 10199 fxxx xx xx fx

Because f( x) = f(x), the function is odd.

Section 2.8 Function Operations and Composition

In exercises 1–10,   1 fxx and   2 gxx 

2 222 2127 fgfg

2 222 2121

2 222 21212 fgfg

2 2 213 2 24 2 ff gg

22 222215

6.     2 2221219gfgfg 

7. f is defined for all real numbers, so its domain is 

8. g is defined for all real numbers, so its domain is   ,. 

9. f + g is defined for all real numbers, so its domain is   ,. 

10. f g is defined for all real numbers except those values that make

 0, gx  so its domain is

,00,.

In Exercises 11–18, 2 ()3fxx and ()26gxx

11.  2 ()(3)(3)(3) (3)32(3)6 12012 fgfg 

fgfg 

12. 2 ()(5)(5)(5) [(5)3][2(5)6] 281644

fgfg

13. 2 ()(1)(1)(1) [(1)3][2(1)6] 484

fgfg 

14. 2 ()(4)(4)(4) [(4)3][2(4)6] 19(2)21

15. 2 ()(4)(4)(4) [43][2(4)6] 19(2)38 fgfg 

16. 2 ()(3)(3)(3) [(3)3][2(3)6] 1212144 fgfg

17. 2 (1)(1)341 (1) (1)2(1)682

18. 2 (5)(5)328 (5)7 (5)2(5)64 ff gg

19. ()34, ()25fxxgxx

()()()() (34)(25)9 fgxfxgx xxx

2 2 ()()()()(34)(25) 615820 6720 fgxfxgxxx xxx xx

()34 () ()25 ffxx x ggxx

The domains of both f and g are the set of all real numbers, so the domains of f + g, f – g, and fg are all   ,.  The domain of f g is the set of all real numbers for which

  0. gx  This is the set of all real numbers except 5 2 , which is written in interval notation as     55 22 , , .  

20. f (x) = 6 – 3x, g(x) = –4x + 1 ()()()() (63)(41) 77 fgxfxgx xx x 



 ()()()() (63)(41)5 fgxfxgx xxx   2 2 ()()()()(63)(41) 246123 12276 fgxfxgxxx xxx xx 

  ()63 () ()41 ffxx x ggxx     

The domains of both f and g are the set of all real numbers, so the domains of f + g, f – g, and fg are all   ,.  The domain of f g is the set of all real numbers for which   0. gx  This is the set of all real numbers except 1 4 , which is written in interval notation as     11 44 –,, .  

21. ()23,22()3 fxxxgxxx  22 2 ()()()() (23)(3) 343 fgxfxgx xxxx xx 

  22 22 2 ()()()() (23)(3) 233 23 fgxfxgx xxxx xxxx xx 





 (continuedonnextpage)

(continued)

()()()() (23)(3) 226339 2599 fgxfxgx xxxx xxxxxx xxxx  

22 43232 432

2 2 ()23 () () 3 ffxxx x ggxxx

The domains of both f and g are the set of all real numbers, so the domains of f + g, f – g, and fg are all   ,.  The domain of f g is the set of all real numbers for which

 0. gx  If 2 30 xx , then by the quadratic formula 111 2 i x   . The equation has no real solutions. There are no real numbers which make the denominator zero. Thus, the domain of f g is also   ,. 

22. ()42,22()32 fxxxgxxx 

22 2 ()()()() (42)(32) 52 fgxfxgx xxxx xx 

22 22 2 ()()()() (42)(32) 4232 352 fgxfxgx xxxx xxxx xx

22 43232 432 ()()()() (42)(32) 4128264 41024 fgxfxgx xxxx xxxxxx xxxx 

2 2 ()42 () () 32 ffxxx x ggxxx

The domains of both f and g are the set of all real numbers, so the domains of f + g, f – g, and fg are all   ,.  The domain of f g is the set of all real numbers x such that 2 320xx . Because 2 32(1)(2)xxxx  , the numbers which give this denominator a value of 0 are x = 1 and x = 2. Therefore, the domain of f g is the set of all real numbers except 1 and 2, which is written in interval notation as (–,1)(1,2)(2,)  

23. 1 ()41, ()fxxgx x  1

()()()()41 fgxfxgxx x  1

()()()()41 fgxfxgxx x 

()()()() 141 41 fgxfxgx x x xx 



1 ()41 ()41 () x ffxx xxx ggx

Because 1 4 41041, xxx  the domain of f is  1 4 ,.    The domain of g is

  ,00,.  Considering the intersection of the domains of f and g, the domains of f + g, f – g, and fg are all  1 4 ,.    Because 1 0 x  for any value of x, the domain of f g is also

1 4 ,.

24. 1 ()54, ()fxxgx x  ()()()() 11 5454 fgxfxgx xx xx 

  ()()()() 11 5454 fgxfxgx xx xx 

()()()() 154 54 fgxfxgx x x xx 

1 ()54 ()54 () x ffxx xxx ggx

Because 4 5 54054, xxx  the domain of f is  4 5 ,.

  The domain of g is     ,00,.  Considering the intersection of the domains of f and g, the domains of f + g, f – g, and fg are all  4 5 ,.



1 0 x  for any value of x, so the domain of f g is also

 4 5 ,.   

25.

MF TMF 

2008280 and2008470, thus 200820082008 280470750(thousand).

26.

MF TMF 

2012390 and2012630, thus 201220122012 3906301020(thousand).

27. Looking at the graphs of the functions, the slopes of the line segments for the period 2008 2012 are much steeper than the slopes of the corresponding line segments for the period 2004 2008. Thus, the number of associate’s degrees increased more rapidly during the period 2008 2012.

28. If 20042012,(), kTkr  and F(k) = s, then M(k) = rs.

29.

It represents the dollars in billions spent for general science in 2000.

30.

It represents the dollars in billions spent on space and other technologies in 2010.

31. Spending for space and other technologies spending decreased in the years 1995 2000 and 2010–2015.

32. Total spending increased the most during the years 2005 2010.

33. (a)

()(1)(1)(1)1(3)4

()(0)(0)(0)0(4)0 fgfg

(d) (1)11 (1) (1)33

34. (a) ()(0)(0)(0)022 fgfg

(b) ()(1)(1)(1) 213 fgfg 

(d) (2)4 (2)2 (2)2 ff gg

35. (a) ()(1)(1)(1)033 fgfg  (b) ()(2)(2)(2) 145 fgfg 

(d) (2)3 (2)undefined (2)0 ff gg

36. (a) ()(1)(1)(1)312 fgfg  (b) ()(0)(0)(0)202 fgfg 

(d) (1)3 (1)3 (1)1 ff gg

37. (a)

222725fgfg 

(b) ()(4)(4)(4)1055 fgfg 

(d) (0)5 (0)undefined (0)0 ff gg

38. (a)

222549fgfg 

(b) ()(4)(4)(4)000 fgfg

(d) (0)8 (0)8 (0)1 ff gg

41. Answers may vary. Sample answer: Both the slope formula and the difference quotient represent the ratio of the vertical change to the horizontal change. The slope formula is stated for a line while the difference quotient is stated for a function f

42. Answers may vary. Sample answer: As h approaches 0, the slope of the secant line PQ approaches the slope of the line tangent of the curve at P.

43.

2 fxx

(a) ()2()2 fxhxhxh

(b) ()()(2)(2) 22 fxhfxxhx xhxh 

1 fxx

(a) ()1()1 fxhxhxh 

(b) ()()(1)(1) 11 fxhfxxhx xhxh  

45.   62fxx

(a) ()6()2662 fxhxhxh 

46.

(b) ()() (662)(62) 662626 fxhfx xhx xhxh

  

  411fxx

(a) ()4()114411 fxhxhxh 

(b) ()() (4411)(411) 44114114 fxhfx xhx xhxh   

47.   25fxx

(a) ()2()5 225 fxhxh xh  

(b) ()() (225)(25) 225252 fxhfx xhx xhxh   

48. ()42fxx

(a) ()4()2 442 fxhxh xh

49. 1 ()fx x 

(a) 1 ()fxh xh

51. 2 () fxx 

(a) ()()2222 fxhxhxxhh  (b) 222 2 ()()2 2 fxhfxxxhhx xhh 

52. 2 () fxx  (a)

2 22 22 ()() 2 2 fxhxh xxhh xxhh  

(b)   222 222 2 ()()2 2 2 fxhfxxxhhx xxhhx xhh  

53. 2 ()1 fxx 

(a) 2 22 22 ()1() 1(2) 12 fxhxh xxhh xxhh

(b) 222 222 2 ()() (12)(1) 121 2 fxhfx xxhhx xxhhx xhh

54. 2 ()12 fxx  (a) 2 22 22 ()12() 12(2) 1242 fxhxh xxhh xxhh

(b)

222 222 2 ()() 124212 124212 42 fxhfx xxhhx xxhhx xhh

55. 2 ()31 fxxx

(a)  2 22 ()31 2331 fxhxhxh xxhhxh 

(b)

22 2 222 2 ()() 2331 31 233131 23 fxhfx xxhhxh xx xxhhxhxx xhhh

56. 2 ()42 fxxx

(a)  2 22 ()42 2442 fxhxhxh xxhhxh

(b)





22 2 222 2 ()() 2442 42 244242 24 fxhfx xxhhxh xx xxhhxhxx xhhh 

2 ()()24 (24) 24 fxhfxxhhh hh hxh h xh

232223431fxxf

2212131fffff

32235gxxg

225532ggggg

65. ()(2)[(2)](3)1 fgfgf 

66. ()(7)[(7)](6)9 fgfgf 

67. ()(3)[(3)](1)9 gfgfg 

68. ()(6)[(6)](9)12 gfgfg 

69. ()(4)[(4)](3)1 fffff 

70. ()(1)[(1)](9)12 ggggg 

71. ()(1)[(1)](9) fgfgf  

However, f (9) cannot be determined from the table given.

72. (())(7)(((7))) ((6))(9)12 gfggfg gfg  

73. (a) ()()(())(57) 6(57)9 304293033 fgxfgxfx x xx    

The domain and range of both f and g are (,)  , so the domain of fg  is (,) 

(b) ()()(())(69) 5(69)7 304573052 gfxgfxgx x xx 

The domain of gf  is (,) 

74. (a) ()()(())(31) 8(31)12 24812244 fgxfgxfx x xx 

The domain and range of both f and g are (,)  , so the domain of fg  is (,) 

(b) ()()(())(812) 3(812)1 243612435 gfxgfxgx x xx 

The domain of gf  is (,) 

75. (a) ()()(())(3)3 fgxfgxfxx  

The domain and range of g are (,)  , however, the domain and range of f are [0,)  . So, 303xx Therefore, the domain of fg  is [3,) 

(b)

()()(())3 gfxgfxgxx  

The domain and range of g are (,)  , however, the domain and range of f are [0,)  .Therefore, the domain of gf  is [0,) 

76. (a) ()()(())(1)1 fgxfgxfxx  

The domain and range of g are (,)  , however, the domain and range of f are [0,)  . So, 101xx . Therefore, the domain of fg  is [1,)  .

(b)   ()()(())1 gfxgfxgxx  

The domain and range of g are (,),  however, the domain and range of f are [0,).  Therefore, the domain of gf  is [0,). 

77. (a) 2 23 ()()(())(31) (31) fgxfgxfxx xx   

The domain and range of f and g are (,),  so the domain of fg  is (,).  (b)  3 2 33 63

()()(())() 31 31 gfxgfxgx xx xx    

The domain and range of f and g are (,),  so the domain of gf  is (,). 

78. (a) 42 42 42 ()()(())(4) 42 2 fgxfgxfxx xx xx  

The domain of f and g is (,),  while the range of f is (,)  and the range of g is   4,, so the domain of fg  is (,). 

(b) 42 ()()(())(2) (2)(2)4 gfxgfxgx xx  

The domain of f and g is (,),  while the range of f is (,)  and the range of g is   4,, so the domain of gf  is (,). 

79. (a) ()()(())(3)31 fgxfgxfxx  

The domain and range of g are (,)  , however, the domain of f is [1,),  while the range of f is [0,).  So, 1 3 310xx . Therefore, the domain of fg  is  1 3 ,    (b)   ()()(())1 31 gfxgfxgx x   

The domain and range of g are (,)  , however, the range of f is [0,)  . So 101xx . Therefore, the domain of gf  is [1,)  .

80. (a) ()()(())(2)22 fgxfgxfxx  

The domain and range of g are (,)  , however, the domain of f is [2,)  . So, 2201 xx . Therefore, the domain of fg  is   1,  .

(b)   ()()(())2 gfxgfxgx  

22 x 

The domain and range of g are (,)  , however, the range of f is [0,)  . So 202xx . Therefore, the domain of gf  is [2,) 

81. (a)

2 1 ()()(())(1) fgxfgxfxx   

The domain and range of g are (,)  , however, the domain of f is (,0)(0,)   . So, 101xx

Therefore, the domain of fg  is (,1)(1,)  

(b)

 22 ()()(())1 gfxgfxgxx  

The domain and range of f is (,0)(0,)   , however, the domain and range of g are (,)  . So 0 x  Therefore, the domain of gf  is (,0)(0,)   .

82. (a) 4 4 ()()(())(4) fgxfgxfxx   

The domain and range of g are (,)  , however, the domain of f is (,0)(0,)   . So, 404xx

Therefore, the domain of fg  is (,4)(4,)  

(b)   44

()()(())4 gfxgfxgxx  

The domain and range of f is (,0)(0,)   , however, the domain and range of g are (,)  . So 0 x  . Therefore, the domain of gf  is (,0)(0,)   .

83. (a)

  11 ()()(()) 2 fgxfgxfxx  

The domain and range of g are (,0)(0,)   , however, the domain of f is [2,)  . So, 1 20 x  1 2 0 or xx (using test intervals). Therefore, the domain of fg  is    1 2 ,0,    

(b)   1 2 ()()(())2 x gfxgfxgx   

The domain of f is [2,)  and its range is [0,).  The domain and range of g are (,0)(0,)   . So 202xx

Therefore, the domain of gf  is (2,)  .

84. (a)   22 ()()(()) 4 fgxfgxfxx  

The domain and range of g are (,0)(0,)   , however, the domain of f is [4,)  . So, 2 40 x  1 2 0 or xx (using test intervals). Therefore, the domain of fg  is    1 2 ,0,    

(b)   2 4 ()()(())4 x gfxgfxgx   

The domain of f is [4,)  and its range is [0,).  The domain and range of g are (,0)(0,)   . So 404xx Therefore, the domain of gf  is (4,) 

85. (a)

 11 55 ()()(()) fgxfgxfxx

The domain of g is (,5)(5,)   , and the range of g is (,0)(0,)  

The domain of f is [0,)  . Therefore, the domain of fg  is   5, 

(b)   1 5 ()()(()) x gfxgfxgx   

The domain and range of f is [0,).  The domain of g is (,5)(5,).  

Therefore, the domain of gf  is [0,). 

86. (a)   33 66 ()()(()) fgxfgxfxx  

The domain of g is (,6)(6,)   , and the range of g is (,0)(0,)  

The domain of f is [0,)  . Therefore, the domain of fg  is   6,.

(b)   3 6 ()()(()) x gfxgfxgx   

The domain and range of f is [0,)  . The domain of g is (,6)(6,)   . Therefore, the domain of gf  is [0,). 

87. (a)   11 1212

()()(()) x fgxfgxfxxx  

The domain and range of g are (,0)(0,)   . The domain of f is (,2)(2,),   and the range of f is (,0)(0,)   . So, 12 00 x x x  or 1 2 0 x  or 1 2 x  (using test intervals). Thus, 0 x  and 1 2 x  . Therefore, the domain of fg  is

 11 22 ,00,,.  

(b)

()()(()) 2 gfxgfxgxx x   

 11 21(2)

The domain and range of g are (,0)(0,).   The domain of f is (,2)(2,),   and the range of f is (,0)(0,).   Therefore, the domain of gf  is (,2)(2,).  

88. (a)   11 14 14 ()()(()) xx x x fgxfgxf 

The domain and range of g are (,0)(0,).   The domain of f is (,4)(4,),   and the range of f is (,0)(0,).   So, 14 00 x x x   or 1 4 0 x  or 1 4 140xx  (using test intervals). Thus, x ≠ 0 and x ≠ 1 4 . Therefore, the domain of fg  is

 11 44 ,00,,.  

(b)   11 41(4) ()()(()) 4 gfxgfxgxx x 

The domain and range of g are (,0)(0,)   . The domain of f is (,4)(4,)   , and the range of f is (,0)(0,)   . Therefore, the domain of gf  is (,4)(4,).  

89.   (2)(1)2 and (3)(2)5 Since (1)7 and (1)3,(3)7. gfggfg gffg   x 

90. ()fx is odd, so (1)(1)(2)2.ff

Because ()gx is even, (1)(1)2gg and (2)(2)0.gg ()(1)1, fg   so (1)1 fg  and (2)1. f  ()fx is odd, so (2)(2)1.ff Thus,  ()(2)(2)(0)0 fgfgf

and  ()(1)(1)(2)1 fgfgf  and  ()(2)(2)(0)0. fgfgf  x –2 –1 0 1 2

91. Answers will vary. In general, composition of functions is not commutative. Sample answer:

233232 692611 322323 64367 fgxfxx

Thus,

fgx  is not equivalent to

3 3 3 7 77 77

3 3 33 3 7 77

In Exercises 97 102, we give only one of many possible answers.

97. 2 ()(62)hxx

Let g(x) = 6x – 2 and 2 () fxx  . 2 ()()(62)(62)() fgxfxxhx 

98. ()(1112)22 hxxx  22 Let ()1112 and ().gxxxfxx  2 22 ()()(1112) (1112)() fgxfxx xxhx 

99. 2 ()1hxx

Let 2 ()1and (). gxxfxx  22 ()()(1)1(). fgxfxxhx 

100. ()(23)3hxx 3

Let ()23 and ().gxxfxx  3 ()()(23)(23)() fgxfxxhx

101. ()612hxx

Let ()6 and ()12.gxxfxx  ()()(6)612() fgxfxxhx  

102. 3 ()234hxx

Let 3 ()23 and ()4.gxxfxx  3 ()()(23)234() fgxfxxhx

103. f(x) = 12x, g(x) = 5280x ()()[()](5280) 12(5280)63,360 fgxfgxfx xx   

The function fg  computes the number of inches in x miles.

104.     3,1760 fxxgxx 

 1760 317605280 fgxfgxfx xx   

  fgx  compute the number of feet in x miles.

105. 2 3 () 4 xx   (a) 33222 (2)(2)(4)3 44 xxxx  

(b) 2 (16)(28)3(8) 643 square units A   

106. (a) 44 4 xxxsss 

(b)   2 2 2 416ysxx 

107. (a) 2 ()4 and () rttrr    22 ()()[()] (4)(4)16 rtrt ttt      

(b) ()() rt   defines the area of the leak in terms of the time t, in minutes.

108. (a) 22 ()()[()] (2)(2)4 rtrt ttt      

(b) It defines the area of the circular layer in terms of the time t, in hours.

109. Let x = the number of people less than 100 people that attend.

(a) x people fewer than 100 attend, so 100 – x people do attend N(x) = 100 – x

(b) The cost per person starts at $20 and increases by $5 for each of the x people that do not attend. The total increase is $5x, and the cost per person increases to $20 + $5x. Thus, G(x) = 20 + 5x.

(d) If 80 people attend, x = 100 – 80 = 20.

201002020520 8020100 80120$9600

110. (a) 1 0.02 yx 

(b) 2 0.015(500)yx

3. A(–6, 3), B(–6, 8) 22 2 (,)[6(6)](83) 05255 dAB  

Midpoint: 6(6)38121111 , , 6, 22222

(d) 12 12 ()(250) (250)(250)

$16.25 yy yy

0.02(250)0.015(250500)

50.015(750)1511.25

111. (a)  1 2 gxx  (b)   1 fxx

112. If the area of a square is 2 x square inches, each side must have a length of x inches. If 3 inches is added to one dimension and 1 inch is subtracted from the other, the new dimensions will be x + 3 and x – 1. Thus, the area of the resulting rectangle is (x) = (x + 3)(x – 1).

Chapter 2 Review Exercises

1. P(3, –1), Q(–4, 5) 22 22 (,)(43)[5(1)] (7)6493685 dPQ

Midpoint: 3(4)15141,,,2 22222

2. M(–8, 2), N(3, –7) 22 22 (,)[3(8)](72) 11(9)12181202

dMN  

Midpoint: 832(7)55 , , 2222

4. Label the points A(5, 7), B(3, 9), and C(6, 8).   22 2 2 (,)3597 22448 dAB   22 22 (,)(65)(87) 11112 dAC  

22 2 2 (,)6389 319110 dBC

Because 

222 8210,  triangle

ABC is a right triangle with right angle at (5, 7).

5. Let B have coordinates (x, y). Using the midpoint formula, we have  610 8,2,22 610 82 22 616104 226 xy xy xy xy

    

 





The coordinates of B are (22, 6).

6. P(–2, –5), Q(1, 7), R(3, 15) 22 22 (,)(1(2))(7(5)) (3)(12)9144 153317

dPQ 



dQR 

 22 22 (,)(31)(157) 2846468217

dPR 

 22 22 (,)(3(2))(15(5)) (5)(20)25400517

 (continuedonnextpage)

(continued)

(,)(,)317217 dPQdQR

517(,), dPR  so these three points are collinear.

7. Center (–2, 3), radius 15 222 222 22 ()() [(2)](3)15 (2)(3)225 xhykr xy xy 

8. Center (5, 7), radius 3

9. Center (–8, 1), passing through (0, 16)

The radius is the distance from the center to any point on the circle. The distance between (–8, 1) and (0, 16) is 2222 (0(8))(161)815 6422528917. r

The equation of the circle is 222 22 [(8)](1)17 (8)(1)289 xy xy 



10. Center (3, –6), tangent to the x-axis

The point (3, –6) is 6 units directly below the x-axis. Any segment joining a circle’s center to a point on the circle must be a radius, so in this case the length of the radius is 6 units.

14. The center of the circle is (5, 6). Use the distance formula to find the radius: 222(45)(96)1910 r 

The equation is 22 (5)(6)10 xy

15. 2246120 xxyy

Complete the square on x and y to put the equation in center-radius form.

 22 22 22 4 6 12 44691249 231 xxyy xxyy xy   

The circle has center (2, –3) and radius 1.

16. 22610300 xxyy

Complete the square on x and y to put the equation in center-radius form. 22 22 (69)(1025)30925 (3)(5)4 xxyy xy 

The circle has center (3, 5) and radius 2. 17.

 22 22 22 22499499 4444 22 73499 4 22444 22 7354 224 2142620 7310 7 3 1 731 xxyy xxyy xxyy xxyy xy xy

  

222 222 22 ()() (3)[(6)]6 (3)(6)36 xhykr xy xy

11. The center of the circle is (0, 0). Use the distance formula to find the radius: 222(30)(50)92534 r 

The equation is 22 34 xy

12. The center of the circle is (0, 0). Use the distance formula to find the radius: 222(20)(30)4913 r 

The equation is 22 13 xy .

13. The center of the circle is (0, 3). Use the distance formula to find the radius: 222(20)(63)4913 r 

The equation is 22(3)13xy

The circle has center   73 22 , and radius 549636 54 42 44 

 22 22 22 221211212525 4444 2 2 5146 11 224 3333150 1150 1150 11 50 xxyy xxyy xxyy xxyy xy

The circle has center   5 11 22 , and radius 146 2 .

19. This is not the graph of a function because a vertical line can intersect it in two points.

domain: [ 6, 6]; range: [ 6, 6]

20. This is not the graph of a function because a vertical line can intersect it in two points.

domain:   ,; range:   0, 

21. This is not the graph of a function because a vertical line can intersect it in two points.

domain:   ,; range: (,1][1,)  

22. This is the graph of a function. No vertical line will intersect the graph in more than one point.

domain:   ,; range:   0, 

23. This is not the graph of a function because a vertical line can intersect it in two points.

domain:   0,;  range:   , 

24. This is the graph of a function. No vertical line will intersect the graph in more than one point.

domain:   ,; range:   , 

25. 2 6 yx 

Each value of x corresponds to exactly one value of y, so this equation defines a function.

26. The equation 2 1 3 xy  does not define y as a function of x. For some values of x, there will be more than one value of y. For example, ordered pairs (3, 3) and (3, –3) satisfy the relation. Thus, the relation would not be a function.

27. The equation 2 yx does not define y as a function of x. For some values of x, there will be more than one value of y. For example, ordered pairs (3, 1) and (3, 1) satisfy the relation.

28. The equation 4 y x  defines y as a function of x because for every x in the domain, which is (–,0)(0,)   , there will be exactly one value of y

29. In the function ()4 fxx  , we may use any real number for x. The domain is   ,. 

30. 8 () 8 fxx x   x can be any real number except 8 because this will give a denominator of zero. Thus, the domain is (–,8)(8,).  

31.   63 fxx 

In the function 63 yx  , we must have 630 x  6306322 xxxx  Thus, the domain is   ,2.

32. (a) As x is getting larger on the interval   2,,  the value of y is increasing.

(b) As x is getting larger on the interval   ,2, the value of y is decreasing.

In exercises 33–36,   2 236.fxxx

33.   2 323336 29336 189615 f   

34.   2 0.520.530.56 20.2530.56 0.51.568 f 

35.  2 0203066 f 

36.   2 236fkkk

37. 2 5 2555251 xyyxyx 

The graph is the line with slope 2 5 and y-intercept (0, –)1. It may also be graphed using intercepts. To do this, locate the x-intercept: 0 y 

 5 2 250525 xxx 

38. 3 7 371473142 xyyxyx 

The graph is the line with slope of 3 7 and y-intercept (0, 2). It may also be graphed using intercepts. To do this, locate the x-intercept by setting y = 0:

 14 3 37014314 xxx 

39. 2 5 252052204 xyyxyx

The graph is the line with slope of 2 5 and y-intercept (0, 4). It may also be graphed using intercepts. To do this, locate the x-intercept: x-intercept: 0 y 

2502022010 xxx

40. 1 3 3 yxyx 

The graph is the line with slope 1 3 and y-intercept (0, 0), which means that it passes through the origin. Use another point such as (6, 2) to complete the graph.

41. f (x) = x

The graph is the line with slope 1 and y-intercept (0, 0), which means that it passes through the origin. Use another point such as (1, 1) to complete the graph.

42. 1 4 48 48 2 xy yx yx

The graph is the line with slope 1 4 and y-intercept (0, –2). It may also be graphed using intercepts. To do this, locate the x-intercept:

04088yxx

43. x = –5

The graph is the vertical line through (–5, 0).

44. f (x) = 3

The graph is the horizontal line through (0, 3).

45. 202yy

The graph is the horizontal line through (0, 2).

46. The equation of the line that lies along the x-axis is y = 0.

47. Line through (0, 5), 2 3 m 

Note that 22 33 m 

Begin by locating the point (0, 5). Because the slope is 2 3 , a change of 3 units horizontally (3 units to the right) produces a change of –2 units vertically (2 units down). This gives a second point, (3, 3), which can be used to complete the graph.

(continuedonnextpage)

48. Line through (2, –4), 3 4 m 

First locate the point (2, –4).

Because the slope is 3 4 , a change of 4 units horizontally (4 units to the right) produces a change of 3 units vertically (3 units up). This gives a second point, (6, –1), which can be used to complete the graph.

49. through (2, –2) and (3, –4)

50. through (8, 7) and   1 2 ,2 21 115 21 2 2 279 8 2186 9 15155 yy m xx

51. through (0, –7) and (3, –7)

77 0 0 303 m

52. through (5, 6) and (5, –2) 21 21 268 550 yy m xx

The slope is undefined.

53. 1123 xy

Solve for y to put the equation in slopeintercept form. 3 11 22 2113yxyx 

Thus, the slope is 11 2

54. 9x – 4y = 2.

Solve for y to put the equation in slopeintercept form. 9 1 42 492yxyx 

Thus, the slope is 9 4

55. 202xx

The graph is a vertical line, through (2, 0). The slope is undefined.

56. x – 5y = 0.

Solve for y to put the equation in slopeintercept form. 1 5 5 yxyx

Thus, the slope is 1 5

57. Initially, the car is at home. After traveling for 30 mph for 1 hr, the car is 30 mi away from home. During the second hour the car travels 20 mph until it is 50 mi away. During the third hour the car travels toward home at 30 mph until it is 20 mi away. During the fourth hour the car travels away from home at 40 mph until it is 60 mi away from home. During the last hour, the car travels 60 mi at 60 mph until it arrived home.

58. (a) This is the graph of a function because no vertical line intersects the graph in more than one point.

(b) The lowest point on the graph occurs in December, so the most jobs lost occurred in December. The highest point on the graph occurs in January, so the most jobs gained occurred in January.

(c) The number of jobs lost in December is approximately 6000. The number of jobs gained in January is approximately 2000.

(d) It shows a slight downward trend.

59. (a) We need to first find the slope of a line that passes between points (0, 30.7) and (12, 82.9) 21 21 82.930.752.2 4.35

Now use the point-intercept form with b = 30.7 and m = 4.35. y = 4.35x + 30.7

The slope, 4.35, indicates that the number of e-filing taxpayers increased by 4.35% each year from 2001 to 2013.

(b) For 2009, we evaluate the function for x = 8. y = 4.35(8) + 30.7 = 65.5

65.5% of the tax returns are predicted to have been filed electronically.

60. We need to find the slope of a line that passes between points (1980, 21000) and (2013, 63800)

21 21 63,80021,000 20131980 42,800 $1297 per year 33 yy m xx  

The average rate of change was about $1297 per year.

61. (a) through (3, –5) with slope –2 Use the point-slope form. 11()

(5)2(3) 52(3) 526 21 yymxx yx yx yx yx

(b) Standard form: 2121yxxy 

62. (a) through (–2, 4) and (1, 3) First find the slope.

341 1(2)3 m 

Now use the point-slope form with 1 11 3 (,)(1,3) and . xym 11 1 3 10 1 33 () 3(1) 3(3)1(1) 391 310 yymxx yx yx yx yxyx 

(b) Standard form: 10 1 33 310 310 yxyx xy  

63. (a) through (2, –1) parallel to 3x – y = 1

Find the slope of 3x – y = 1. 313131 xyyxyx 

The slope of this line is 3. Because parallel lines have the same slope, 3 is also the slope of the line whose equation is to be found. Now use the point-slope form with 11 (,)(2,1) and 3. xym 11() (1)3(2) 13637 yymxx yx yxyx 

(b) Standard form: 373737yxxyxy 

64. (a) x-intercept (–3, 0), y-intercept (0, 5)

Two points of the line are (–3, 0) and (0, 5). First, find the slope.

The slope is 5 3 and the y-intercept is (0, 5). Write the equation in slopeintercept form: 5 3 5 yx

(b) Standard form: 5 3 53515

65. (a) through (2, –10), perpendicular to a line with an undefined slope

A line with an undefined slope is a vertical line. Any line perpendicular to a vertical line is a horizontal line, with an equation of the form y = b. The line passes through (2, –10), so the equation of the line is y = –10.

(b) Standard form: y = –10

66. (a) through (0, 5), perpendicular to 8x + 5y = 3

Find the slope of 8x + 5y = 3. 83 55 853583 xyyx yx  

The slope of this line is 8 5 The slope of any line perpendicular to this line is 5 8 , because   85 58 1.

The equation in slope-intercept form with slope 5 8 and y-intercept (0, 5) is 5 8 5. yx

(b) Standard form: 5 8 58540 58405840 yxyx xyxy  

67. (a) through (–7, 4), perpendicular to y = 8

The line y = 8 is a horizontal line, so any line perpendicular to it will be a vertical line. Because x has the same value at all points on the line, the equation is x = –7. It is not possible to write this in slopeintercept form.

(b) Standard form: x = –7

68. (a) through (3, –5), parallel to y = 4

This will be a horizontal line through (3, –5). Because y has the same value at all points on the line, the equation is y = –5.

(b) Standard form: y = –5

69. ()3fxx

The graph is the same as that of , yx  except that it is translated 3 units downward.

70. () fxx 

The graph of () fxx  is the reflection of the graph of yx  about the x-axis.

71. 2 ()13fxx

The graph of 2 ()13fxx is a translation of the graph of 2 yx  to the left 1 unit, reflected over the x-axis and translated up 3 units.

72. ()2fxx

The graph of ()2fxx is the reflection of the graph of yx  about the x-axis, translated down 2 units.

73.  ()3fxx

To get y = 0, we need 031 x  34. x  To get y = 1, we need 132 x  45. x 

Follow this pattern to graph the step function.

74. 3 ()212fxx

The graph of 3 ()212fxx is a translation of the graph of 3 yx  to the left 1 unit, stretched vertically by a factor of 2, and translated down 2 units.

75.  42 if 1 () 35 if 1 fxxx xx   

Draw the graph of y = –4x + 2 to the left of x = 1, including the endpoint at x = 1. Draw the graph of y = 3x – 5 to the right of x = 1, but do not include the endpoint at x = 1. Observe that the endpoints of the two pieces coincide.

76. 2 3 if 2 () 4 if 2 fxxx xx      

Graph the curve 2 3 yx to the left of x = 2, and graph the line y = –x + 4 to the right of x = 2. The graph has an open circle at (2, 7) and a closed circle at (2, 2).

77. if 3 () 6 if 3 fxxx xx   

Draw the graph of yx  to the left of x = 3, but do not include the endpoint. Draw the graph of y = 6 – x to the right of x = 3, including the endpoint. Observe that the endpoints of the two pieces coincide.

78. Because x represents an integer,   .xx  Therefore,   2.xxxxx 

79. True. The graph of an even function is symmetric with respect to the y-axis.

80. True. The graph of a nonzero function cannot be symmetric with respect to the x-axis. Such a graph would fail the vertical line test

81. False. For example, 2 () fxx  is even and (2, 4) is on the graph but (2, –4) is not.

82. True. The graph of an odd function is symmetric with respect to the origin.

83. True. The constant function   0 fx  is both even and odd. Because     0,fxfx  the function is even. Also

00,fxfx  so the function is odd.

84. False. For example, 3 () fxx  is odd, and (2, 8) is on the graph but (–2, 8) is not.

85. 2 10 xy

Replace x with –x to obtain 2 ()10. xy 

The result is not the same as the original equation, so the graph is not symmetric with respect to the y-axis. Replace y with –y to obtain 22 ()1010.xyxy  The result is the same as the original equation, so the graph is symmetric with respect to the x-axis. Replace x with –x and y with –y to obtain 22 ()()10()10. xyxy 

The result is not the same as the original equation, so the graph is not symmetric with respect to the origin. The graph is symmetric with respect to the x-axis only.

86. 22 5530 yx

Replace x with –x to obtain 2222 55()305530. yxyx 

The result is the same as the original equation, so the graph is symmetric with respect to the y-axis. Replace y with –y to obtain 2222 5()5305530. yxyx 

The result is the same as the original equation, so the graph is symmetric with respect to the x-axis. The graph is symmetric with respect to the y-axis and x-axis, so it must also be symmetric with respect to the origin. Note that this equation is the same as 22 6 yx , which is a circle centered at the origin.

87. 23 xy 

Replace x with –x to obtain 2323 () . xyxy  The result is the same as the original equation, so the graph is symmetric with respect to the y-axis. Replace y with –y to obtain 2323 (). xyxy 

The result is not the same as the original equation, so the graph is not symmetric with respect to the x-axis. Replace x with –x and y with –y to obtain 2323 ()(). xyxy 

The result is not the same as the original equation, so the graph is not symmetric with respect to the origin. Therefore, the graph is symmetric with respect to the y-axis only.

88. 3 4 yx

Replace x with –x to obtain 3 4 yx

The result is not the same as the original equation, so the graph is not symmetric with respect to the y-axis. Replace y with –y to obtain 33 ()44 yxyx  3 4 yx

The result is not the same as the original equation, so the graph is not symmetric with respect to the x-axis. Replace x with –x and y with –y to obtain 333 ()()444. yxyxyx 

The result is not the same as the original equation, so the graph is not symmetric with respect to the origin. Therefore, the graph has none of the listed symmetries.

89. 64 xy

Replace x with –x to obtain 6()4 xy  64 xy  . The result is not the same as the original equation, so the graph is not symmetric with respect to the y-axis. Replace y with –y to obtain 6()464 xyxy  . The result is not the same as the original equation, so the graph is not symmetric with respect to the x-axis. Replace x with –x and y with –y to obtain 6()()4 xy  64 xy  . This equation is not equivalent to the original one, so the graph is not symmetric with respect to the origin. Therefore, the graph has none of the listed symmetries.

90. yx 

Replace x with –x to obtain (). yxyx  The result is not the same as the original equation, so the graph is not symmetric with respect to the y-axis. Replace y with –y to obtain yxyx  The result is the same as the original equation, so the graph is symmetric with respect to the x-axis. Replace x with –x and y with –y to obtain (). yxyx  The result is not the same as the original equation, so the graph is not symmetric with respect to the origin. Therefore, the graph is symmetric with respect to the x-axis only.

91. y = 1

This is the graph of a horizontal line through (0, 1). It is symmetric with respect to the y-axis, but not symmetric with respect to the x-axis and the origin.

92. xy 

Replace x with –x to obtain xyxy

The result is the same as the original equation, so the graph is symmetric with respect to the y-axis. Replace y with –y to obtain xyxy  The result is the same as the original equation, so the graph is symmetric with respect to the x-axis. Because the graph is symmetric with respect to the xaxis and with respect to the y-axis, it must also by symmetric with respect to the origin.

93. 22 0 xy

Replace x with x to obtain 2 22200xyxy  The result is the same as the original equation, so the graph is symmetric with respect to the y-axis. Replace y with y to obtain 2 22200. xyxy The result is the same as the original equation, so the graph is symmetric with respect to the x-axis. Because the graph is symmetric with respect to the x-axis and with respect to the y-axis, it must also by symmetric with respect to the origin.

94. 2 2 24 xy

Replace x with x to obtain

 22 2 2 2424.xyxy 

The result is the same as the original equation, so the graph is symmetric with respect to the y-axis. Replace y with y to obtain

2 2 24. xy The result is not the same as the original equation, so the graph is not symmetric with respect to the x-axis. Replace x with –x and y with –y to obtain

 22 2 2 2424,xyxy  which is not equivalent to the original equation. Therefore, the graph is not symmetric with respect to the origin.

95. To obtain the graph of () gxx  , reflect the graph of () fxx  across the x-axis.

96. To obtain the graph of ()2hxx , translate the graph of () fxx  down 2 units.

97. To obtain the graph of ()24kxx , translate the graph of () fxx  to the right 4 units and stretch vertically by a factor of 2.

98. If the graph of ()34fxx is reflected about the x-axis, we obtain a graph whose equation is (34)34.yxx 

99. If the graph of ()34fxx is reflected about the y-axis, we obtain a graph whose equation is ()3()434.yfxxx 

100. If the graph of ()34fxx is reflected about the origin, every point (x, y) will be replaced by the point (–x, –y). The equation for the graph will change from 34yx to 3()4yx  34yx  34.yx

101. (a) To graph ()3,yfx translate the graph of y = f (x), 3 units up.

(b) To graph (2),yfx translate the graph of y = f (x), 2 units to the right.

(d) To graph ()yfx  , keep the graph of y = f (x) as it is where 0 y  and reflect the graph about the x-axis where y < 0.

102. No. For example suppose   2 fxx and   2.gxx  Then ()()(())(2)22 fgxfgxfxx  

The domain and range of g are (,)  , however, the domain of f is [2,)  . So, 2201 xx . Therefore, the domain of fg  is   1,  . The domain of g, (,),  is not a subset of the domain of   ,1,. fg  

For Exercises 103–110, 2 ()34fxx=− and 2 ()34.gxxx=−−

103. 22 4322 432 ()()()() (34)(34) 391241216 39161216 fgxfxgx xxx xxxxx xxxx

fgfg 

 

104. 22 ()(4)(4)(4) (344)(4344) (3164)(16344) (484)(16124) 44044

 

fgfg 

105. 22 ()(4)(4)(4) [3(4)4][(4)3(4)4] [3(16)4][163(4)4] [484][16124] 442468

  





106. 22 22 22 2 ()(2)(2)(2) [3(2)4][(2)3(2)4] [3(4)4][43(2)4] (124)(464) 1668 fgkfkgk kkk kkk kkk kk





107. 2 2 (3)334394 (3) (3)9334 3334 2742323 99444

108.

(1)

109. The domain of (fg)(x) is the intersection of the domain of f (x) and the domain of g(x). Both have domain

,,  so the domain of (fg)(x) is

110. 22 2 3434 () (1)(4) 34 fxx x gxx xx

Because both   fx and

 gx have domain

,,  we are concerned about values of x that make 

0. gx  Thus, the expression is undefined if (x + 1)(x – 4) = 0, that is, if x = –1 or x = 4. Thus, the domain is the set of all real numbers except x = –1 and x = 4, or (–,–1)(–1,4)(4,).  

111.   29fxx

()2()9229 fxhxhxh 

  Thus, ()()2 2. fxhfxh hh  

()()(229)(29) 229292 fxhfxxhx xhxh

112. 2 ()53 fxxx 2 22 ()()5()3 2553 fxhxhxh xxhhxh





 

 222 222 2 ()() (2553)(53) 255353 25 fxhfx xxhhxhxx xxhhxhxx xhhh

 

 2 ()()25 (25) 25 fxhfxxhhh hh hxh xh h

For Exercises 113–118, 2 ()2 and ().fxxgxx  113.

2 ()()[()]2 22 gfxgfxgx xx 

114. 22 ()()[()]()2 fgxfgxfxx

2, so33211.fxxf

Therefore,

331gfgfg

2 11.

116.

2 2 , so 6636. gxxg Therefore,

 36234

6636fgfgf 

11123gfgfgg

Because 3 is not a real number, 

1 gf  is not defined.

118. To find the domain of , fg  we must consider the domain of g as well as the composed function, . fg  Because

2 2 fgxfgxx  

 we need to determine when 2 20. x 

Step1: Find the values of x that satisfy 2 20. x  2 22xx

Step2: The two numbers divide a number line into three regions.

Step3 Choose a test value to see if it satisfies the inequality, 2 20. x 

Interval Test Value Is 2 20 x  true or false?   ,2 2 2 220?

20 True     2,2 0 2 020?

20 False     2,  2 2 220?

The domain of fg  is



,22,.

20 True  

119.

111718fgfg

120. ()(3)(3)(3)990 fgfg 

121.

()(1)(1)(1)326 fgfg

122. (0)5 (0)undefined (0)0 ff gg

123. ()(2)[(2)](1)2 gfgfg

124. ()(3)[(3)](2)1 fgfgf

125. ()(2)[(2)](2)1 fgfgf

126. ()(3)[(3)](4)8 gfgfg 

127. Let x = number of yards. f (x) = 36x, where f ( x) is the number of inches. g(x) = 1760x, where g(x) is the number of yards. Then  ()()()1760(36) gfxgfxx  63,360. x

There are 63,360x inches in x miles

128. Use the definition for the perimeter of a rectangle. P= length + width + length + width ()226 Pxxxxxx  This is a linear function.

129. If 3 4 3 () Vrr   and if the radius is increased by 3 inches, then the amount of volume gained is given by 4433 33 ()(3)()(3). g VrVrVrrr 

130. (a) 2 Vrh   If d is the diameter of its top, then h = d and 2 . rd  So,     3 2 2 244 ()()(). dddVddd   

(b) 2 22 Srrh  

 2 222 2 2 222 23 222 ()22()ddd ddd Sddd 

Chapter 2 Test

1. (a) The domain of   3 fxx occurs when 0. x  In interval notation, this correlates to the interval in D,   0,. 

(b) The range of   3 fxx is all real numbers greater than or equal to 0. In interval notation, this correlates to the interval in D,   0,. 

(c) The domain of   2 3 fxx is all real numbers. In interval notation, this correlates to the interval in C,   ,. 

(d) The range of   2 3 fxx is all real numbers greater than or equal to 3. In interval notation, this correlates to the interval in B,   3,. 

(e) The domain of   3 3 fxx is all real numbers. In interval notation, this correlates to the interval in C, 

,. 

(f) The range of   3 3 fxx is all real numbers. In interval notation, this correlates to the interval in C,   ,. 

(g) The domain of   3 fxx is all real numbers. In interval notation, this correlates to the interval in C,   ,. 

(h) The range of   3 fxx is all real numbers greater than or equal to 0. In interval notation, this correlates to the interval in D,   0,. 

(i) The domain of 2 xy  is 0 x  because when you square any value of y, the outcome will be nonnegative. In interval notation, this correlates to the interval in D,   0,. 

(j) The range of 2 xy  is all real numbers. In interval notation, this correlates to the interval in C,   ,. 

2. Consider the points  2,1 and   3,4. 413 3(2)5 m 

3. We label the points A  2,1 and B   3,4. 22 22 (, )[3(2)](41) 5325934 dAB

4. The midpoint has coordinates 231415 , 2222,. 

5. Use the point-slope form with 3 11 5 (,)(2,1) and . xym

11 3 5 3 5 () 1[(2)] 1(2)513(2) 55365311 35113511 yymxx yx yxyx yxyx xyxy

6. Solve 3x – 5y = –11 for y. 3 11 55 3511 5311 xy yx yx

Therefore, the linear function is 3 11 55 ().fxx

7. (a) The center is at (0, 0) and the radius is 2, so the equation of the circle is 22 4 xy

(b) The center is at (1, 4) and the radius is 1, so the equation of the circle is 22 (1)(4)1 xy

8. 22 410130xyxy

Complete the square on x and y to write the equation in standard form:

The circle has center ( 2, 5) and radius 4.

9. (a) This is not the graph of a function because some vertical lines intersect it in more than one point. The domain of the relation is [0, 4]. The range is [–4, 4].

(b) This is the graph of a function because no vertical line intersects the graph in more than one point. The domain of the function is (–,–1)(–1,).   The range is (–,0)(0,).   As x is getting larger on the intervals  ,1 and   1,, the value of y is decreasing, so the function is decreasing on these intervals. (The function is never increasing or constant.)

10. Point A has coordinates (5, –3).

(a) The equation of a vertical line through A is x = 5.

(b) The equation of a horizontal line through A is y = –3.

11. The slope of the graph of 32yx is –3.

(a) A line parallel to the graph of 32yx has a slope of –3. Use the point-slope form with 11 (,)(2,3) and 3. xym 11() 33(2) 33639 yymxx yx yxyx

(b) A line perpendicular to the graph of 32yx has a slope of 1 3 because

12. (a)

(d)

13. To graph   21 fxx , we translate the graph of , yx  2 units to the right and 1 unit down.

14.  ()1fxx

To get y = 0, we need 011 x 

10. x  To get y = 1, we need 112 x  01. x  Follow this pattern to graph the step function.

15. 1 2 3 if 2 () 2 if 2 x fxxx      

For values of x with x < –2, we graph the horizontal line y = 3. For values of x with 2, x  we graph the line with a slope of 1 2 and a y-intercept of (0, 2). Two points on this line are (–2, 3) and (0, 2).

16. (a) Shift f (x), 2 units vertically upward.

(b) Shift f (x), 2 units horizontally to the left.

(d) Reflect f ( x), across the y-axis.

(e) Stretch f (x), vertically by a factor of 2.

17. Starting with , yx  we shift it to the left 2 units and stretch it vertically by a factor of 2. The graph is then reflected over the x-axis and then shifted down 3 units.

18. 22 323 xy

(a) Replace y with –y to obtain 2222 32()3323. xyxy

The result is the same as the original equation, so the graph is symmetric with respect to the x-axis.

(b) Replace x with –x to obtain 2222 3()23323. xyxy 

The result is the same as the original equation, so the graph is symmetric with respect to the y-axis.

(c) The graph is symmetric with respect to the x-axis and with respect to the y-axis, so it must also be symmetric with respect to the origin.

19. 2 ()232, ()21fxxxgxx 

(a)

 2 2 2 ()()()() 23221 23221 21 fgxfxgx xxx xxx xx 

(b) 2 ()232 () ()21 ffxxx x ggxx

(c) We must determine which values solve the equation 210. x  1 2 21021 xxx  Thus, 1 2 is excluded from the domain, and the domain is     11 22 ,,.  

(d) 2 ()232 fxxx

2 22 22 232 22332 242332 fxhxhxh xxhhxh xxhhxh 

22 2 22 2 2 ()() (242332) (232) 2423 32232 423 fxhfx xxhhxh xx xxhhx hxx xhhh    

 2 ()()423 (423) 423 fxhfxxhhh hh hxh h xh

fgfg 

(e) 2 ()(1)(1)(1) (21312)(211) (21312)(211) (232)(21) 1(1)0

(f) 2 ()(2)(2)(2) (22322)(221) (24322)(221) (862)(41) 4(3)12

210201gxxg

011. Therefore,

2 00 121312 21312 2321 fgfg f

For exercises 20 and 21,   1 fxx and 

27.gxx 20.

 27 (27)126 fgfgxfx xx 

The domain and range of g are (,)  , while the domain of f is [0,)  . We need to find the values of x which fit the domain of f: 2603 xx . So, the domain of fg  is [3,)  .

21. 

 1 217 gfgfxgx x 

 

The domain and range of g are (,)  , while the domain of f is [0,)  . We need to find the values of x which fit the domain of f: 101xx . So, the domain of gf  is [1,)  .

22. (a) C(x) = 3300 + 4.50x (b) R(x) = 10.50x (c) ()()() 10.50(33004.50) 6.003300

 

PxRxCx xx x

 (d) ()0 6.0033000 6.003300

550 Px x x x    

She must produce and sell 551 items before she earns a profit.

Chapter 2 Graphs and Functions

Section 2.1 Rectangular Coordinates and Graphs

Classroom Example 1 (page 184)

(a) (transportation, $12,153)

(b) (health care, $4917)

Classroom Example 2 (page 186)

25169194

Classroom Example 3 (page 186)

2 2 2 2 22 (,)5012 25934 (,)4032 162541 (,)4531 81485

The longest side has length 85

 222 ? 344185 344185

The triangle formed by the three points is not a right triangle.

Classroom Example 4 (page 187)

The distance between (2,5) and (0,3) is 205344822

Classroom Example 5 (page 188)

(a) The coordinates of M are 7(2)5139,,4 222

(b) Let (x, y) be the coordinates of Q. Use the midpoint formula to find the coordinates:

820 22,4,4 8 4880 2 20 420812 2 xy x xx y yy

The coordinates of Q are (0, 12).

Classroom Example 6 (page 188)

The year 2011 lies halfway between 2009 and 2013, so we must find the coordinates of the midpoint of the segment that has endpoints (2009, 124.0) and (2013, 137.4)

The estimate of $130.7 billion is $0.1 billion more than the actual amount.

Classroom Example 7 (page 189)

Choose any real number for x, substitute the value in the equation and then solve for y. Note that additional answers are possible.

(a) x y = 2x + 5

1 2(1)57 y 

The distance between (0,3) and (8,5) is 80536464 12882

The distance between (2,5) and (8,5) is 2855100100 200102 PQ QR PR

22 22 2 2

Because 2282102  , the points are collinear.

0 2(0)55 y 

3 2(3)51 y 

Three ordered pairs that are solutions are ( 1, 7), (0, 5), and (3, 1). Other answers are possible.

(b) y 3 1 xy

9 339182 x 

2 332111 x 

1 3 3 1100 x 

0 3 3 0111 x 

7 337182 x 

Ordered pairs that are solutions are ( 2, 9), ( 1, 2), (0, 1), (1, 0) and (2, 7).

Section 2.3 Functions

Classroom Example 1 (page 204)

4,0,3,1,3,1

M is a function because each distinct x value has exactly one y value.

2,3,3,2,4,5,5,4

N is a function because each distinct x value has exactly one y value.

4,3,0,6,2,8,4,3

P is not a function because there are two y-values for x = 4.

Classroom Example 2 (page 205)

(a) Domain: { 4, 1, 1, 3}

Range: { 2, 0, 2, 5}

The relation is a function.

(b) Domain: {1, 2, 3}

Range: {4, 5, 6, 7}

The relation is not a function because 2 maps to 5 and 6.

Range: {5}

The relation is a function.

Classroom Example 3 (page 206)

(a) Domain: { 2, 4}; range: {0, 3}

(b) Domain: (,)  ; range: (,) 

(d) Domain: (,)  ; range: (,4] 

Classroom Example 4 (page 207)

(a)

Not a function

Function

Classroom Example 5 (page 208)

(a)

25yx represents a function because y is always found by multiplying x by 2 and subtracting 5. Each value of x corresponds to just one value of y. x can be any real number, so the domain is all real numbers or (,) 

Because y is twice x, less 5, y also may be any real number, and so the range is also all real numbers, (,) 

(b) For any choice of x in the domain of 2 3 yx , there is exactly one corresponding value for y, so the equation defines a function. The function is defined for all values of x, so the domain is (,)  . The square of any number is always positive, so the range is [3,)  .

(c) For any choice of x in the domain of xy  , there are two possible values for y. Thus, the equation does not define a function. The domain is [0,)  while the range is (,) 

(d) By definition, y is a function of x if every value of x leads to exactly one value of y. Substituting a particular value of x, say 1, into yx  corresponds to many values of y. The ordered pairs (0, 2) (1, 1) (1, 0) (3, 1) and so on, all satisfy the inequality. This does not represent a function. Any number can be used for x or for y, so the domain and range of this relation are both all real numbers, (,) 

(e) For 3 2 y x   , we see that y can be found by dividing x + 2 into 3. This process produces one value of y for each value of x in the domain. The domain includes all real numbers except those that make the denominator equal to zero, namely x = 2. Therefore, the domain is (,2)(2,)   . Values of y can be negative or positive, but never zero. Therefore the range is (,0)(0,)  

Classroom Example 6 (page 210)

(a) 2 (3)(3)6(3)413 f 

(b) 2 ()64 frrr

Classroom Example 7 (page 210)

(a) 2 (1)2(1)97 f 

(b) (1)6 f 

(d) (1)0 f 

Classroom Example 8 (page 211)

(a) 2 2 2 ()23 (5)(5)2(5)312 ()23 fxxx f fttt

  

(b) 2 2362 3 2 ()2 3 216 (5)(5)233 2 ()2 3 xyyx fxx f ftt    

Classroom Example 9 (page 213)

The function is increasing on (,1),  decreasing on (–1, 1) and constant on (1,). 

Classroom Example 10 (page 213)

The example refers to the following figure.

(a) The water level is changing most rapidly from 0 to 25 hours.

(b) The water level starts to decrease after 50 hours.

Section 2.4 Linear Functions

Classroom Example 1 (page 220)

3 2 ()6fxx ; Use the intercepts to graph the function.

  3 2 (0)066: f  y-intercept

33 22 0664: xxx  x-intercept

Domain: (,),  range: (,) 

Section 2.4 Linear Functions 25

Classroom Example 2 (page 220)

()2fx  is a constant function. Its graph is a horizontal line with a y-intercept of 2.

Domain: (,),  range: {2}

Classroom Example 3 (page 221)

x = 5 is a vertical line intersecting the x-axis at (5, 0).

Domain:{5}, range: (,) 

Classroom Example 4 (page 221)

340; xy Use the intercepts.

  3040400:-intercept yyyy

  3400300:-intercept xxxx 

The graph has just one intercept. Choose an additional value, say 4, for x



  34401240 4123 yy yy



Graph the line through (0, 0) and (4, 3).

Domain: (,),  range: (,) 

Classroom Example 5 (page 223)

(a) 4(6)105 2242 m 

(b) 880 0 5(3)8 m 

Classroom Example 6 (page 224)

2x 5y = 10

Solve the equation for y 2510

The slope is 2 , 5 the coefficient of x.

Classroom Example 7 (page 224)

First locate the point ( 2, 3). Because the slope is 4 3 , a change of 3 units horizontally (3 units to the right) produces a change of 4 units vertically (4 units up). This gives a second point, (1, 1), which can be used to complete the graph.

Classroom Example 8 (page 225)

The average rate of change per year is 86589547889 296.33 million 201320103

The graph confirms that the line through the ordered pairs fall from left to right, and therefore has negative slope. Thus, the amount spent by the federal government on R&D for general science decreased by an average of $296.33 million (or $296,330,000) each year from 2010 to 2013.

Classroom Example 9 (page 226)

(a) ()1202400Cxx (b) ()150 Rxx  (c) ()()() 150(1202400) 302400 PxRxCx xx x

(d) ()0302400080 Pxxx 

At least 81 items must be sold to make a profit.

Section 2.5 Equations of Lines and Linear Models

Classroom Example 1 (page 234) (5)2(3) 526 21 yx yx yx

Classroom Example 2 (page 234)

First find the slope: 3(1)4 459 m 

Now use either point for 11 (,) xy : 4 9 3[(4)] 9(3)4(4) 927416 9411 4911 yx yx yx yx xy

Classroom Example 3 (page 235)

Write the equation in slope-intercept form: 3 4 341243123 xyyxyx

The slope is 3 4 , and the y-intercept is (0, 3).

Classroom Example 4 (page 236)

First find the slope: 421 222 m 

Now, substitute 1 2 for m and the coordinates of one of the points (say, (2, 2)) for x and y into the slopeintercept form y = mx + b, then solve for b: 1 2 223bb . The equation is 1 2 3 yx

Classroom Example 5 (page 236)

The example refers to the following figure:

(a) The line rises 5 units each time the x-value increases by 2 units. So the slope is 5 2 . The y-intercept is (0, 5), and the x-intercept is ( 2, 0).

(b) An equation defining f is 5 2 ()5fxx

Classroom Example 6 (page 238)

(a) Rewrite the equation 325 xy in slopeintercept form to find the slope: 35 22 325xyyx  The slope is 3 2

The line parallel to the equation also has slope 3 2 . An equation of the line through (2, 4) that is parallel to 325 xy is 33 22 (4)(2)43yxyx  3 2 7 yx or 3x 2y = 14.

(b) The line perpendicular to the equation has slope 2 3 . An equation of the line through (2, 4) that is perpendicular to 325 xy is 224 (4)(2)4333 yxyx  8 2 33yx or 2x +3y = 8.

Classroom Example 7 (page 240)

(a) First find the slope: 77036695 504 31 m 

Now use either point for 11 (,) xy : 6695504(1) 6695504504 5046191 yx yx yx

(b) The year 2015 is represented by x = 6. 504(6)61919215 y 

According to the model, average tuition and fees for 4-year colleges in 2015 will be about $9215.

Classroom Example 8 (page 242)

Write the equation as an equivalent equation with 0 on one side.

32(5)238 xxx  32(5)2380 xxx 

Now graph the equation to find the x-intercept.

The solution set is { 4}.

Section 2.6 Graphs of Basic Functions

Classroom Example 1 (page 249)

(a) The function is continuous over (,0)(0,)  

(b) The function is continuous over its entire domain (,). 

Classroom Example 2 (page 252)

(a) Graph each interval of the domain separately. If x < 1, the graph of f(x) = 2x + 4 has an endpoint at (1, 6), which is not included as part of the graph. To find another point on this part of the graph, choose x = 0, so y = 4. Draw the ray starting at (1, 6) and extending through (0, 4). Graph the function for x ≥ 1, f(x) = 4 x similarly. This part of the graph has an endpoint at (1, 3), which is included as part of the graph. Find another point, say (4, 0), and draw the ray starting at (1, 3) which extends through (4, 0).

(b) Graph each interval of the domain separately. If x ≤ 0, the graph of f(x) = x 2 has an endpoint at (0, 2), which is included as part of the graph. To find another point on this part of the graph, choose x = 2, so y = 0. Draw the ray starting at (0, 2) and extending through ( 2, 0). Graph the function for x > 0, 2 ()2fxx similarly. This part of the graph has an endpoint at (0, 2), which is not included as part of the graph. Find another point, say (2, 2), and draw the curve starting at (0, 2) which extends through (2, 2). Note that the two endpoints coincide, so (0, 2) is included as part of the graph.

Classroom Example 3 (page 254)

Create a table of sample ordered pairs:

Classroom Example 4 (page 254)

For x in the interval (0, 2], y = 20. For x in (2, 3], y = 20 + 2 = 22. For x in (3, 4], y = 22 + 2 = 24. For x in (4, 5], y = 24 + 2 = 26. For x in (5, 6], y = 26 + 2 = 28.

Section 2.7 Graphing Techniques

Classroom Example 1 (page 260)

Use this table of values for parts (a) (c)

Classroom Example 2 (page 262)

Use this table of values for parts (a) and (b)

Classroom Example 3 (page 263)

(a) xy 

Replace x with –x to obtain xy . The result is not the same as the original equation, so the graph is not symmetric with respect to the y-axis. Replace y with –y to obtain xyxy  The result is the same as the original equation, so the graph is symmetric with respect to the xaxis. The graph is symmetric with respect to the x-axis only.

(b) 3 yx

Replace x with –x to obtain 3 yx 3 yx . The result is the same as the original equation, so the graph is symmetric with respect to the y-axis. Replace y with –y to obtain 3 yx  . The result is not the same as the original equation, so the graph is not symmetric with respect to the x-axis. Therefore, the graph is symmetric with respect to the y-axis only.

Replace x with –x to obtain 2()6 xy 

26 xy  . The result is not the same as the original equation, so the graph is not symmetric with respect to the y-axis. Replace y with –y to obtain 2()6 xy

26 xy . The result is not the same as the original equation, so the graph is not symmetric with respect to the x-axis. Therefore, the graph is not symmetric with respect to either axis.

(d) 22 25 xy

Replace x with –x to obtain 22 ()25 xy  22 25 xy . The result is the same as the original equation, so the graph is symmetric with respect to the y-axis. Replace y with –y to obtain 22()25xy 22 25 xy . The result is the same as the original equation, so the graph is symmetric with respect to the x-axis. Therefore, the graph is symmetric with respect to both axes. Note that the graph is a circle of radius 5, centered at the origin.

Classroom Example 4 (page 265)

(a) 3 2 yx 

Replace x with –x and y with –y to obtain 333 ()2()22 yxyxyx  . The result is the same as the original equation, so the graph is symmetric with respect to the origin.

(b) 2 2 yx 

Replace x with –x and y with –y to obtain 222 ()2()22 yxyxyx  The result is not the same as the original equation, so the graph is not symmetric with respect to the origin.

Classroom Example 5 (page 266)

(a) 53 ()23 gxxxx 

Replace x with –x to obtain 53 53 53 ()()2()3() 23 (23)() gxxxx xxx xxxgx

   g(x) is an odd function.

(b) 2 ()23hxx

Replace x with –x to obtain 22 ()2()323() hxxxhx  h(x) is an even function.

Replace x with –x to obtain 2 2 ()()6()9 69() and () kxxx xxkxkx 

k(x) is neither even nor odd.

Classroom Example 6 (page 267)

Compare a table of values for 2 () gxx  with 2 ()2fxx . The graph of f(x) is the same as the graph of g(x) translated 2 units up.

x 2 () gxx  2 ()2fxx

Classroom Example 7 (page 268)

Compare a table of values for 2 () gxx  with 2 ()(4)fxx . The graph of f(x) is the same as the graph of g(x) translated 4 units left. x 2 () gxx  2 ()(4)fxx

Classroom Example 8 (page 269)

(a) 2 ()(1)4fxx

This is the graph of 2 () gxx  , translated one unit to the right, reflected across the x-axis, and then translated four units up.

x 2 () gxx  2 ()(1)4fxx 2 4 5

(continuedonnextpage)

continued)

(b) ()23fxx

This is the graph of () gxx  , translated three units to the left, reflected across the x-axis, and then stretched vertically by a factor of two. x () gxx  ()23fxx

Classroom Example 9 (page 270)

The graphs in the exercises are based on the following graph.

(a) ()()2gxfx

This is the graph of f(x) translated two units down.

This is the graph of () gxx  , translated two units to the left, shrunk vertically by a factor of 2, and then translated 3 units down.

(b) ()(2)hxfx

This is the graph of f(x) translated two units right.

This is the graph of f(x) translated one unit left, and then translated two units up.

(d) ()() Fxfx 

This is the graph of f(x) reflected across the y-axis.

Section 2.8 Function Operations and Composition

Classroom Example 1 (page 278)

For parts (a) (d), ()34fxx and 2 ()21gxx

(a) (0)3(0)44 f  and 2 (0)2(0)11 g  , so ()(0)415 fg

(b) (4)3(4)48 f  and 2 (4)2(4)131 g  , so ()(4)83123 fg

(d) (3)3(3)45 f  and 2 (3)2(3)117 g  , so 5 (3) 17 f g

Classroom Example 2 (page 279)

For parts (a) (e), 2 ()3 fxxx  and ()45gxx

(a) 22 ()()(3)(45)5

(d) 2 3 () 45 fxx x gx

(e) The domains of f and g are both (,)  . So, the domains of f + g, fg, and fg are the intersection of the domains of f and g, (,)  . The domain of f g includes those real numbers in the intersection of the domains of f and g for which ()450gxx 5 4 x  . So the domain of f g is

Classroom Example 3 (page 280)

From the figure, we have f(1) = 3 and g(1) = 1, so (f + g)(1) = 3 + 1 = 4. f(0) = 4 and g(0) = 0, so (fg)(0) = 4 0 = 4.

f( 1) = 3 and g( 1) = 1, so (fg)( 1) = (3)(1) = 3

f( 2) = 0 and g( 2) = 2, so 

0 2 (2)0 f g

xf(x) g(x)

From the table, we have f(1) = 1 and g(1) = 6, so (f + g)(1) = 1 + 6 = 7. f(0) = 1 and g(0) = 4, so (fg)(1) = 1 4 = 5.

f( 1) = 3 and g( 1) = 2, so (fg)( 1) = ( 3)(2) = 6

f( 2) = 5 and g( 2) = 0, so

  5 0 (2) f g  f g is undefined.

From the formulas, we have (1)3(1)47 f  and (1)11 g  , so

(f + g)(1) = 7 + ( 1) = 6. (0)3(0)44 f  and (0)00 g  , so

(fg)(1) = 4 0 = 4. (1)3(1)41 f  and (1)11 g  , so (fg)( 1) =(1)( 1) = 1. (2)3(2)42 f  and (2)22 g  , so 2 (2)1. 2 f g

Classroom Example 4 (page 281)

Step1: Find ()fxh  : 2 22 22 ()3()2()4 3(2)224 363224 fxhxhxh xxhhxh xxhhxh

Step2: Find ()() fxhfx  : 222 2 ()() (363224)(324) 632 fxhfx xxhhxhxx xhhh

Step3: Find the difference quotient: 2 ()()632 632fxhfxxhhhxh hh 

Classroom Example 5 (page 283)

For parts (a) and (b), ()4fxx and 2 ()gx x 

(a) First find g(2): 2 (2)1 2 g  . Now find ()(2)((2))(1)145 fgfgf 

(b) First find f(5): (5)5493 f  . Now find 2 3 ()(5)((5))(3) gfgfg 

The screens show how a graphing calculator evaluates the expressions in this classroom example.

Classroom Example 6 (page 283)

For parts (a) and (b), ()1fxx and ()25gxx

(a) ()()(())(25)124 fgxfgxxx  

The domain and range of g are both (,) 

However, the domain of f is [1,)  . Therefore, ()gx must be greater than or equal to 1: 2512 xx . So, the domain of fg  is [2,)  .

(b) ()()(())215 gfxgfxx 

The domain of f is [1,)  , while the range of f is [0,)  . The domain of g is (,)  . Therefore, the domain of () gf  is restricted to that portion of the domain of g that intersects with the domain of f, that is [1,) 

Classroom Example 7 (page 284)

For parts (a) and (b), 5 () 4 fx x   and 2 ()gx x 

(a)  55 ()()(()) 2424 fgxfgxx xx   

The domain and range of g are both all real numbers except 0. The domain of f is all real numbers except 4. Therefore, the expression for ()gx cannot equal 4. So, 21 4 2 x x  . So, the domain of fg  is the set of all real numbers except for 1 2 , and 0. This is written       11 22 ,,00,  

(b) 228 ()()(()) 5(4)5 gfxgfxx x    

The domain of f is all real numbers except 4, while the range of f is all real numbers except 0. The domain and range of g are both all real numbers except 0, which is not in the range of f. So, the domain of gf  is the set of all real numbers except for 4. This is written

Classroom Example 8 (page 285)

()25fxx and 2 ()3 gxxx  2 2 2 22 2

()()(3)2(3)5 625 gfxgxxx xxx xx fgxfxxxx

()()(25)3(25)(25) 3(42025)25 125870

In general, 22 125870625, xxxx  so

()()()() gfxfgx   .

Classroom Example 9 (page 286)

2 ()()4(32)5(32)8 fgxxx

Answers may vary. Sample answer: 2 ()458 fxxx and ()32gxx

Chapter 2 Graphs and Functions

2.1 Rectangular Coordinates and Graphs

■ Ordered Pairs ■ The Rectangular Coordinate System ■ The Distance Formula

■ The Midpoint Formula ■ Equations in Two Variables

Key Terms: ordered pair, origin, x-axis, y-axis, rectangular (Cartesian) coordinate system, coordinate plane (xy-plane), quadrants, coordinates, conditional statement, collinear, graph of an equation, x-intercept, y-intercept

Ordered Pairs

CLASSROOM EXAMPLE 1 Writing Ordered Pairs

Use the table to write ordered pairs to express the relationship between each category and the amount spent on it.

(a) transportation

(b) health care

The

Rectangular Coordinate System

The x-axis and y-axis together make up a ____________ ____________system, or ____________ ____________ system. The plane into which the coordinate system is introduced is the ____________ ____________, or ____________ ____________.

The x-axis and y-axis divide the plane into four regions, or ____________.

The Distance Formula

Distance Formula

Suppose that ( ) 11 , Pxy and 22 (,)Rxy are two points in a coordinate plane. The distance between P and R , written (,)dPR , is given by the following formula.

( ) , dPR = ________________________________.

The distance between two points in a coordinate plane is the ____________ ____________ of the ____________ of the square of the difference between their x-coordinates and the square of the difference between their y-coordinates.

CLASSROOM EXAMPLE 2 Using the Distance Formula

Find the distance between () 3,5 P and (2,8). Q

If the sides a, b, and c of a triangle satisfy 222 abc += then the triangle is a ____________ ____________ with ____________ having lengths a and b and ____________ having length c.

CLASSROOM EXAMPLE 3 Applying the Distance Formula

Determine whether the points R(0, 2), S(5, 1), and T( 4, 3) are the vertices of a right triangle.

Section 2.1 Rectangular Coordinates and Graphs 2-3

CLASSROOM EXAMPLE 4 Applying the Distance Formula

Determine whether the points ()()() 2, 5, 0,3, and 8,5 PQR are collinear.

The Midpoint Formula

MIDPOINT FORMULA

The coordinates of the midpoint M of the line segment with endpoints 11 (,)Pxy and 22 (,)Qxy are given by the following.

That is, the x-coordinate of the midpoint of a line segment is the _______________ of the x-coordinates of the segment’s endpoints, and the y-coordinate is the _______________ of the y-coordinates of the segment’s endpoints.

CLASSROOM EXAMPLE 5 Using the Midpoint Formula

Use the midpoint formula to do each of the following.

(a) Find the coordinates of the midpoint M of the line segment with endpoints ( 7, 5) and ( 2, 13).

(b) Find the coordinates of the other endpoint Q of a line segment with one endpoint (8,20) P and midpoint (4,4). M

CLASSROOM EXAMPLE 6 Applying the Midpoint Formula

Total revenue of full-service restaurants in the United States increased from $124.0 billion in 2009 to $137.4 billion in 2013. Use the midpoint formula to estimate the total revenue for 2011, and compare this to the actual figure of $130.6 billion. (Source: National Restaurant Association.)

Equations in Two Variables

CLASSROOM EXAMPLE 7 Finding Ordered-Pair Solutions of Equations

For each equation, find at least three ordered pairs that are solutions.

(a) 25yx=−+

(b) 3 1 xy=+

Graphing an Equation by Point Plotting

Step 1 Find the intercepts.

Step 2 Find as many additional ordered pairs as needed.

Step 3 Plot the ordered pairs from Steps 1 and 2.

Step 4 Join the points from Step 3 with a smooth line or curve.

Chapter 2 Graphs and Functions

CLASSROOM EXAMPLE 8 GRAPHING EQUATIONS

Graph each of the equations here, from Classroom Example 7.

(a) 25yx=−+

(b) 3 1 xy=+

Section 2.2 Circles 2-7

2.2 Circles

■ Center-Radius Form ■ General Form ■ An Application

Key Terms: circle, radius, center of a circle

Center-Radius Form

By definition, a circle is the set of all points in a plane that lie a given distance from a given point. The given distance is called the ____________________ and the given point is the

Center-Radius Form of the Equation of a Circle

A circle with center (h, k) and radius r has the equation

222 ()(), xh y kr−+−=

which is the center-radius form of the equation of a circle. A circle with center (0, 0) and radius r has the following equation.

222 x y r +=

CLASSROOM EXAMPLE 1 Finding the Center-Radius Form

Find the center-radius form of the equation of each circle described. (a) center (1, 2), radius 3 (b) center (0, 0), radius 2

CLASSROOM EXAMPLE 2 Graphing Circles

Graph each circle discussed in Classroom Example 1.

(a) ()() 22 129xy−++= (b) 22 4 xy+=

General

Form

General Form of the Equation of a Circle

For some real numbers D, E, and F, the equation 22

can have a graph that is a circle or a point, or is nonexistent.

CLASSROOM EXAMPLE 3 Finding the Center and Radius by Completing the Square

Show that 2248440 xxyy++−−= has a circle as its graph. Find the center and radius.

CLASSROOM EXAMPLE 4 Finding the Center and Radius by Completing the Square

Show that 22 222645 xyxy++−= has a circle as its graph. Find the center and radius.

Section 2.2 Circles 2-9

CLASSROOM EXAMPLE 5 Determining Whether a Graph is a Point or Nonexistent

The graph of the equation 2262120 xxyy−+++= is either a point or is nonexistent. Which is it?

An Application

CLASSROOM EXAMPLE 6 Locating the Epicenter of an Earthquake

If three receiving stations at (1, 4), ( 6, 0), and (5, 2) record distances to an earthquake epicenter of 4 units, 5 units, and 10 units, respectively, show algebraically that the epicenter lies at ( 3, 4).

2-10 Chapter 2 Graphs and Functions

2.3 Functions

■ Relations and Functions ■ Domain and Range

■ Determining Whether Relations are Functions ■ Function Notation

■ Increasing, Decreasing, and Constant Functions

Key Terms: dependent variable, independent variable, relation, function, input, output, input-output (function machine), domain, range, increasing function, decreasing function, constant function

Relations and Functions

One quantity can sometimes be described in terms of another.

• The letter grade a student receives in a mathematics course depends on a ___________________.

• The amount paid for gas at a gas station depends on the number of _________________.

Relation and Function

A relation is a set of ___________________ _______________. A function is a relation in which, for each distinct value of the first component of the ordered pairs, there is ____________________ __________ value of the second component.

If the value of the second component y depends on the value of the first component x, then y is the ___________________ _______________ and x is the ___________________ _______________.

CLASSROOM EXAMPLE 1 Deciding Whether Relations Define Functions

Decide whether each relation defines a function.

4,0, 3,1, 3, 1 2,3, 3, 2, 4,5, 5,4 4, 3, 0,6, 2, 8,4,3

In a function, no two ordered pairs can have the same _________________ and different

In a function, there is exactly one value of the variable, the second component, for each value of the _________________ variable, the first component.

Section 2.3 Functions 2-11

Domain and Range

For every relation consisting of ordered pairs (x, y), there are two important sets of elements.

• The set of all values of the independent variable (x) is the _______________.

• The set of all values of the dependent variable (y) is the _________________.

CLASSROOM EXAMPLE 2 Finding Domains and Ranges of Relations

Give the domain and range of each relation. Tell whether the relation defines a function.

(a) {()()()()} 4,2, 1,0, 1,2, 3,5

(b)

CLASSROOM EXAMPLE 3 Finding Domains and Ranges from Graphs

Give the domain and range of each relation.

(a) (b)

(c)

(d)

Chapter 2 Graphs and Functions

Agreement on Domain

Unless specified otherwise, the domain of a relation is assumed to be all real numbers that produce real numbers when substituted for the independent variable.

In general, the domain of a function defined by an algebraic expression is ____________________________________, except those numbers that lead to ____________________________________or

Determining Whether Relations Are Functions

Vertical Line Test

If every vertical line intersects the graph of a relation in no more than __________ __________, then the relation is a function.

CLASSROOM EXAMPLE 4 Using the Vertical Line Test

Use the vertical line test to determine whether each relation graphed in Classroom Example 3 is a function.

CLASSROOM EXAMPLE 5 Identifying Functions, Domains, and Ranges

Decide whether each relation defines y as a function of x, and give the domain and range.

(a) (b)

(c) (d)

Section 2.3 Functions 2-13

Variations of the Definition of Function

1. A function is a relation in which, for each distinct value of the first component of the ordered pairs, there is exactly one value of the second component.

2. A function is a set of ordered pairs in which no first component is repeated.

3. A function is a rule or correspondence that assigns exactly one range value to each distinct domain value.

Function Notation

When a function f is defined with a rule or an equation using x and y for the independent and dependent variables, we say, “yisafunctionofx” to emphasize that ydependsonx. We use the notation

()yfx =

called function notation, to express this and read ()fx as “f of x,” or “f at x.”

Note that () f x is just another name for the _____________ _____________ y.

The symbol () f x does not indicate “f times x,” but represents the _____________ associated with the indicated _____________ _____________.

CLASSROOM EXAMPLE 6 Using Function Notation

Let () 2 64fxxx=−−+ and () 31.gxx=+ Find each of the following.

(a) (3) f

(b) ()fr

2-14 Chapter 2 Graphs and Functions

CLASSROOM EXAMPLE 7 Using Function Notation

For each function , find ( ) 1. f (a) () 2 29fxx=−

{()()()()} 4,0,1,6,0,8,2,2 f =−−

Finding an Expression for () f x

Consider an equation involving x and y. Assume that y can be expressed as a function f of x.

To find an expression for ( ) , fx use the following steps.

Step 1 Solve the equation for ___________.

Step 2 Replace ___________ with ___________.

CLASSROOM EXAMPLE 8 Writing Equations Using Function Notation

Assume that y is a function f of x. Rewrite each equation using function notation. Then find (5) f and ().ft (a) 2 23yxx=+− (b) 236 xy−=

Section 2.3 Functions 2-15

Increasing, Decreasing, and Constant Functions

Suppose that a function f is defined over an open interval I and 1x and 2x are in I.

(a) f increases over I if, whenever ( ) ( ) 1212,. xxfxfx <<

(b) f decreases over I if, whenever ( ) ( ) 1212,. xxfxfx <>

To decide whether a function is increasing, decreasing, or constant over an interval, ask yourself, ______________________________________________________________

Our definition of increasing, decreasing, and constant function behavior applies to open intervals of the ___________, not to ___________ ___________.

CLASSROOM EXAMPLE 9 Determining Intervals over Which a Function Is

Increasing, Decreasing, or Constant

The figure shows the graph of a function. Determine the largest open intervals of the domain over which the function is increasing, decreasing, or constant.

Chapter 2 Graphs and Functions

CLASSROOM EXAMPLE 10 Interpreting a Graph

The figure shows the relationship between the number of gallons, (),gt of water in a small swimming pool and time in hours, t. By looking at this graph of the function, we can answer questions about the water level in the pool at various times. For example, at time 0 the pool is empty. The water level then increases, stays constant for a while, decreases, then becomes constant again. Use the graph to respond to the following.

(a) Over what period of time is the water level changing most rapidly?

(b) After how many hours does the water level start to decrease?

Section 2.4 Linear Functions 2-17

2.4 Linear Functions

■ Basic Concepts of Linear Functions ■ Standard Form Ax + By = C ■ Slope ■ Average Rate of Change ■ Linear Models

Key Terms: linear function, constant function, standard form, relatively prime, slope, average rate of change, mathematical modeling, linear cost function, cost, fixed cost, revenue function, profit function

Basic Concepts of Linear Functions

Linear Function

A function f is a linear function if, for real numbers a and b, ( ) f xaxb. =+

If 0, a ≠ the domain and range of a linear function are both ( ) ,. −∞∞

CLASSROOM EXAMPLE 1 Graphing a Linear Function Using Intercepts

Graph () 3 6. 2 fxx=+ Give the domain and range.

2-18 Chapter 2 Graphs and Functions

CLASSROOM EXAMPLE 2 Graphing a Horizontal Line

Graph ( ) 2. fx = Give the domain and range.

CLASSROOM EXAMPLE 3 Graphing a Vertical Line

Graph 5. x = Give the domain and range of this relation.

Section 2.4 Linear Functions 2-19

Standard Form Ax + By = C

CLASSROOM EXAMPLE 4 Graphing Ax + By = C (C = 0)

Graph 340. xy+= Give the domain and range.

Slope

The slope m of the line through the points 11 (,) xy and 22 (,) xy is given by the following.

That is, the slope of a line is the change in __________ divided by the corresponding change in __________, where the change in __________ is not 0.

When using the slope formula, it makes no difference which point is () 11 , x y or ()22,. x y However, be consistent. Be sure to write the difference of the __________ in the numerator and the difference of the __________ in the denominator.

Undefined Slope

The slope of a vertical line is _______________________.

2-20 Chapter 2 Graphs and Functions

CLASSROOM EXAMPLE 5 Finding Slopes With The Slope Formula

Find the slope of the line through the given points.

(a) ()2,4, (2, 6) (b) ()3,8, (5,8) (c) ()() 4,10, 4,10

Slope Equal to Zero

The slope of a horizontal line is _____.

Slope is the same no matter which pair of distinct points on the line are used to find it.

CLASSROOM EXAMPLE 6 Finding Slope from an Equation

Find the slope of the line 2510. xy−=

CLASSROOM EXAMPLE 7 Graphing a Line Using a Point and the Slope

Graph the line passing through the point (2,3) and having slope 4 3 .

• Alinewithapositivesloperisesfrom_________to_________.

• Alinewithanegativeslopefallsfrom_________to_________.

• Alinewith________________________neitherrisesnorfalls.

• Theslopeofa________________________isundefined.

Slope gives the average rate of change in per unit of change in , where the value of y depends on the value of x.

Average Rate of Change

CLASSROOM EXAMPLE 8 Interpreting Slope as Average Rate of Change

In 2010, the federal government spent $9547 million on research and development for general science. In 2013, $8658 million was spent. Assume a linear relationship, and find the average rate of change in the amount of money spent on R&D per year. Graph as a line segment, and interpret the result. (Source: National Science Foundation.)

2-22 Chapter 2 Graphs and Functions

Linear Models

A linear cost function has the form ___________________, where x represents the number of items produced, m represents the ___________________ per item and b represents the ___________________.

The revenue function for selling a product depends on the price per item p and the number of items sold x. It is given by the function ___________________.

The profit function is ___________________.

CLASSROOM EXAMPLE 9 Writing Linear Cost, Revenue, and Profit Functions

Assume that the cost to produce an item is a linear function and all items produced are sold. The fixed cost is $2400, the variable cost per item is $120, and the item sells for $150. Write linear functions to model each of the following.

(a) cost (b) revenue (c) profit

(d) How many items must be sold for the company to make a profit?

Section 2.5 Equations of Lines and Linear Models 2-23

2.5 Equations of Lines and Linear Models

■ Point-Slope Form ■ Slope-Intercept Form ■ Vertical and Horizontal Lines

■ Parallel and Perpendicular Lines ■ Modeling Data

■ Graphical Solution of Linear Equations in One Variable

Key Terms: point-slope form, slope-intercept form, negative reciprocals, scatter diagram linear regression, zero (of a function)

Point-Slope Form

The point-slope form of the equation of the line with slope m passing through the point ( ) 11 , xy is given as follows.

CLASSROOM EXAMPLE 1 Using the Point-Slope Form (Given a Point and the Slope)

Write an equation of the line through the point () 3,5 having slope 2.

CLASSROOM EXAMPLE 2 Using the Point-Slope Form (Given Two Points)

Write an equation of the line through the points () 4,3 and ()5,1. Write the result in standard form . AxByC +=

2-24 Chapter 2 Graphs and Functions

Slope-Intercept Form

The slope-intercept form of the equation of the line with slope m and y-intercept (0, b) is given as follows. y mxb=+

CLASSROOM EXAMPLE 3 Finding Slope and y-Intercept from an Equation of a Line

Find the slope and y-intercept of the line with equation 3412. xy−=

CLASSROOM EXAMPLE 4 Using the Slope-Intercept Form (Given Two Points)

Write an equation of the line through the points ( 2, 4) and (2, 2). Then graph the line using the slope-intercept form.

Section 2.5 Equations of Lines and Linear Models 2-25

CLASSROOM EXAMPLE 5 Finding an Equation from a Graph

Use the graph of the linear function f shown in the figure to complete the following.

(a) Identify the slope, y-intercept, and x-intercept.

(b) Write an equation that defines f

Vertical and Horizontal Lines

Equations of Vertical and Horizontal Lines

An equation of the vertical line through the point (, ) ab is _____________.

An equation of the horizontal line through the point (, ) ab is _____________.

Parallel and Perpendicular Lines

Parallel Lines

Two distinct nonvertical lines are parallel if and only if they have the same ________.

Perpendicular Lines

Two lines, neither of which is vertical, are perpendicular if and only if their slopes have a product of _______. Thus, the slopes of perpendicular lines, neither of which is vertical, are

CLASSROOM EXAMPLE 6 Finding Equations of Parallel and Perpendicular Lines

Write an equation in both slope-intercept and standard form of the line that passes through the point (2, –4) and satisfies the given condition.

(a) parallel to the line 254 xy+=

2-26 Chapter 2 Graphs and Functions

(b) perpendicular to the line 254 xy+=

Summary of Forms of Linear Equations

Equation

=+ y mxb

Description When to Use

Slope-Intercept Form

Slope is _____.

y-intercept is _____.

11()−=−yy mxx Point-Slope Form

Slope is _____.

Line passes through _______.

+=AxBy C Standard Form

(If the coefficients and constant are rational, then A, B, and C are expressed as relatively prime integers, with A ≥ 0)

Slope is _____ ( ) 0. B ≠

x-intercept is _____ ( ) 0. A ≠

y-intercept is _____ ( ) 0. B ≠

= yb Horizontal Line

Slope is _____.

y-intercept is _______.

= x a Vertical Line

Slope is ________________.

x-intercept is _______.

The slope and y-intercept can be easily identified and used to quickly graph the equation. This form can also be used to find the equation of a line given a point and the slope.

This form is ideal for finding the equation of a line if the slope and a point on the line or two points on the line are known.

The x- and y-intercepts can be found quickly and used to graph the equation. The slope must be calculated.

If the graph intersects only the y-axis, then y is the only variable in the equation.

If the graph intersects only the x-axis, then x is the only variable in the equation.

Modeling Data

CLASSROOM EXAMPLE 7 Finding an Equation of a Line That Models Data

Average annual tuition and fees for in-state students at public four-year colleges are shown in the table for selected years and graphed as ordered pairs of points in the figure, where x = 0 represents 2009, x = 1 represents 2010, and so on, and y represents the cost in dollars. This graph of ordered pairs of data is called a scatter diagram.

(a) Find an equation that models the data using the data points (1, 6695) and (3, 7703).

(b) Use the equation from part (a) to predict the cost of tuition and fees at public four-year colleges in 2015.

Guidelines for Modeling

Step 1 Make a __________ __________ of the data.

Step 2 Find an __________ that models the data. For a line, this involves selecting __________ data points and finding the __________ of the __________ through them.

2-28 Chapter 2 Graphs and Functions

Graphical Solution of Linear Equations in One Variable

CLASSROOM EXAMPLE 8 Solving an Equation with a Graphing Calculator

Use a graphing calculator to solve () 325238. xxx −+−=+

Section 2.6 Graphs of Basic Functions 2-29

2.6 Graphs of Basic Functions

■ Continuity ■ The Identity, Squaring, and Cubing Functions

■ The Square Root and Cube Root Functions ■ The Absolute Value Function

■ Piecewise-Defined Functions ■ The Relation 2 x y =

Key Terms: continuous function, parabola, vertex, piecewise-defined function, step function

Continuity

Continuity (Informal Definition)

A function is continuous over an interval of its domain if its hand-drawn graph over that interval can be sketched without lifting the ____________ from the ____________.

If a function is not continuous at a ____________, then it has a ____________there.

CLASSROOM EXAMPLE 1 Determining Intervals of Continuity

Describe the intervals of continuity for each function.

The Identity, Squaring, and Cubing Functions

Identity Function () fxx =

Domain: __________

Range: __________

• ( ) fxx = is __________ on its entire domain.

• It is __________ on its entire domain.

(a)

(b)

2-30 Chapter 2 Graphs and Functions

Squaring Function () 2 fxx =

Domain: __________

Range: __________

• ( ) 2 fxx = decreases on the open interval __________ and increases on the open interval __________.

• It is __________ on its entire domain.

Cubing Function () 3 f xx =

Domain: __________

Range: __________

• ( ) 3 fxx = __________ on its entire domain.

• It is __________ on its entire domain.

The Square Root and Cube Root Functions

Square Root Function () f xx =

Domain: __________

Range: __________

• ( ) fxx = __________ on the open interval __________.

• It is __________ on its entire domain.

Cube Root Function () 3 f xx =

Domain: __________

Range: __________

• ( ) 3 fxx = __________ on its entire domain.

• It is __________ on its entire domain __________.

2-32 Chapter 2 Graphs and Functions

The Absolute Value Function

The absolute value function is defined as follows.

Absolute Value Function () fxx =

Domain: __________

Range: __________

• ( ) fxx = decreases on the open interval __________ and increases on the open interval __________.

• It is __________ on its entire domain __________.

Piecewise-Defined Functions

CLASSROOM EXAMPLE 2 Graphing Piecewise-Defined Functions

Graph each function.

Section 2.6 Graphs of Basic Functions 2-33

The greatest integer function, ( )   fxx = , pairs every real number x with the greatest integer __________ ____________ or _____________ ______________ x .

Greatest Integer Function () fxx =

Domain: __________

Range: ________________________________________________________________

• ( )   fxx = is constant on the open intervals …, _____________________________, …

• It is __________ at all integer values in its domain __________.

Chapter 2 Graphs and Functions

CLASSROOM EXAMPLE 3 Graphing a Greatest Integer Function

=−

Graph () 1 2 3

CLASSROOM EXAMPLE 4 Applying a Greatest Integer Function

An express mail company charges $20 for a package weighing up to 2 lb. For each additional pound or fraction of a pound, there is an additional charge of $2. Let ( ) yCx = represent the cost to send a package weighing x pounds. Graph ()yCx = for x in the interval (0, 6] .

Section 2.6 Graphs of Basic Functions 2-35

The Relation xy =

Recall that a function is a relation where every domain value is paired with ________ ________ ________ ________ range value.

2 x y =

Domain: __________

Range: ____________

2.7 Graphing Techniques

■ Stretching and Shrinking ■ Reflecting ■ Symmetry ■ Even and Odd Functions

■ Translations

Key Terms: symmetry, even function, odd function, vertical translation, horizontal translation

Stretching and Shrinking

CLASSROOM EXAMPLE 1 Stretching or Shrinking Graphs

Graph each function.

Vertical Stretching or Shrinking of the Graph of a Function

Suppose that a > 0. If a point (x, y) lies on the graph of ( ) , yfx = then the point (x, ay) lies on the graph of () . yafx =

(a) If a > 1, then the graph of ( ) yafx = is a _____________ _____________ of the graph of ( ) yfx =

(b) If 0 < a < 1, then the graph of ( ) yafx = is a _____________ _____________ of the graph of ( ) . yfx =

Horizontal Stretching or Shrinking of the Graph of a Function

Suppose that a > 0. If a point (x, y) lies on the graph of ( ) , yfx = then the point ( ) , x ay lies on the graph of () . yfax =

(a) If 0 < a < 1, then the graph of ( ) yfax = is a _____________ _____________ of the graph of ( ) yfx =

(b) If a > 1, then the graph of ( ) yfax = is a _____________ _____________ of the graph of ( ) . yfx =

Reflecting

CLASSROOM EXAMPLE 2 Reflecting Graphs Across Axes Graph each function.

(a) () gxx =− (b) () hxx =−

2-38 Chapter 2 Graphs and Functions

Reflecting across an Axis

The graph of ()yfx =− is the same as the graph of ( ) yfx = reflected across the __________-_____________. (If a point (x, y) lies on the graph of ( ) , yfx = then _____________ lies on this reflection.)

Symmetry

For a graph to be symmetric with respect to the y-axis, the point ___________ must be on the graph whenever the point (x, y) is on the graph.

For a graph to be symmetric with respect to the x-axis, the point ___________ must be on the graph whenever the point (x, y) is on the graph.

Symmetry with Respect to an Axis

The graph of an equation is symmetric with respect to the __________-_____________ if the replacement of x with x results in an equivalent equation.

The graph of an equation is symmetric with respect to the __________-_____________ if the replacement of y with y results in an equivalent equation.

CLASSROOM EXAMPLE 3 Testing for Symmetry with Respect to an Axis Test for symmetry with respect to the x-axis and the y-axis.

Symmetry with Respect to the Origin

The graph of an equation is symmetric with respect to the origin if the replacement of both x with –x and y with –y at the same time results in an _____________ _____________.

CLASSROOM EXAMPLE 4 Testing for Symmetry with Respect to the Origin

Determine whether the graph of each equation is symmetric with respect to the origin?

3 2 yx =−

2 2 yx =−

Summary of Tests for Symmetry

Symmetry with Respect to: x-axis y-axis Origin

Equation is unchanged if: _______ is replaced with _______ _______ is replaced with _______ _______ is replaced with _______ and _______ is replaced with _______

Example:

A graph symmetric with respect to the __________-_____________ does not represent a function.

Chapter 2 Graphs and Functions

Even and Odd Functions

A function f is called an even function if __________________ for all x in the domain of f . (Its graph is symmetric with respect to the _______________.)

A function f is called an odd function if __________________ for all x in the domain of f . (Its graph is symmetric with respect to the _______________.)

CLASSROOM EXAMPLE 5 Determining Whether Functions Are Even, Odd, or Neither

Determine whether each function defined is even,odd, or neither.

If a function defined by a polynomial in x has only even exponents on x (including the case of a constant where 0 x is understood to have the __________________ exponent __________________), it will always be an __________________ function. Similarly, if only odd exponents appear on x, the function will be an __________________ function.

Translations

CLASSROOM EXAMPLE 6 Translating a Graph Vertically

Graph () 2 2. fxx=+

Vertical Translations

Given a function g defined by ()() , gxfxc =+ where c is a real number:

• For every point (x, y) on the graph of f, there will be a corresponding point __________________ on the graph of g.

• The graph of g will be the same as the graph of f, but translated c units ______________ if c is positive or c units ______________ if c is negative.

The graph of g is a __________________ __________________ of the graph of f.

CLASSROOM EXAMPLE 7 Translating a Graph Horizontally

Graph ()() 2 4. fxx=+

Horizontal Translations

Given a function g defined by ()() , gxfxc =− where c is a real number:

• For every point (x, y) on the graph of f, there will be a corresponding point __________________ on the graph of g.

• The graph of g will be the same as the graph of f, but translated c units ______________ if c is positive or c units ______________ if c is negative.

The graph of g is a __________________ __________________ of the graph of f

2-42 Chapter 2 Graphs and Functions

Summary of Translations

(c > 0)

To Graph: Shift the Graph of ( ) yfx = by c Units:

( ) yfxc =+

( ) yfxc =−

( ) yfxc =+

( ) yfxc =−

CLASSROOM EXAMPLE 8 Using More Than One Transformation Graph each function.

(a) ()() 2 14 fxx=−−+

(b) () 23gxx=−+

CLASSROOM EXAMPLE 9

Graphing Translations and Reflections of a Given Graph

A graph of a function defined by ( ) yfx = is shown in the figure. Use this graph to sketch each of the following graphs.

(a) ()() 2 gxfx=−

(b) ()() 2 hxfx=−

(d) ( ) ( ) Fxfx =−

Summary of Graphing Techniques

In the descriptions that follow, assume that a > 0, h > 0, and k > 0. In comparison with the graph of () : yfx =

1. The graph of () yfxk =+ is translated ____________ units ____________.

2. The graph of () yfxk =− is translated ____________ units ____________.

3. The graph of ()yfxh =+ is translated ____________ units ____________.

4. The graph of ()yfxh =− is translated ____________ units ____________.

5. The graph of ()yafx = is a ____________ ____________ of the graph of ( ) yfx = if a > 1. It is a ____________ ____________ if 0 < a < 1.

6. The graph of ()yfax = is a ____________ ____________ of the graph of ( ) yfx = if 0 < a < 1. It is a ____________ ____________ if a > 1.

7. The graph of ()yfx =− is ____________ across the __________-axis.

8. The graph of ()yfx =− is ____________ across the __________-axis.

2.8 Function Operations and Composition

■ Arithmetic Operations on Functions ■ The Difference Quotient

■ Composition of Functions and Domain

Key Terms: difference quotient, secant line, composite function (composition)

Arithmetic Operations on Functions

Operations on Functions and Domains

Given two functions f and , g then for all values of x for which both ()fx and ()gx are defined, the functions ,,, fgfgfg +− and f g are defined as follows.

Sum function

Difference function

Product function

Quotient function

The domains of ,, fgfg +− and fg include all real numbers in the intersection of the domains of f and g, while the domain of f g includes those real numbers in the intersection of the domains of f and g for which ( ) 0. gx ≠

CLASSROOM EXAMPLE 1 Using Operations on Functions

Let () 34fxx=− and () 2 21.gxx=− Find each of the following.

(a) ()() 0 fg +

(b) ()() 4 fg

(d) () 3 f g

Chapter 2 Graphs and Functions

CLASSROOM EXAMPLE 2 Using Operations on Functions and Determining Domains

Let () 2 3 fxxx =− and () 45.gxx=+ Find each of the following.

(a) ( ) ()fgx + (b) ( ) ()fgx

( ) () fgx

(e) Give the domains of the functions in parts (a)–(d).

CLASSROOM EXAMPLE 3 Evaluating Combinations of Functions

If possible, use the given representations of functions f and g to evaluate ()()()()()() () 1,0,1,and2.fgfgfgf g

+−−−

(a)

Chapter 2 Graphs and Functions

The Difference Quotient

CLASSROOM EXAMPLE 4 Finding The Difference Quotient

Let () 2 324.fxxx=−+ Find and simplify the expression for the difference quotient.

Composition of Functions and Domain

If f and g are functions, then the composite function, or composition, of f and g is defined by ( ) ( ) ( ) ( ) . fg x fg x = 

The domain of ( ) f g  is the set of all numbers x in the domain of g such that ()gx is in the domain of . f

CLASSROOM EXAMPLE 5 Evaluating Composite Functions

Let () 4 fxx=+ and () 2 . gx x =

(a) Find ( ) (2) fg  (b) Find ( ) (5) gf 

CLASSROOM EXAMPLE 6 Determining Composite Functions and Their Domains

Given that () 1 fxx=− and () 25,gxx=+ find each of the following.

(a) ( ) ()fgx  and its domain (b) ( ) ()gfx  and its domain

CLASSROOM EXAMPLE 7 Determining Composite Functions and Their Domains

Given that () 5 4 fx x = + and () 2 , gx x = find each of the following.

(a) ( ) ()fgx  and its domain (b) ( ) ()gfx  and its domain

2-50 Chapter 2 Graphs and Functions

CLASSROOM EXAMPLE 8 Showing That ( ) ( ) gf x  Is Not Equivalent to ( ) ( ) fg x  Let () 25fxx=− and () 2 3. gxxx =+ Show that ( ) ( ) ( ) ().gfxfgx

CLASSROOM EXAMPLE 9 Finding Functions That Form a Given Composite Find functions f and g such that