A Level Further Mathematics for AQA Student Book 2
After the enlargement: 2 y x +1 −4 3 3 + 9 16
( ) ( )
⇔
For the enlargement, the x and y inside the brackets need to be replaced. You can see that this curve has the same shape as the one in part a but is in a different position.
2
=1
(x + 3) 2 ( y − 12) 2 + =1 144 81
WORKED EXAMPLE 3.7
pl e
You also need to be able to identify the sequence of transformations that maps one curve to another. You have already seen that sometimes there are several transformations giving the same result for a given curve (for example, because of the symmetry of the ellipse, a 90° rotation gives the same result as a reflection in the line y = x). It is also possible that transformations can be carried out in a different order. Therefore there may be more than one correct answer to a question.
Sa m
The parabola y2 = 2x is transformed to the parabola x2 = −6y by a rotation followed by a horizontal stretch. a Describe fully both transformations. b The rotation from part a can be replaced by a reflection and still result in the same curve. What is the reflection? c The same curve can also be obtained by a horizontal stretch followed by a rotation. Find the scale factor of the stretch. a The rotation is 90° clockwise about the origin. The equation after the rotation is x2 = −2y. After the enlargement:
() x k
2
= −2y
⇔ x2 = −2k2y Hence k2 = 3. The enlargement has centre at the origin and scale factor √3. b The reflection is in the line y = −x.
c After the stretch: y2 = 2 x k
()
54
The rotation replaces x by −y (and y by x or −x). The horizontal stretch with scale factor k replaces x by x . k
Compare the last equation to x2 = −6y.
The reflection needs to replace x by −y and y by x or −x. (See the table in Key point 3.1) The horizontal stretch with scale factor k replaces x by x . k
After the rotation:
The rotation still needs to be 90° clockwise.
x2 = 2 (−y) = − 2 y k k 2 Hence = 6, so the scale factor is 1 . 3 k
Compare this to x2 = −6y. © Cambridge University Press