Linear Functions & Inequalities Linear Equation:

Notes # Alpha 3

A linear equation has the form +Ax + By = C, where A and B are both not zero. The graph of a linear equation is always a line.

Solution of a Linear Equation: An ordered pair that makes the equation true. Each ordered pair corresponds to a point in the coordinate plane. Linear Function: A linear function is defined by f(x) = mx + b, where m and b are real numbers. Constant Function: A function f is a constant function if f(x) = b. The graph is a horizontal line. A constant function either has no zeros (b ≠ 0) or every value of x is a zero (b = 0). Linear Inequality: A linear inequality has the form Ax + By < C, Ax + By > C, Ax + By > C, or Ax + By < C, where A and B are both not zero. The graph of linear inequality consists of a boundary and the shading of a region. Ex A: Write an inequality that describes each graph.

#1)

#2)

#3)

y = -.8x x = -3

-3x + y = 10 -3x + y = 5 x=3

Inequality:

Inequality:

Inequality:

Ex B: Graph each inequality or equation.

#1)

y = 4x + 2

#2)

2y < x – 5

#3)

y = |3x – 2|

Note: When graphing, you must label all x- and y-int. You must also label vertices. You must also write the equation on the curve. Shade as needed.

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Linear Functions & Inequalities Zeros of the f: values of x for which f(x) = 0. (zeros of a function are the x-intercepts.) X-intercept: The point at which a graph crosses the x-axis. In a linear function, the xintercept will have coordinates b ,0 . Thus, b is the x-intercept. m

m

Ex C: Find the zero of each function.

#1)

f(x) = 0.5x + 4

#2)

f(x) = 10x

#3)

f(x) = 7x + 3

Note: You can either sub 0 for f(x) and solve for x. Or you can use the formula b ,0 m

Zero =

Zero =

Zero =

Ex D: Complete the following word problem.

Mike can spend up to $40 per day plus $0.35 per mile when renting a car to use on company business. The total cost of the daily rental (C) is a function of the total miles driven. a. Write a linear inequality that expresses the acceptable daily rental car cost.

b. Graph the inequality.

Note: When graphing make sure x is the independent variable and y is the dependant variable.

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