1 section 1.2 linear functions and graphs. 2 example x (miles) c (cost) 100$60 150$75 200$90 250$105...

1

Section 1.2

Linear Functions

and Graphs

2

EXAMPLE

x

(Miles)

C

(Cost)

100 $60

150 $75

200 $90

250 $105

The following table shows the cost per day for a rental car depending on how many miles you drive a day.

3

LINEAR FUNCTIONS

Definition: A linear function is a function of the form

f (x) = mx + b.

NOTE: The letters “m” and “b” represent coefficients (numbers).

4

STRAIGHT LINES ANDLINEAR GRAPHS

The graph of the linear function f (x) = mx + b is the straight line consisting of all the points (x,y) in the xy-plane that satisfy the equation

y = mx + b.

5

SLOPE AND y-INTERCEPTOF A LINE

• The y-value where a line crosses the y-axis is called the y-intercept.

• The slope of a line in the xy-plane is defined by

slo p erise

ru n

6

SLOPE-INTERCEPTEQUATION

• The constant b is the y-intercept of the line.

• The coefficient m of x is the slope of the line.

• This equation is called the slope-intercept equation for a line.

In the equation y = mx + b,

7

POINT-SLOPE FORMULAFOR A LINE

If (x0, y0) is a fixed point on a line, and (x, y) is any other point on the line, we can find the slope by

my y

x x

slo p e 0

0

.

The equation y y m x x 0 0( )is called the point-slope equation for the line with slope m that passes through the point (x0, y0).

8

GRAPHS OF EQUATIONS

Definition: The graph of an equation involving two variables x and y consists of all points in the xy-plane whose coordinates (x, y) satisfy the equation.

9

GRAPHS OF LINES

• Equations whose graphs are vertical lines. These have the form x = constant.

• Equations whose graphs are horizontal lines. These have the form y = constant.

• Equations whose graphs are slanted lines (lines that are neither vertical or horizontal). These have the form y = mx + b.

There are three types of equations whose graphs are straight lines:

10

GRAPH OF A FUNCTION

Definition: The graph of the function f is the graph of the equation y = f (x).

11

THE VERTICAL LINE TEST

Recall that a function assigns to each number x a specific number f (x). Thus, a graph is the graph of a function if and only if no vertical line intersects the graph at more than one point.

12

EXAMPLE

x

(Miles)

C

(Cost)

100 $60

150 $75

200 $90

250 $105

The following table shows the cost per day for a rental car depending on how many miles you drive a day.

(a)How many miles did you drive if the cost was $80?

(b)What do the slope and y-intercept mean in practical terms?