1. 1 Section 1.2 Linear Functions and Graphs

2. 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. 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. 4 STRAIGHT LINES AND LINEAR 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. 5 SLOPE AND y-INTERCEPT OF 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

6. 6 SLOPE-INTERCEPT EQUATION 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. 7 POINT-SLOPE FORMULA FOR A LINE If (x 0, y 0 ) is a fixed point on a line, and (x, y) is any other point on the line, we can find the slope by The equation is called the point-slope equation for the line with slope m that passes through the point (x 0, y 0 ).

8. 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. 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. 10 GRAPH OF A FUNCTION Definition: The graph of the function f is the graph of the equation y = f (x).

11. 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. 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?