1. © 2010 Pearson Prentice Hall. All rights reserved. CHAPTER 7 Algebra: Graphs, Functions, and Linear Systems

2. © 2010 Pearson Prentice Hall. All rights reserved. 2 7.2 Linear Functions and Their Graphs

3. © 2010 Pearson Prentice Hall. All rights reserved. Objectives 1.Use intercepts to graph a linear equation. 2.Calculate slope. 3.Use the slope and y-intercept to graph a line. 4.Graph horizontal and vertical lines. 5.Interpret slope as a rate of change. 6.Use slope and y-intercept to model data. 3

4. © 2010 Pearson Prentice Hall. All rights reserved. Graphing Using Intercepts All equations of the form Ax + By = C are straight lines when graphed, as long as A and B are not both zero, and are called linear equations in two variables. 4

5. © 2010 Pearson Prentice Hall. All rights reserved. Example 1: Using Intercepts to Graph a Linear Equation Graph: 3x + 2y = 6. Solution: Find the x-intercept by letting y = 0 and solving for x. 3x + 2y = 6 3x + 2 · 0 = 6 3x = 6 x = 2 Find the y-intercept by letting x = 0 and solving for y. 3x + 2y = 6 3 · 0 + 2y = 6 2y = 6 y = 3 5

6. © 2010 Pearson Prentice Hall. All rights reserved. The x-intercept is 2, so the line passes through the point (2,0). The y-intercept is 3, so the line passes through the point (0,3). Example 1continued Now, we verify our work by checking for x = 1. Plug x = 1 into the given linear equation. We leave this to the student. For x = 1, the y-coordinate should be 1.5. 6

7. © 2010 Pearson Prentice Hall. All rights reserved. Slope The slope of the line through the distinct points (x 1,y 1 ) and (x 2,y 2 ) is where x 2 – x 1 ≠ 0. 7

8. © 2010 Pearson Prentice Hall. All rights reserved. Example 2: Using the Definition of Slope Find the slope of the line passing through the pair of points: (−3, −1) and (−2, 4). Solution: Let (x 1, y 1 ) = (−3, −1) and (x 2, y 2 ) = (−2, 4). We obtain the slope such that Thus, the slope of the line is 5. 8

9. © 2010 Pearson Prentice Hall. All rights reserved. The Slope-Intercept Form of the Equation of a Line The slope-intercept form of the equation of a nonvertical line with slope m and y-intercept b is y = mx + b. 9

10. © 2010 Pearson Prentice Hall. All rights reserved. Graphing y = mx + b using the slope and y-intercept: 1.Plot the point containing the y-intercept on the y- axis. This is the point (0,b). 2.Obtain a second point using the slope m. Write m as a fraction, and use rise over run, starting at the point containing the y-intercept, to plot this point. 3.Use a straightedge to draw a line through the two points. Draw arrowheads at the end of the line to show that the line continues indefinitely in both directions. The Slope-Intercept Form of the Equation of a Line 10

11. © 2010 Pearson Prentice Hall. All rights reserved. Graph the linear function by using the slope and y-intercept. Solution: Since the graph is given in slope-intercept form we can easily find the slope and y-intercept. Example 3: Graphing by Using the Slope and y-intercept 11

12. © 2010 Pearson Prentice Hall. All rights reserved. Step 1 Plot the point containing the y-intercept on the y-axis. The y-intercept is (0, 2). Step 2 Obtain a second point using the slope, m. The slope as a fraction is already given: We plot the second point at (3, 4). Step 3 Use a straightedge to draw a line through the two points. Example 3 continued 12

13. © 2010 Pearson Prentice Hall. All rights reserved. Graph the linear function 2x + 5y = 0 by using the slope and y-intercept. Solution: We put the equation in slope-intercept form by solving for y. Example 4: Graphing by Using the Slope and the y-intercept slope-intercept form 13

14. © 2010 Pearson Prentice Hall. All rights reserved. Next, we find the slope and y-intercept: Start at y-intercept (0, 0) and obtain a second point by using the slope. We obtain (5, −2) as the second point and use a straightedge to draw the line through these points. Example 4 continued 14

15. © 2010 Pearson Prentice Hall. All rights reserved. Equations of Horizontal and Vertical Lines The graph of y = b or f(x) = b is a horizontal line. The y-intercept is b. The graph of x = a is a vertical line. The x-intercept is a. 15

16. © 2010 Pearson Prentice Hall. All rights reserved. Graph y = −4 in the rectangular coordinate system. Solution: All ordered pairs have y-coordinates that are −4. Any value can be used for x. We graph the three ordered pairs in the table: (−2,−4), (0, −4), and (3,−4). Then use a straightedge to draw the horizontal line. Example 5: Graphing a Horizontal Line Graph y = −4 in the rectangular coordinate system. Solution: All ordered pairs have y-coordinates that are −4. Any value can be used for x. We graph the three ordered pairs in the table: (−2,−4), (0, −4), and (3,−4). Then use a straightedge to draw the horizontal line. 16

17. © 2010 Pearson Prentice Hall. All rights reserved. The graph of y = −4 or f(x) = −4. Example 5 continued 17

18. © 2010 Pearson Prentice Hall. All rights reserved. Graph x = 2 in the rectangular coordinate system. Solution: All ordered pairs have the x-coordinate 2. Any value can be used for y. We graph the ordered pairs (2,−2), (2,0), and (2,3). Drawing a line that passes through the three points gives the vertical line. Example 6: Graphing a Vertical Line 18

19. © 2010 Pearson Prentice Hall. All rights reserved. Example 6 continued The graph of x = 2. No vertical line represents a linear function. All other lines are graphs of functions. 19

20. © 2010 Pearson Prentice Hall. All rights reserved. Horizontal and Vertical Lines 20

21. © 2010 Pearson Prentice Hall. All rights reserved. Slope as Rate of Change Slope is defined as a ratio of a change in y to a corresponding change in x. Slope can be interpreted as a rate of change in an applied situation. 21

22. © 2010 Pearson Prentice Hall. All rights reserved. Example 7: Slope as a Rate of Change The graph shows cost of entitlement programs, in billions of dollars, from 2007 with projections through 2016. Find the slope of the line segment representing Social Security. Round to one decimal place. Describe what the slope represents. 22

23. © 2010 Pearson Prentice Hall. All rights reserved. Solution: Let x represent a year and y the cost, in billions of dollars, of Social Security in that year. The two points shown in the line segment have the following coordinates. Example 7 continued 23

24. © 2010 Pearson Prentice Hall. All rights reserved. Now we compute the slope. The slope indicates that for the period from 2007 through 2016, the cost of Social Security is projected to increase by approximately $43.1 billion per year. The rate of change is approximately $43.1 billion per year. Example 7 continued 24

25. © 2010 Pearson Prentice Hall. All rights reserved. Modeling Data with the Slope-Intercept Form of the Equation of a Line Linear functions are useful for modeling data that fall on or near a line. 25