1. 7.2 Linear Functions And Their Graphs

2. Objectives Use intercepts to graph a linear equation. Calculate slope.
Use the slope and y-intercept to graph a line.
Graph horizontal and vertical lines.
Interpret slope as a rate of change.
Use slope and y-intercept to model data.

3. Using Intercepts to Graph Lines Equation
Equation of a Line: Ax + By = C E.g., 2x + 3y = 4
x-intercept—point where the line crosses the x-axis
Y-intercept—point where the line crosses the y-axis
To find x-intercept, let y = 0 and solve for x.
To find y-intercept, let x = 0 and solve for y.
(0, y0)
(x0, 0)

4. Example: Using Intercepts to Graph Lines 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. Example: Using Intercepts to Graph Lines Equation
The x-intercept is 2; line passes through (2,0). The y-intercept is 3; line passes through (0,3).
Now, we verify our work by checking for x = 1. Plug x = 1 into the given linear equation.
For x = 1, the y-coordinate should be 1.5.

6. Slope of a Line
Slope of the line through points (x1,y1) and (x2,y2): where x2 – x1 ≠ 0.

7. Example
Find the slope of the line passing through points: (−3, −1) and (−2, 4).
Solution: Let (x1, y1) = (−3, −1) and (x2, y2) = (−2, 4).
Thus, the slope of the line is 5.

8. The Slope-intercept form of the Equation of a Line
Recall: Ax + By = C is an equation of a line. Solving for y, we get By = -Ax + C y = (-Ax + C)/B y = (-A/B)x + C/B When x = 0, y = C/B y = 0, x = (C/B)/(A/B) = C/A C/B – C/B C -A m = = = --- · = -A/B – C/A -C/A B C
(0, C/B)
(C/A, 0)

9. The Slope-intercept form of the Equation of a Line
y = (-A/B)x + C/B m = -A/B y = mx + b where m = slope b = y-intercept E.g., y = (3/2)x m = 3/ b = 4
(0, C/B)
(C/A, 0)

10. Example Graph: 2x + 5y = 0 using the slope and y- intercept.
Solution: 2x + 5y = 0 5y = -2x y = (-2/5)x m = -2/5 b = 0

11. Your Turn
Graph the lines using the slope-intercept form of the equation.
y = (-3/4)x – 5
2x + 3y + 4 = 0

12. Equation of Horizontal and Vertical Lines
y = b or f(x) = b horizontal line. The y-intercept is b.
x = a vertical line. The x-intercept is a.

13. Interpretation of the Slope of a Line
Slope: ration of a change in y to a corresponding change in x m = Δx/Δy
Slope can be interpreted as a rate of change in the vertical value as the corresponding horizontal value changes.

14. Interpretation of the Slope of a Line
The graph shows cost of entitlement programs, in billions of dollars, from 2007 with projections through Find the slope of the line segment representing Social Security. Round to one decimal place.
Describe what the slope represents.

15. Example
Solution: Let x represent a year and y the cost (in $109).

16. Example (cont.)
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.