Solving Equations by Graphing

1. Section 2-7
Solving Equations by Graphing

2. Essential Questions
• How do you find x- and y-intercepts of
functions?
• How do you solve equations by examining
graphs of the related functions?

3. Vocabulary
1. Intercepts:
2. y-intercept:
3. x-intercept:
4. Root of the Equation:
5. Zero of a Function:

4. Vocabulary
1. Intercepts: Points where a graph crosses an axis
2. y-intercept:
3. x-intercept:
4. Root of the Equation:
5. Zero of a Function:

5. Vocabulary
1. Intercepts: Points where a graph crosses an axis
2. y-intercept: Point where a graph crosses the y-axis
3. x-intercept:
4. Root of the Equation:
5. Zero of a Function:

6. Vocabulary
1. Intercepts: Points where a graph crosses an axis
2. y-intercept: Point where a graph crosses the y-axis
3. x-intercept: Point where a graph crosses the x-axis
4. Root of the Equation:
5. Zero of a Function:

7. Vocabulary
1. Intercepts: Points where a graph crosses an axis
2. y-intercept: Point where a graph crosses the y-axis
3. x-intercept: Point where a graph crosses the x-axis
4. Root of the Equation: The solution to an equation
5. Zero of a Function:

8. Vocabulary
1. Intercepts: Points where a graph crosses an axis
2. y-intercept: Point where a graph crosses the y-axis
3. x-intercept: Point where a graph crosses the x-axis
4. Root of the Equation: The solution to an equation
5. Zero of a Function: Another name for the root of
the equation; The value of x for which f(x) = 0

9. Example 1
Find the x- and y-intercepts of the graph.
y =
4
5
x 2
+
12
5
x − 8

10. Example 1
Find the x- and y-intercepts of the graph.
y =
4
5
x 2
+
12
5
x − 8
x-intercepts:

11. Example 1
Find the x- and y-intercepts of the graph.
y =
4
5
x 2
+
12
5
x − 8
x-intercepts:
(-5, 0), (2, 0)

12. Example 1
Find the x- and y-intercepts of the graph.
y =
4
5
x 2
+
12
5
x − 8
x-intercepts:
y-intercept:
(-5, 0), (2, 0)

13. Example 1
Find the x- and y-intercepts of the graph.
y =
4
5
x 2
+
12
5
x − 8
x-intercepts:
y-intercept:
(-5, 0), (2, 0)
(0, -8)

14. Example 2
Find the root of each equation by graphing the related
function.
a. 0 =
1
3
x + 2

15. Example 2
Find the root of each equation by graphing the related
function.
a. 0 =
1
3
x + 2
x
y

16. Example 2
Find the root of each equation by graphing the related
function.
a. 0 =
1
3
x + 2
x
y

17. Example 2
Find the root of each equation by graphing the related
function.
a. 0 =
1
3
x + 2
x
y

18. Example 2
Find the root of each equation by graphing the related
function.
a. 0 =
1
3
x + 2
x
y

19. Example 2
Find the root of each equation by graphing the related
function.
a. 0 =
1
3
x + 2
x
y

20. Example 2
Find the root of each equation by graphing the related
function.
a. 0 =
1
3
x + 2
x
y

21. Example 2
Find the root of each equation by graphing the related
function.
a. 0 =
1
3
x + 2
x
y

22. Example 2
Find the root of each equation by graphing the related
function.
a. 0 =
1
3
x + 2
x
y

23. Example 2
Find the root of each equation by graphing the related
function.
a. 0 =
1
3
x + 2
x
y

24. Example 2
Find the root of each equation by graphing the related
function.
a. 0 =
1
3
x + 2
x
y
The root is at x = -6

25. Example 2
Find the root of each equation by graphing the related
function.
a. 0 =
1
3
x + 2
x
y
The root is at x = -6
The x-intercept is at (-6, 0)

26. Example 2
Find the root of each equation by graphing the related
function.
b. 0 = 5x −15

27. Example 2
Find the root of each equation by graphing the related
function.
b. 0 = 5x −15
x
y

28. Example 2
Find the root of each equation by graphing the related
function.
b. 0 = 5x −15
x
y

29. Example 2
Find the root of each equation by graphing the related
function.
b. 0 = 5x −15
x
y

30. Example 2
Find the root of each equation by graphing the related
function.
b. 0 = 5x −15
x
y

31. Example 2
Find the root of each equation by graphing the related
function.
b. 0 = 5x −15
x
y

32. Example 2
Find the root of each equation by graphing the related
function.
b. 0 = 5x −15
x
y

33. Example 2
Find the root of each equation by graphing the related
function.
b. 0 = 5x −15
x
y
The root is at x = 3

34. Example 2
Find the root of each equation by graphing the related
function.
b. 0 = 5x −15
x
y
The root is at x = 3
The x-intercept is at (3, 0)

35. Example 3
Matt Mitarnowski found a service that allows
him to download albums to his smart phone.
Each album costs $1.25 to download. He also
paid a subscription fee of $8 to access or just
stream the music. Someone needs to inform
him of Spotify Premium, where there is no extra
fee on top of the subscription fee to sync music.
If the total cost for the download and
subscription fee was $15.50, how many albums
did he download? Solve by graphing the related
function.

36. Example 3

37. Example 3
a = albums

38. Example 3
8 +1.25a = 15.50
a = albums

39. Example 3
8 +1.25a = 15.50
a = albums
−15.50−15.50

40. Example 3
8 +1.25a = 15.50
a = albums
−15.50−15.50
1.25a − 7.5 = 0

41. Example 3
8 +1.25a = 15.50
a = albums
−15.50−15.50
1.25a − 7.5 = 0
5
4
a −
15
2
= 0

42. Example 3
8 +1.25a = 15.50
a = albums
−15.50−15.50
1.25a − 7.5 = 0
5
4
a −
15
2
= 0
a
c

43. Example 3
8 +1.25a = 15.50
a = albums
−15.50−15.50
1.25a − 7.5 = 0
5
4
a −
15
2
= 0
a
c

44. Example 3
8 +1.25a = 15.50
a = albums
−15.50−15.50
1.25a − 7.5 = 0
5
4
a −
15
2
= 0
a
c

45. Example 3
8 +1.25a = 15.50
a = albums
−15.50−15.50
1.25a − 7.5 = 0
5
4
a −
15
2
= 0
a
c

46. Example 3
8 +1.25a = 15.50
a = albums
−15.50−15.50
1.25a − 7.5 = 0
5
4
a −
15
2
= 0
a
c

47. Example 3
8 +1.25a = 15.50
a = albums
−15.50−15.50
1.25a − 7.5 = 0
5
4
a −
15
2
= 0
a
c
Matt bought 6 albums