The document discusses linear functions and piecewise linear graphs. It provides examples of linear equations modeling real-world situations involving salaries, allowances, and weight over time. Key concepts explained include slope, slope-intercept form, linear functions, and piecewise linear graphs, which have at least two different constant rates of change. Worked examples calculate slope and solve linear equations to find values like number of dirty dishes based on allowance amount.

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CHAPTER 3
LINEAR FUNCTIONS

2. SECTION 3-1
Constant-Increase and Constant-Decrease Situations

3. WARM-UP
Look at the four graphs on page 139 as such:
a. Constant Increase
b. Linear Combination
c. Point-Slope
d. Step Function
1. Name at least two points on each graph.

4. WARM-UP
2. Give the domain and range of each function

5. WARM-UP
2. Give the domain and range of each function
a. D = Set of whole numbers; R = {n: n = 3, 3.2, 3.4, ...}

6. WARM-UP
2. Give the domain and range of each function
a. D = Set of whole numbers; R = {n: n = 3, 3.2, 3.4, ...}
b. D = {A: A = 0, 3, 6}; R = {S: S = 0, 7, 14}

7. WARM-UP
2. Give the domain and range of each function
a. D = Set of whole numbers; R = {n: n = 3, 3.2, 3.4, ...}
b. D = {A: A = 0, 3, 6}; R = {S: S = 0, 7, 14}
c. D = {W: W ≥ 0}; R = {L: L ≥ 7}

8. WARM-UP
2. Give the domain and range of each function
a. D = Set of whole numbers; R = {n: n = 3, 3.2, 3.4, ...}
b. D = {A: A = 0, 3, 6}; R = {S: S = 0, 7, 14}
c. D = {W: W ≥ 0}; R = {L: L ≥ 7}
d. D = {w: w > 0}; R = {C: C = .33, .55, .77, ...}

9. Linear Equation:

10. Linear Equation: Equation that gives a graph of a line

11. EXAMPLE 1
Matt Mitarnowski sells sports cars. He gets a base salary of
$30,000 per year plus 2% of his sales. If Matt’s sales for the
year totaled D dollars, what is his salary S?

12. EXAMPLE 1
Matt Mitarnowski sells sports cars. He gets a base salary of
$30,000 per year plus 2% of his sales. If Matt’s sales for the
year totaled D dollars, what is his salary S?
S = 30,000 + .02D

13. EXAMPLE 1
Matt Mitarnowski sells sports cars. He gets a base salary of
$30,000 per year plus 2% of his sales. If Matt’s sales for the
year totaled D dollars, what is his salary S?
S = 30,000 + .02D
Let’s look at the table and graph

14. EXPLORE
Calculate the slope for the situation in Example 1.

15. EXPLORE
Calculate the slope for the situation in Example 1.
(0, 30,000), (1,000,000, 50,000)

16. EXPLORE
Calculate the slope for the situation in Example 1.
(0, 30,000), (1,000,000, 50,000)
50, 000 − 30, 000
m=
1, 000, 000 − 0

17. EXPLORE
Calculate the slope for the situation in Example 1.
(0, 30,000), (1,000,000, 50,000)
50, 000 − 30, 000
m=
1, 000, 000 − 0
20, 000
=
1, 000, 000

18. EXPLORE
Calculate the slope for the situation in Example 1.
(0, 30,000), (1,000,000, 50,000)
50, 000 − 30, 000
m=
1, 000, 000 − 0
20, 000
=
1, 000, 000
2
=
100

19. EXPLORE
Calculate the slope for the situation in Example 1.
(0, 30,000), (1,000,000, 50,000)
50, 000 − 30, 000
m=
1, 000, 000 − 0
20, 000
=
1, 000, 000
2 1
= =
100 50

20. EXPLORE
Calculate the slope for the situation in Example 1.
(0, 30,000), (1,000,000, 50,000)
50, 000 − 30, 000
m=
1, 000, 000 − 0
20, 000
=
1, 000, 000
2 1
= = = .02
100 50

21. Slope-intercept Form:

22. Slope-intercept Form: y = mx + b, where m = slope and
b = the y-coordinate of the y-intercept

23. Slope-intercept Form: y = mx + b, where m = slope and
b = the y-coordinate of the y-intercept
*In this form, the slope will ALWAYS be with the
independent variable, and the y-coordinate of the y-
intercept will ALWAYS be by itself

24. Slope-intercept Form: y = mx + b, where m = slope and
b = the y-coordinate of the y-intercept
*In this form, the slope will ALWAYS be with the
independent variable, and the y-coordinate of the y-
intercept will ALWAYS be by itself
Linear Function:

25. Slope-intercept Form: y = mx + b, where m = slope and
b = the y-coordinate of the y-intercept
*In this form, the slope will ALWAYS be with the
independent variable, and the y-coordinate of the y-
intercept will ALWAYS be by itself
Linear Function: A function of the form y = mx + b

26. Slope-intercept Form: y = mx + b, where m = slope and
b = the y-coordinate of the y-intercept
*In this form, the slope will ALWAYS be with the
independent variable, and the y-coordinate of the y-
intercept will ALWAYS be by itself
Linear Function: A function of the form y = mx + b
Euler notation:

27. Slope-intercept Form: y = mx + b, where m = slope and
b = the y-coordinate of the y-intercept
*In this form, the slope will ALWAYS be with the
independent variable, and the y-coordinate of the y-
intercept will ALWAYS be by itself
Linear Function: A function of the form y = mx + b
Euler notation: f(x) = mx + b

28. Slope-intercept Form: y = mx + b, where m = slope and
b = the y-coordinate of the y-intercept
*In this form, the slope will ALWAYS be with the
independent variable, and the y-coordinate of the y-
intercept will ALWAYS be by itself
Linear Function: A function of the form y = mx + b
Euler notation: f(x) = mx + b
Mapping notation:

29. Slope-intercept Form: y = mx + b, where m = slope and
b = the y-coordinate of the y-intercept
*In this form, the slope will ALWAYS be with the
independent variable, and the y-coordinate of the y-
intercept will ALWAYS be by itself
Linear Function: A function of the form y = mx + b
Euler notation: f(x) = mx + b
Mapping notation: f:x mx + b

30. EXAMPLE 2
Fuzzy Jeff gets an allowance of $15 per week. Whenever
his parents pick up a dirty dish he left out, Jeff losts $.30.
a. Write an equation modeling this situation

31. EXAMPLE 2
Fuzzy Jeff gets an allowance of $15 per week. Whenever
his parents pick up a dirty dish he left out, Jeff losts $.30.
a. Write an equation modeling this situation
A = Allowance; d = Dirty Dishes

32. EXAMPLE 2
Fuzzy Jeff gets an allowance of $15 per week. Whenever
his parents pick up a dirty dish he left out, Jeff losts $.30.
a. Write an equation modeling this situation
A = Allowance; d = Dirty Dishes
A = 15 - .3d

33. EXAMPLE 2
Fuzzy Jeff gets an allowance of $15 per week. Whenever
his parents pick up a dirty dish he left out, Jeff losts $.30.
a. Write an equation modeling this situation
A = Allowance; d = Dirty Dishes
A = 15 - .3d
b. Graph the equation

34. EXAMPLE 2
c. If Jeff gets no allowance, how many dishes did he leave
out?

35. EXAMPLE 2
c. If Jeff gets no allowance, how many dishes did he leave
out?
A = 15 - .3d

36. EXAMPLE 2
c. If Jeff gets no allowance, how many dishes did he leave
out?
A = 15 - .3d
0 = 15 - .3d

37. EXAMPLE 2
c. If Jeff gets no allowance, how many dishes did he leave
out?
A = 15 - .3d
0 = 15 - .3d
-15

38. EXAMPLE 2
c. If Jeff gets no allowance, how many dishes did he leave
out?
A = 15 - .3d
0 = 15 - .3d
-15 -15

39. EXAMPLE 2
c. If Jeff gets no allowance, how many dishes did he leave
out?
A = 15 - .3d
0 = 15 - .3d
-15 -15
-15 = -.3d

40. EXAMPLE 2
c. If Jeff gets no allowance, how many dishes did he leave
out?
A = 15 - .3d
0 = 15 - .3d
-15 -15
-15 = -.3d
d = 50

41. EXAMPLE 2
c. If Jeff gets no allowance, how many dishes did he leave
out?
A = 15 - .3d
0 = 15 - .3d
-15 -15
-15 = -.3d
d = 50 dishes

42. Piecewise Linear Graph:

43. Piecewise Linear Graph: When the rate of change switches
from one constant value to another

44. Piecewise Linear Graph: When the rate of change switches
from one constant value to another
*Made up of two or more segments or rays

45. EXAMPLE 3
The graph below describes Shecky’s weight over the first
16 weeks of his life. Write out an explanation of each piece
of the piecewise linear function.

46. EXAMPLE 3

47. EXAMPLE 3
Shecky weighed 9 pounds at birth.

48. EXAMPLE 3
Shecky weighed 9 pounds at birth.
In his first week alive, he lost one pound.

49. EXAMPLE 3
Shecky weighed 9 pounds at birth.
In his first week alive, he lost one pound.
Over the next four weeks, Shecky gained a pound a week.

50. EXAMPLE 3
Shecky weighed 9 pounds at birth.
In his first week alive, he lost one pound.
Over the next four weeks, Shecky gained a pound a week.
In the following three weeks, Shecky’s weight stayed the
same.

51. EXAMPLE 3
Shecky weighed 9 pounds at birth.
In his first week alive, he lost one pound.
Over the next four weeks, Shecky gained a pound a week.
In the following three weeks, Shecky’s weight stayed the
same.
Over the last 8 weeks, Shecky gained half of a pound per
week.

52. HOMEWORK

53. HOMEWORK
p. 143 #1 - 26
“Fortune favors the brave.” - Publius Terence