Here is a table and graph describing a function with an initial value of 30 and a rate of change of 5: Table: x y 0 30 1 35 2 40 3 45 Graph: y 45 40 35 30 0 1 2 3 x This function has an initial value (y-intercept) of 30, since when x = 0, y = 30. The rate of change is 5, since each time x increases by 1, y increases by 5.

1. Course 3, Lesson 4-6
1. A greyhound can run at a speed of 55 feet per second.
The table shows the speed of a cheetah. Compare their
rates of change.
2. Jonas saves at the rate shown in the table. His friend
Thomas saves at a rate of $100 per month. Compare
their rates.

2. Course 3, Lesson 4-6
ANSWERS
1. The speed of a greyhound is 55 feet per second. The
speed of a cheetah is 103 feet per second. Since
103 > 55, the cheetah has a greater rate of change.
2. Jonas saves at a rate of $120 per month, so Jonas
has a greater rate of change than Thomas.

3. HOW can we model relationships
between quantities?
Functions
Course 3, Lesson 4-6

4. Course 3, Lesson 4-6 Common Core State Standards © Copyright 2010. National Governors Association Center for Best Practices and Council
of Chief State School Officers. All rights reserved.
Functions
• 8.F.4
Construct a function to model a linear relationship between two
quantities. Determine the rate of change and initial value of the function
from a description of a relationship or from two (x, y) values, including
reading these from a table or from a graph. Interpret the rate of change
and initial value of a linear function in terms of the situation it models,
and in terms of its graph or a table of values.
Mathematical Practices
1 Make sense of problems and persevere in solving them.
3 Construct viable arguments and critique the reasoning of others.
4 Model with mathematics.

5. To find and interpret the
• rate of change of a function,
• initial value of a function
Course 3, Lesson 4-6
Functions

6. 1
Need Another Example?
2
3
4
Step-by-Step Example
1. A shoe store offers free points when you sign
up for their rewards card. Then, for each pair of
shoes purchased, you earn an additional
number of points. The graph shows the total
points earned for several pairs of shoes. Find
and interpret the rate of change and initial value.
To find the rate of change, choose two points from the graph.
Next find the initial value or the y-value when x = 0. Recall this value is called
the y-intercept. Extend the line so it intersects the y-axis. The value for y when
x = 0 is 30. So, the initial number of points earned is 30.
The rate of change is 15, so the amount of points earned per
pair of shoes is 15.

7. Answer
Need Another Example?
A pottery studio charges a certain amount per
hour, plus a firing fee to fire the pottery. The
graph shows the total cost of using the studio for
different amounts of time. Find and interpret the
rate of change and initial value.
The firing fee is $10. The hourly fee is $15.

8. 1
Need Another Example?
2
3
4
5
6
Step-by-Step Example
2. Joan has some photos in her photo album. Each week she plans to add
12 photos. Joan had 120 photos after 8 weeks. Assume the relationship
is linear. Find and interpret the rate of change and initial value.
Since each week Joan adds 12 photos to her photo album the rate
of change is 12. To find the initial value, use slope-intercept form
to find the y-intercept.
The y-intercept is 24. So, the initial amount of photos is 24.
y = mx + b Slope-intercept form
y = 12x + b Replace m with the rate of change, 12.
120 = 12(8) + b Replace y with 120 and x with 8
24 = b Solve for b.

9. Answer
Need Another Example?
Each week, Sam works the same number of
hours. Her first week included an orientation. The
total number of hours that she worked the 2nd,
3rd, and 4th weeks was 20 hours, 28 hours, and
36 hours. Assume the relationship is linear. Find
and interpret the rate of change and initial value.
She works 8 hours each week.
The orientation was 4 hours long.

10. 1
Need Another Example?
2
3
4
5
6
Step-by-Step Example
3. The table shows how much money Ava has
saved. Assume the relationship between the two
quantities is linear. Find and interpret the rate of
change and initial value.
Choose any two points from the table to find the rate of
change. The rate of change is or 20, so Ava
saves $20 each month.
To find the initial value, use the slope-intercept form to find
the y-intercept.
y = mx + b Slope-intercept form
y = 20x + b Replace m with the rate of change, 20.
110 = 20(3) + b Use the point (3, 110). x = 3, y = 110
50 = b Solve for b.
The y-intercept is 50, so Ava had initially saved $50.

11. Answer
Need Another Example?
The table shows the weight of a puppy in weeks
4, 5, 6 and 7. Assume the relationship between
the two quantities is linear. Find and interpret the
rate of change and initial value.
The puppy gains 3 pounds per week. At birth,
the puppy weighed 5 pounds.

12. How did what you learned
today help you answer the
HOW can we model relationships
between quantities?
Course 3, Lesson 4-6
Functions

13. How did what you learned
today help you answer the
HOW can we model relationships
between quantities?
Course 3, Lesson 4-6
Functions
Sample answers:
• The initial value of a real-world function is the y-
intercept, or the y-value when x = 0.
• You can interpret the rate of change in a real-world
function by finding the slope.

14. Describe a function that has
an initial value of 30 by
making a table or drawing a
graph. Be sure to include
the rate of change of your
function.
Ratios and Proportional RelationshipsFunctions
Course 3, Lesson 4-6