1. Lesson 3.4 Constant Rate of Change (linear functions)
3.3.2: Proving Average Rate of Change

2. Introduction
A rate of change is a ratio that describes how much one quantity changes with respect to the change in another quantity of the function.
With linear functions the rate of change is called the slope. The slope of a line is the ratio of the change in y-values to the change in x-values. Formula: m =
Linear functions have a constant rate of change, meaning values increase or decrease at the same rate over a period of time.
3.3.2: Proving Average Rate of Change

3. Constant Rate of Change (slope)
Recall…..
The rate of change between any two points of a linear function will be equal.
Calculating
Constant Rate of Change (slope)
from a Table
Choose two points from the table.
Assign one point to be (x1, y1) and the other point to be (x2, y2).
Substitute the values into the slope formula: 𝑚= 𝑦 2 − 𝑦 1 𝑥 2 − 𝑦 2
The result is the rate of change for the interval between the two points chosen.
3.3.2: Proving Average Rate of Change

4. Number of Customers (f(x))
Guided Practice
Example 1
To raise money, students
plan to hold a car wash.
They ask some adults how
much they would pay for
a car wash. The table on
the right shows the results
of their research. What is the
rate of change for their results?
Carwash Price (x)
Number of Customers (f(x)) $4
120 $6
105 $8 90
$10 75
3.3.2: Proving Average Rate of Change

5. Number of Customers (f(x))
Guided Practice: Example 1, continued
Choose two points from the table.
(4, 120) and (10, 75)
2. Assign one point to
be (x1, y1) and the other
to be (x2, y2). It doesn’t matter which is which. Let (4, 120) be (x1, y1) and
(10,76) be (x2, y2).
Carwash Price (x)
Number of Customers (f(x)) $4
120 $6
106 $8 92
$10 78
3.3.2: Proving Average Rate of Change

6. Guided Practice: Example 1, continued
3. Substitute (0, 1.5) and (155, 0) into the slope formula to calculate the rate of change.
Slope formula
Substitute (4, 120) and (10, 78) for (x1, y1) and (x2, y2).
Simplify as needed.
= 78−120 10−4
= −42 6
= -7
The rate of change for this function is -7 customers per dollar. For every dollar the carwash price increases, 7 customers are lost.
3.3.3: Recognizing Average Rate of Change

7. Constant Rate of Change (slope)
Recall….
The rate of change between any two points of a linear function will be equal.
Estimating
Constant Rate of Change (slope)
from a Graph
Pick two points from the graph.
Identify (x1, y1) as one point and (x2, y2) as the other point.
Substitute (x1, y1) and (x2, y2) into the slope formula to calculate the rate of change.
The result is the estimated constant rate of change (slope) for the graph.
m =
3.3.2: Proving Average Rate of Change

8. Calculate the constant rate of change (slope) for these tables.
You Try
Calculate the constant rate of change (slope) for these tables. 1) 2) x
f(x) 1 -6 2
-11 3
-16 4
-21 x
f(x) -3 -5 -4 3 6 -2
3.3.2: Proving Average Rate of Change

9. Guided Practice Example 2 The graph to the right
compares the distance a
small motor scooter can
travel in miles to the amount
of fuel used in gallons.
What is the rate of change
for this scenario?
3.3.3: Recognizing Average Rate of Change

10. Guided Practice: Example 2, continued Pick two points from the graph.
The function is linear, so the rate of change will be constant for any interval (continuous portion) of the function.
Choose points on the graph with coordinates that are easy to estimate. For example, (0, 1.5) and (155,0)
2. Identify (x1, y1) as one point and (x2, y2) as the other point. It doesn’t matter which is which.
Let’s have (0, 1.5) be (x1, y1) and (155,0) be (x2, y2)
3.3.3: Recognizing Average Rate of Change

11. Guided Practice: Example 2, continued
3. Substitute (0, 1.5) and (155, 0) into the slope formula to calculate the rate of change.
Slope formula
Substitute (0,1.5) and (155, 0) for (x1, y1) and (x2, y2).
Simplify as needed.
= 0− −0
= −
≈ -0.01
3.3.3: Recognizing Average Rate of Change

12. Calculate the constant rate of change (slope) for these graphs.
You Try
Calculate the constant rate of change (slope) for these graphs. 1) 2)
3.3.2: Proving Average Rate of Change