1. Types of Functions and Their Rates of Change
Chapter 1.4
Types of Functions and Their Rates of Change

2. Identify linear functions
Interpret slope as a rate of change
Identify nonlinear functions
Identify where a function is increasing or decreasing
Use interval notation
Use and interpret average rate of change
Calculate the difference quotient

3. Teaching Example 1 Solution
Find the slope of the line passing through (−1, 2) and (3, −4). Explain what the slope indicates about the line.
Solution
The line falls 3/2 units for each unit increase in x.

4. Teaching Example 2 Solution
The number of gallons of water remaining in a 100-gallon tank after x-minutes is given by G(x) = 100 – 5x. a. Evaluate G(5) and interpret the result. b. Find the slope of the graph of G. Interpret this slope as a rate of change.
Solution
a. G(5) = 100 – 5(5) = 100 – 25 = 75
After 5 minutes the tank contains 75 gallons of water. 4

5. Teaching Example 2 (cont)
The number of gallons of water remaining in a 100-gallon tank after x-minutes is given by G(x) = 100 – 5x. a. Evaluate G(5) and interpret the result. b. Find the slope of the graph of G. Interpret this slope as a rate of change.
Solution
b. m = –5. Water is leaving the tank at a rate of 5 gal/min. 5

6. Teaching Example 3 Solution
In 1985, $95 billion was spent on advertising in the United States, and in 2005, $270 billion was spent. Find and interpret the slope of the line passing through (1985, 95) and (2005, 270).
Solution
This means that advertising spending increased, on average, by $8.75 billion per year from 1985 to 2005. 6

7. Teaching Example 4 Solution
Sketch a graph of y = x – 1. Determine the slope; y-intercept, x-intercept, formula for f(x) and any zeros.
Solution
slope =
y-intercept = –1
x-intercept = 1
f(x) = x – 1
The zero of the function is 1. 7

8. Teaching Example 5 Solution
For each function f, determine where f is increasing, and where f is decreasing. a. f(x) = x2 b. f(x) = 1 – 2x c. f(x) = |x – 2|
Solution
a. Increasing when x > 0
Decreasing when x < 0 8

9. Teaching Example 5 (cont)
For each function f, determine where f is increasing, and where f is decreasing. a. f(x) = x2 b. f(x) = 1 – 2x c. f(x) = |x – 2|
Solution
b. never increasing
Decreasing for all real numbers 9

10. Teaching Example 5 (cont)
For each function f, determine where f is increasing, and where f is decreasing. a. f(x) = x2 b. f(x) = 1 – 2x c. f(x) = |x – 2|
Solution
c. Increasing when x > 2
Decreasing when x < 2 10

11. Teaching Example 6 Solution
Determine where is increasing or decreasing.
Solution
f(x) is increasing on and
decreasing on 11

12. Teaching Example 7 Solution
Let f(x) = x2. Find the average rate of change from x = 2 to x = 4.
Solution 12

13. Teaching Example 8 Solution
If f(1) = 12 and f(4) = −3, find the average rate of change of f from 1 to 4.
Solution
The average rate of change of f from 1 to 4 is −5.

14. Teaching Example 9 Solution
Find the difference quotient for f(x) = –2x + 5
Solution

15. Teaching Example 10
Find the difference quotient for
Solution 15