1. 1.4 Types of Functions and Their Rates of Change
Identify linear functions
Interpret slope as a rate of change
Identify nonlinear functions
Identify where a function is increasing or decreasing
Use and interpret average rate of change
Calculate the difference quotient

2. Linear Function
A function f represented by f(x) = mx + b, where m and b are constants, is a linear function.

3. Recognizing Linear Functions
A car initially located 30 miles north of the Texas border, traveling north at 60 miles per hour is represented by the function f(x) = 60x and has the graph:

4. Rate of Change of a Linear Function (1 of 2)
In a linear function f, each time x increases by one unit, the value of f(x) always changes by an amount equal to m. That is, a linear function has a constant rate of change. The constant rate of change m is equal to the slope of the graph of f.

5. Constant Function
A function f represented by f(x) = b, where b is a constant (fixed number), is a constant function.

6. Rate of Change of a Linear Function (2 of 2)
In our car example:
Elapsed time (hours) 1 2 3 4 5
Distance (miles) 30 90
150
210
270
330
Throughout the table, as x increases by 1 unit, y increases by 60 units. That is, the rate of change or the slope is 60.

7. Slope of Line as a Rate of Change
The slope m of the line passing through the points (x1, y1) and (x2, y2) is

8. Positive Slope
If the slope of a line is positive, the line rises from left to right. Slope 2 indicates that the line rises 2 units for every unit increase in x.

9. Negative Slope
If the slope of a line is negative, the line falls from left to right.

10. Slope of 0
Slope 0 indicates that the line is horizontal.

11. Slope is Undefined
When x1 = x2, the line is vertical and the slope is undefined.

12. Example: Calculating the slope of a line given two points (1 of 2)
Find the slope of the line passing through the points (−2, 3) and (1, −2). Plot these points together with the line. Explain what the slope indicates about the line. Solution

13. Example: Calculating the slope of a line given two points (2 of 2)

14. Zero of a Function
Let ƒ be any function. Then any number c for which ƒ(c) = 0 is called a zero of the function ƒ.

15. Four Representations of a Linear Function f

16. Nonlinear Functions
If a function is not linear, then it is called a nonlinear function.
The following are characteristics of a nonlinear function:
Graph is not a (straight) line.
Does not have a constant rate of change.
Cannot be written as ƒ(x) = mx + b.
Can have any number of zeros.

17. Graphs of Nonlinear Functions (1 of 2)
There are many nonlinear functions.

18. Graphs of Nonlinear Functions (2 of 2)
Here are two other common nonlinear functions:

19. Rock Music’s Share of All U.S. Sales (Percentage)

20. Increasing and Decreasing Functions (1 of 2)

21. Increasing and Decreasing Functions (2 of 2)
Suppose that a function f is defined over an interval I on the number line. If x1 and x2 are in I, a. f increases on I if, whenever x1 < x2, f(x1) < f(x2); b. f decreases on I if, whenever x1 < x2, f(x1) > f(x2).

22. Example: Determining where a function is increasing or decreasing

23. Average Rate of Change (1 of 2)
Graphs of nonlinear functions are not straight lines, so we speak of average rate of change.
The line L is referred to as the secant line, and the slope of L represents the average rate of change of f from x1 to x2. Different values of x1 and x2 usually yield a different secant line and a different average rate of change.

24. Average Rate of Change (2 of 2)
Let (x1, y1) and (x2, y2) be distinct points on the graph of a function f. The average rate of change of f from x1 to x2 is
That is, the average rate of change from x1 to x2 equals the slope of the line passing through (x1, y1) and (x2, y2).

25. Example: Finding an average rate of change (1 of 2)
Let f(x) = 2x². Find the average rate of change from x = 1 to x = 3. Solution Calculate f(1) and f(3) f(1) = 2(1)² = 2 f(3) = 2(3)² = 18 The average rate of change equals the slope of the line passing through the points (1, 2) and (3, 18).

26. Example: Finding an average rate of change (2 of 2)
(1, 2) and (3, 18)
The average rate of change from x = 1 to x = 3 is 8.

27. Difference Quotient (1 of 2)

28. Difference Quotient (2 of 2)
The difference quotient of a function f is an expression of the form

29. Example: Finding a difference quotient (1 of 2)
Let f(x) = 3x − 2. a. Find f(x + h) b. Find the difference quotient of f and simplify the result. Solution a. To find f(x + h), substitute (x + h) for x in the expression 3x – 2.

30. Example: Finding a difference quotient (2 of 2)
b. The difference quotient can be calculated as follows: