Chapter 1 Introduction to Functions and Graphs

1. Chapter 1 Introduction to Functions and Graphs

2. Types of Functions and Their Rates of Change 1.4 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. Linear Function A function f represented by f(x) = mx + b, where m and b are constants, is a linear function.

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

5. 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 functionf(x) = 60x + 30and has the graph:

6. Rate of Change of a Linear Function 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.

7. Rate of Change of a Linear Function In our car example: 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.

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

9. 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.

10. Negative Slope If the slope of a line is negative, the line falls from left to right. Slope –1/2 indicates that the line falls 1/2 unit for every unit increase in x.

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

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

13. Example: Calculating the slope of a line Find the slope of the line passing through the points (2, 3) and (1, –2). Plot these points together with the line. Interpret the slope. Solution

14. Example: Calculating the slope of a line The slope –5/3 indicates that the line falls 5/3 units for each unit increase in x, or equivalently, the line falls 5 units for each 3-unit increase in x.

15. Nonlinear Functions If a function is not linear, then it is called a nonlinear function. • The graph of a nonlinear function is not a (straight) line. • Nonlinear functions cannot be written in the form f(x) = mx + b.

16. Graphs of Nonlinear Functions There are many nonlinear functions. Square Function Square Root Function

17. Graphs of Nonlinear Functions Here are two other common nonlinear functions: Cube Function Absolute Value Function

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

19. Graphs of Increasing and Decreasing Functions when x1 < x2, then f(x1) > f(x2) and f is decreasing when x1 < x2, then f(x1) < f(x2) and f is increasing

20. Graphs of Increasing and Decreasing Functions If could walk from left to right along the graph of an increasing function, it would be uphill. For a decreasing function, we would walk downhill.

21. Number Line Graph

22. Interval Notation A convenient notation for number line graphs is called interval notation. Instead of drawing number lines . . .

23. Closed and Open Intervals When a set includes the endpoints, the interval is a closedinterval and brackets are used. When a set does not include the endpoints, the interval is an openinterval and parentheses are used.

24. Half-open Intervals When a set includes one endpoint and not the other, the interval is a half-open and 1 bracket and 1 parenthesis is used. This represents the interval

25. Union Symbol An inequality in the form x < 1 orx > 3 indicates the set of real numbers that are either less than 1 or greater than 3. The union symbol U can be used to write this inequality in interval notation as

26. Interval Notation

27. Increasing, Decreasing, and Endpoints The concepts of increasing and decreasing apply only to intervals of the real number line and NOT to individual points. Decreasing: (–∞, 0] Increasing: [0, ∞) Do NOT say that the function f both increases and decreases at the point (0, 0).

28. Example: Determining where a function is increasing or decreasing and Use the graph of interval notation to identify where f is increasing or decreasing. Solution Decreasing: Increasing:

29. Average Rate of Change 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 Lrepresents the average rate of change fromx1 to x2. Different values of x1 and x2 usually yield different secant lines and different average rates of change.

30. Average Rate of Change 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).

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

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

33. The difference quotient of a function f is an expression of the form where h ≠ 0. The Difference Quotient

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

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