1. Chapter 1 Functions and Their Graphs

2. 1.3.1 Graphs of Functions Objectives:  Find the domains and ranges of functions & use the Vertical Line Test for functions.  Determine intervals on which functions are increasing, decreasing, or constant.  Determine relative maximum and relative minimum values of functions. 2

3. Vocabulary Vertical Line Test Increasing, Decreasing, and Constant Functions Relative Minimum and Relative Maximum 3

4. Warm Up 1.3.1 A hand tool manufacturer produces a product for which the variable cost is $5.35 per unit and the fixed costs are $16,000. The company sells the product for $8.20 and can sell all that it produces. a. Write the total cost C as a function of x the number of units produced. b. Write the profit P as a function of x. c. How many units need to be sold for the company to be profitable? 4

5. Example 1 Use the graph of f to find: a. The domain of f. b. The function values f (–1) and f (2). c. The range of f. 5 (-1, -5) (2, 4) (4, 0)

6. How Do We Know It’s a Function? Vertical Line Test If any vertical line cuts the graph of a relation in more than one place, then the relation is not a function. 6

7. Example 2 Function or not? a. b. 7

8. Increasing, Decreasing, and Constant Functions A function is increasing on an interval if, for any x 1 and x 2 in the interval, x 1 < x 2 implies f (x 1 ) < f (x 2 ). A function is decreasing on an interval if, for any x 1 and x 2 in the interval, x 1 f (x 2 ). A function is constant on an interval if, for any x 1 and x 2 in the interval, f (x 1 ) = f (x 2 ). 8

9. Picture = 10 3 Words 9

10. Example 3a Determine where the function is increasing, decreasing, or constant. 10

11. Example 3b Determine where the function is increasing, decreasing, or constant. 11

12. Relative Minimum and Relative Maximum A function value f (a) is a relative minimum of f if there exists an interval (x 1, x 2 ) that contains a such that x 1 < x < x 2 implies f (a) ≤ f (x). A function value f (a) is a relative maximum of f if there exists an interval (x 1, x 2 ) that contains a such that x 1 < x < x 2 implies f (a) ≥ f (x). 12

13. Picture = 10 3 More Words 13

14. Example 4 Use your graphing calculator to approximate the relative minimum of the function given by: f (x) = –x 3 + x. 14

15. Example 5 During a 24-hour period, the temperature y (in °F) of a certain city can be approximated by the model y = 0.0026x 3 – 1.03x 2 + 10.2x + 34, 0 ≤ x ≤ 24 where x represents the time of day, with x = 0 corresponding to 6 A.M. Approximate the maximum and minimum temperatures during this 24-hour period. 15

16. Homework 1.3.1 Worksheet 1.3.1  # 1 – 33 odd 16

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