Chapter 1 Functions and Their Graphs

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

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

Warm Up 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

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)

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

Example 2 Function or not? a. b. 7

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

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

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

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

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

Example 5 During a 24-hour period, the temperature y (in °F) of a certain city can be approximated by the model y = x 3 – 1.03x x + 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

Homework Worksheet  # 1 – 33 odd 16

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