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Pre-Calculus

Pre-Calculus . 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.

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Pre-Calculus

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  1. Pre-Calculus 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.

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

  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. • Write the total cost C as a function of x the number of units produced. • Write the profit P as a function of x. • How many units need to be sold for the company to be profitable?

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

  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.

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

  8. Increasing, Decreasing, and Constant Functions • A function is increasing on an interval if, for any x1 and x2 in the interval, x1 < x2 implies f (x1) < f (x2). • A function is decreasing on an interval if, for any x1 and x2 in the interval, x1 < x2 implies f (x1) > f (x2). • A function is constant on an interval if, for any x1 and x2 in the interval, f (x1) = f (x2).

  9. Picture = 103 Words

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

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

  12. Relative Minimum and Relative Maximum • A function value f (a) is a relative minimum of f if there exists an interval (x1, x2) that contains a such that x1 < x < x2 implies f (a) ≤ f (x). • A function value f (a) is a relative maximum of f if there exists an interval (x1,x2) that contains a such that x1 < x < x2 implies f (a) ≥ f (x).

  13. Picture = 103 More Words

  14. Example 4 • Use your graphing calculator to approximate the relative minimum of the function given by: f (x) = –x3 + x.

  15. Example 5 • During a 24-hour period, the temperature y (in °F) of a certain city can be approximated by the model y = 0.0026x3 – 1.03x2 + 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.

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

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