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2.3 Polynomial and Rational Functions

2.3 Polynomial and Rational Functions. Identify a Polynomial Function Identify a Rational Function Find Vertical and Horizontal Asymptotes for Rational Functions Review finding x and y intercepts of graphs.

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2.3 Polynomial and Rational Functions

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  1. 2.3 Polynomial and Rational Functions • Identify a Polynomial Function • Identify a Rational Function • Find Vertical and Horizontal Asymptotes for Rational Functions • Review finding x and y intercepts of graphs

  2. Polynomial and rational functions are often used to express relationships in application problems.

  3. Scary Math Definition for Polynomial

  4. Be sure to know the end behavior properties (2 and 3 below).

  5. Scary Math Definition for Rational Function

  6. Go forward a few slides to see the easy-to-understand explanation.

  7. DEFINITION: • The line x = a is a vertical asymptote if any of the following limit statements are true: • We will learn about limits in section 3.1

  8. If c makes the denominator zero, but doesn’t make the numerator zero, then x = c is a vertical asymptote. • If c makes both the denominator and the numerator zero, then there is a hole at x=c

  9. Example of hole

  10. Example 2: Determine the vertical asymptotes of the function given by

  11. Example 2 There are Vertical Asymptotes at x = 1 and x = -1. There isn‘t a vertical asymptote at x = 0. Since 0 makes both the numerator and denominator equal zero, there is a hole where x = 0.

  12. Since x = 1 and x = –1 make the denominator 0, but don’t make the numerator 0, x = 1 and x = –1 are vertical asymptotes. • x=0 is not a vertical asymptote since it makes both the numerator and denominator 0.

  13. The line y = b is a horizontal asymptote if either or both of the following limit statements are true: or We will learn about limits in section 3.1.

  14. The graph of a rational function may or may not cross a horizontal asymptote. Horizontal asymptotes are found by comparing the degree of the numerator to the degree of the denominator. 3 cases Same: y = leading coefficient/leading coefficient BOB: y = 0 (bottom degreebigger) TUB: undefined-no H.A. (top degreebigger Bob and tub are not in the textbook.

  15. Determine the horizontal asymptote of the function given by

  16. Example of vertical and horizontal asymptotes

  17. Intercepts • The x-intercepts occur at values for which y = 0. For a fraction to = 0, the numerator must equal 0. Since 8 ≠ 0, there are no x-intercepts. • To find the y-intercept, let x = 0. y-intercept (0, 8/5)

  18. Suppose the average cost per unitin dollars, to produce x units of a product is given by (a) find (b) How much would 10 units cost? (c) Identify any intercepts & asymptotes. Graph the function to verify your answers.

  19. $12.50, $6.25, $3.85 • $12.50 x 10 = $125.00 • V.A. x = -30 H.A. y = 0 no x-intercepts y-intercept (0, 50/3)

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