1 / 20

SAT Question of the Day

SAT Question of the Day. 8.2 Rational Functions and Their Graphs. Objectives: Identify and evaluate rational functions Graph a rational function, find its domain, write equations for its asymptotes, and identify any holes in its graph. acid. solution. add x milliliters of distilled water.

lawton
Download Presentation

SAT Question of the Day

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. SAT Question of the Day

  2. 8.2 Rational Functions and Their Graphs Objectives: Identify and evaluate rational functions Graph a rational function, find its domain, write equations for its asymptotes, and identify any holes in its graph

  3. acid solution add x milliliters of distilled water Example 1 William begins with 75 milliliters of a 15% acid solution. He adds x milliliters of distilled water to the container holding the acid solution. a) Write a function, C, that represents the acid concentration of the solution in terms of x. 15% of 75 = 11.25

  4. Example 1 William begins with 75 milliliters of a 15% acid solution. He adds x milliliters of distilled water to the container holding the acid solution. b) What is the acid concentration of the solution if 35 milliliters of distilled water is added?

  5. Example 2 Find the domain of Find the values of x for which the denominator equals 0. x2 – 9x – 36 = 0 (x – 12)(x + 3) = 0 x = 12 or -3 The domain is all real numbers except 12 and -3.

  6. Vertical Asymptote In a rational function R, if x – a is a factor of the denominator but not a factor of the numerator, x = a is vertical asymptote of the graph of R. Hole In a rational function R, if x – a is a factor of the denominator AND a factor of the numerator, x = a is a hole in the graph of R.

  7. Example 3 Identify all vertical asymptotes of Factor the denominator. Equations for the vertical asymptotes are x = 2 and x = 1.

  8. R(x) = is a rational function; P and Q are polynomials P Q Horizontal Asymptote • If degree of P < degree of Q, thenthe horizontal asymptote of R is y = 0.

  9. R(x) = is a rational function; P and Q are polynomials P a Q b • If degree of P = degree of Q and a and b are the leading coefficients of P and Q, thenthe horizontal asymptote of R is y = . Horizontal Asymptote

  10. R(x) = is a rational function; P and Q are polynomials P Q Horizontal Asymptote • If degree of P > degree of Q, thenthere is no horizontal asymptote

  11. Homework Lesson 8.2 exercises 11-39 odd

  12. SAT Problem of the Day

  13. 8.2.2 Rational Functions and Their Graphs Objectives: Identify and evaluate rational functions Graph a rational function, find its domain, write equations for its asymptotes, and identify any holes in its graph

  14. Example 1 Let . Identify all vertical asymptotes and all horizontal asymptotes. Equations for the vertical asymptotes are x = -5 and x = 4. Because the degree of the numerator is greater than the degree of the denominator, the graph has no horizontal asymptotes.

  15. Example 2 Let . Identify all vertical asymptotes and all horizontal asymptotes. 1 Vertical asymptotes: x = -3 and x = 3 Horizontal asymptotes: numerator and denominator have the same degree leading coefficients y = 2

  16. Holes in Graphs In a rational function R, if x – b is a factor of the numerator and the denominator, there is a hole in the graph of R when x = b (unless x = b is a vertical asymptote).

  17. f(x) = 2x2 + 2x x2 – 1 2x(x + 1) f(x) = (x + 1)(x – 1) Example 3 Identify all asymptotes and holes in the graph of the rational function. factor: hole in the graph:x = –1 vertical asymptote:x = 1 horizontal asymptote:y = 2

  18. Example 4 Use asymptotes to graph the rational function. Write equations for the asymptotes, and graph them as dashed lines. horizontal asymptote:y = 2 vertical asymptote:x = -4 Use a table to help obtain an accurate plot.

  19. Collins Type 2 Explain how to use the asymptotes of the graph of g(x) = x - 5 x - 3 to sketch the graph of the function.

  20. Homework Worksheet “More problems on rational function graphing”

More Related