1 / 19

PreCalculus

PreCalculus. 2.6 – Rational Functions. Example 1 – Finding the Domain of a Rational Function. Find the domain of f(x) = 1/x and discuss the behavior of f near any excluded x-values. Horizontal and Vertical Asymptotes.

filia
Download Presentation

PreCalculus

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. PreCalculus 2.6 – Rational Functions

  2. Example 1 – Finding the Domain of a Rational Function Find the domain of f(x) = 1/x and discuss the behavior of f near any excluded x-values.

  3. Horizontal and Vertical Asymptotes • Asymptotes – A line that the graph of a function approaches more and more closely without ever touching.

  4. Asymptotes of a Rational Function • The graph of f has vertical asymptotes at the zeros of the denominator of the function. • The graph of f has one or no horizontal asymptote determined by comparing the degrees of the numerator and denominator. • If the degree of the numerator is less than the denominator, the graph of f has the line y = 0 (the x-axis) as a horizontal asymptote.

  5. If the degree of the numerator is equal to the denominator, the graph of f has the line resulting in dividing the leading term of the numerator by the leading term of the denominator as the horizontal asymptote. • If the degree of the numerator is greater than the degree of the denominator, the graph of f has no horizontal asymptote.

  6. Example 2 – Finding Horizontal and Vertical Asymptotes • Find all vertical and horizontal asymptotes of the graph of each rational function. A) f (x) = B) f (x) =

  7. Example 3 – Finding Horizontal and Vertical Asymptotes • Find all vertical and horizontal asymptotes of the graph of each rational function. A) f (x) = B) f (x) =

  8. Analyzing Graphs of Rational Functions • Let f(x) = g(x) / h(x) , where g(x) and h(x) are polynomials with no common factors. • Find and plot the y-intercept (if any) by evaluating f(0) • Find the zeros of the numerator (if any) by solving the equation g(x) = 0. Then plot the corresponding x-intercepts. • Find the zeros of the denominator (if any) by solving the equation h(x) = 0. Then sketch the corresponding vertical asymptotes.

  9. Find and sketch the horizontal asymptote (if any) by using the rule for finding the horizontal asymptote of a rational function. • Test for symmetry • Plot at least one point between and one point beyond each x-intercept and vertical asymptote. • Use smooth curves to complete the graph between and beyond the vertical asymptotes.

  10. Example 4 – Sketching the Graph of a Rational Function Sketch the graph of g(x) = and state its domain.

  11. Example 5 – Sketching the Graph of a Rational Function Sketch the graph of f(x) =

  12. Example 6 – Sketching the Graph of a Rational Function Sketch the graph of f(x) =

  13. Example 7 – Sketching the Graph of a Rational Function Sketch the graph of f(x) =

  14. Slant Asymptotes • If the degree of the numerator is exactly one more than the degree of the denominator, the graph of the function has a slant (or oblique) asymptote. • To find the equation of a slant asymptote, use long division.

  15. Example 8 – A Rational Function with a Slant Asymptote Find any slant asymptotes for each of the rational functions below. A) f (x) = B) f (x) =

More Related