1 / 18

Rational Functions and Asymptotes

Rational Functions and Asymptotes. Let’s find: vertical , horizontal, and slant asymptotes when given a rational function. Get Started. MAIN MENU. All done?. Example A. Example B. You try. What is a Rational Function?. A function that is the ratio of two polynomials. A few examples…….

topanga-fernandez
Download Presentation

Rational Functions and Asymptotes

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. Rational Functions and Asymptotes Let’s find: vertical, horizontal, and slant asymptotes when given a rational function. Get Started

  2. MAIN MENU All done? Example A Example B You try

  3. What is a Rational Function? A function that is the ratio of two polynomials. A few examples…… It is “Rational” because one is divided by the other, like a ratio. Return to Main Menu

  4. What is an asymptote? A line that a curve approaches but never reaches, or “touches” Return to main menu

  5. Vertical Asymptotes To find the vertical asymptote of a function, we must set the denominator equal to zero and solve for x. There will be a vertical asymptote, x = 2. Back to Main Menu

  6. Horizontal Asymptotes • Compare the degrees of the numerator and denominator. We will let the denominator degree be “d” and the numerator degree be “n.” There are 3 cases Case 1 n < d Case 2 n > d Case 3 n = d

  7. Case 1 If n < d, then y = 0 is a horizontal asymptote(HA) of the function. HA: y = 0 Return to main menu

  8. Case 2 If n > d, then there is no horizontal asymptote (HA). HA : none Return to main menu

  9. Case 3 If n = d, then the horizontal asymptote is the ratio of the numerator leading coefficient over the denominator leading coefficient. HA: Return to main menu

  10. Slant Asymptote If the degree of the numerator (n) is EXACTLY one greater that the degree of the denominator (d), there will be a slant asymptote. Since the degree of the numerator is 2 and the degree of the denominator is 1, there will be a slant asymptote. Return to Main Menu Lets find the slant asymptote

  11. Let’s find the slant asymptote We must divide the denominator into the numerator The line y = x – 2 is the slant asymptote of the rational function. Please note, if there is a remainder upon dividing, we discard it as it will have no affect on the rational function as x approaches infinity. Return to Main Menu

  12. Example A Find all asymptotes of the function g(x). Vertical Asymptotes Set the denominator equal to zero and solve. Horizontal Asymptotes Compare degrees…. Numerator-degree of 1 Denominator-degree of 2 Since n < d, HA: y = 0 Return to Main Menu VA: x =2, x=-2

  13. Example B Find all asymptotes of the function f(x). Vertical Asymptotes: Set the denominator equal to zero and solve Horizontal Asymptotes: Compare degrees…. Numerator-degree of 2 Denominator-degree of 2 Since n = d, HA: y = 2 Return to Main Menu VA: x = -3, x = 2

  14. You Try Find all asymptotes for A.) VA: none, HA: y = 1 B) VA: x = -1, HA: y=0 Return to Main Menu C) VA: x = 1, x = -1, HA: y = 0

  15. Hmmmm, better try again Back to choices Return to Main Menu

  16. Correct ! Back to choices Return to Main Menu

  17. All Done? Let’s Review! • To find vertical asymptotes - set the denominator equal to zero and solve. • To find horizontal asymptotes - compare the degree of the numerator to the degree of the denominator. • To find slant asymptotes - divide the numerator by the denominator. Return to Main Menu References

  18. References Retrieved 9/22/13 Dreamstime http://www.animationlibrary.com animation/22939/Yellow_dog_thinks/ Retrieved 9/22/13 Dreamstime http/www.animationlibrary.com/animation/22938/Yellow_dog_sings/ Larson, R. (2011). Algebra and Trigonometry. Belfast, CA: Cengage Asymptotes. Retrieved from Mathisfun.com Return to Main Menu

More Related