1 / 12

Rational Graphs

Rational Graphs. Today we are going to look further at the behavior of the graphs of different functions. Now we are going to determine if the graphs have horizontal and/or vertical asymptotes. Vertical Asymptotes.

tocho
Download Presentation

Rational Graphs

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 Graphs Today we are going to look further at the behavior of the graphs of different functions. Now we are going to determine if the graphs have horizontal and/or vertical asymptotes.

  2. Vertical Asymptotes Imaginary vertical lines to which a graph gets closer to but usually never crosses. These are represented by dotted lines on the graph. Horizontal Asymptotes Imaginary horizontal lines to which a graph gets closer to but usually never crosses. These are represented by dotted lines on the graph.

  3. Vertical Asymptotes A vertical asymptote comes from the denominator. It exists at the value(s) that will make the denominator zero. Example: f(x)= 9 x – 2 If there are no variables in the denominator, there are no vertical asymptotes. Example: f(x) = x2 – 9 6

  4. Horizontal Asymptotes To determine horizontal asymptotes, there are 3 rules that you must memorize.

  5. Rule #1 If the degree of the numerator is < than the degree of the denominator, then the HA is y = 0. Example: f(x) = 5 x - 2 ← degree is 0 ← degree is 1 Example:  HA @ y = 0 f(x) = 2x x2 - 5 ← degree is 1 ← degree is 2  HA @ y = 0

  6. Rule #2 • If the degree of the numerator = degree of denominator, then divide the leading coefficient of the numerator by the leading coefficient of the denominator. Example: f(x) = 6x 5x + 10 ← degree is 1 ← degree is 1  HA @ y = 6/5 and VA @ x = -2

  7. Example: f(x) = x – 2 x + 2 f(x) = 2(x2 – 9) x2 – 4 ← degree is 1 ← degree is 1  HA @ y = 1 and VA @ x = -2 Example: ← degree is 2 ← degree is 2  HA @ y = 2 and VA @ x = ±2

  8. Rule #3 If the degree of the numerator  the degree of denominator , then there are no horizontal asymptotes. In this case there may be asymptotes, but they are oblique or parabolic. In calculus you will learn another way to find horizontal asymptotes when studying the limits of functions.

  9. Let’s graph some rational graphs: y = 3 x - 1 VA: x – 1 = 0 x = 1 HA: y = 0 y 0 x 1 Range: Domain:

  10. Let’s graph some rational graphs: y = 3x x + 2 VA: x + 2= 0 x = -2 HA: y = 3 y 3 x -2 Range: Domain:

  11. Let’s graph some rational graphs: y = 2 - x x - 1 VA: x - 1= 0 x = 1 HA: y = -1 y -1 x 1 Range: Domain:

  12. Let’s graph some rational graphs: y = 1 x2 + 3 VA: x2 + 3= 0 x2 = -3 none 1 HA: y = 0 -1 1 -1 y 0 (-,) Range: Domain:

More Related