Download presentation

1
**Graphs of Rational Functions**

2.7 Copyright © Cengage Learning. All rights reserved.

2
**What You Should Learn Analyze and sketch graphs of rational functions.**

Sketch graphs of rational functions that have slant asymptotes.

3
**The Graph of a Rational Functions**

4
**The Graph of a Rational Functions**

To sketch the graph of a rational function, use the following guidelines.

5
**The Graph of a Rational Function**

6
**Example 2 – Sketching the Graph of a Rational Function**

Sketch the graph of by hand. Solution: y-intercept: because g(0) = x-intercepts: None because 3 0. Vertical asymptote: x = 2, zero of denominator Horizontal asymptote: y = 0, because degree of N (x) < degree of D (x)

7
**Example 2 – Solution Additional points:**

cont’d Additional points: By plotting the intercept, asymptotes, and a few additional points, you can obtain the graph shown in Figure Confirm this with a graphing utility. Figure 2.46

8
Slant Asymptotes

9
Slant Asymptotes Consider a rational function whose denominator is of degree 1 or greater. If the degree of the numerator is exactly one more than the degree of the denominator, then the graph of the function has a slant (or oblique) asymptote.

10
**Slant Asymptotes For example, the graph of**

has a slant asymptote, as shown in Figure To find the equation of a slant asymptote, use long division. Figure 2.50

11
**Slant Asymptotes For instance, by dividing x + 1 into x2 – x, you have**

f (x) = x – 2 + As x increases or decreases without bound, the remainder term approaches 0, so the graph of approaches the line y = x – 2, as shown in Figure 2.50 Slant asymptote ( y = x 2)

12
**Example 6 – A Rational Function with a Slant Asymptote**

Sketch the graph of Solution: First write f (x) in two different ways. Factoring the numerator enables you to recognize the x-intercepts.

13
**Example 6 – A Rational Function with a Slant Asymptote**

Long division enables you to recognize that the line y = x is a slant asymptote of the graph.

14
**Example 6 – Solution y-intercept: (0, 2), because f (0) = 2**

cont’d y-intercept: (0, 2), because f (0) = 2 x-intercepts: (–1, 0) and (2, 0) Vertical asymptote: x = 1, zero of denominator Horizontal asymptote: None, because degree of N (x) > degree of D (x) Additional points:

15
**Example 6 – Solution The graph is shown in Figure 2.51. cont’d**

Similar presentations

OK

Bellwork 1.Identify any vertical and horizontal asymptotes, or holes in the graphs of the following functions. 2. Write a polynomial function with least.

Bellwork 1.Identify any vertical and horizontal asymptotes, or holes in the graphs of the following functions. 2. Write a polynomial function with least.

© 2018 SlidePlayer.com Inc.

All rights reserved.

Ads by Google

Ppt on video teleconferencing companies Ppt on minimum wages act 2014 Teamwork for kids ppt on batteries Ppt on radio station Ppt on conservation of momentum example Ppt on fourth and fifth state of matter Ppt on wireless networking technology Ppt on area of parallelogram and triangles geometry Ppt on the history of space flight Pdf to ppt online