Download presentation

Presentation is loading. Please wait.

Published byJaxson jay Edgar Modified over 2 years ago

1
1 Graph Sketching: Asymptotes and Rational Functions OBJECTIVES Find limits involving infinity. Determine the asymptotes of a function’s graph.

2
2 A rational function is a function f that can be described by where P(x) and Q(x) are polynomials, with Q(x) not the zero polynomial. The domain of f consists of all inputs x for which Q(x) ≠ 0. DEFINITION:

3
3 ASYMPTOTES A vertical asymptote is a vertical line that a function approaches, but never reaches. If a is a value for x that makes the denominator = 0, the line x = a is a vertical asymptote. The graph will never cross a vertical asymptote

4
4 The line x = a is a vertical asymptote if any of the following limit statements are true: x = a ∞ ∞ -∞

5
5 ASYMPTOTES A horizontal asymptote behaves differently. We determine what happens to the function as x approaches positive or negative infinity. The graph of the function can cross the horizontal asymptote. y = 0

6
6 EXAMPLE 1: Graph the function Does not exist x-intercepty-intercept does not exist There is a vertical asymptote at x = 1

7
7 EXAMPLE 1: Graph the function Now lets look at the one-sided limits around x = 1 x 1.00011.0011.01 1.2 1.523 y 30 0003000300 15 631.5 x -300.50.90.990.999 y -0.75-1.5-3-6-30-300-3 000 1 -∞-∞ 1 ∞

8
8 EXAMPLE 1: Graph the function There is a horizontal asymptote at y = 0 x = 1 y = 0 (0, -3) ∞ -∞

9
9 EXAMPLE 2: Graph the function x-intercepty-intercept

10
10 EXAMPLE 2: Graph the function does not exist There is a vertical asymptote at x = 3 x = 3

11
11 Now lets look at the one-sided limits around x = 3 ∞ -∞

12
12 Look for Horizontal Asymptotes There is a horizontal asymptote at y = 2 x = 3 y = 2 -∞ ∞

13
13 EXAMPLE 2: Graph the function x = 3 y = 2 -∞ ∞

14
14 EXAMPLE 3: Graph the function x- intercepty-intercept Domain: x ≠ -1, 3

15
15 EXAMPLE 3: cont There is a 'hole' at (-1, ¾ )

16
16 EXAMPLE 3: cont DNE There is a vertical asymptote at x = 3 x = 3

17
17 f(3.001) = 1001 ∞ f(2.999) = -999 - ∞ Look at one-sided limits around x = 3 From the left From the right x = 3 -∞ ∞

18
18 Is there a horizontal asymptote? There is a horizontal asymptote at y = 1 x = 3 -∞ ∞ y = 1

19
19 (2, 0) y = 1 x = 3 ∞ - ∞

Similar presentations

© 2017 SlidePlayer.com Inc.

All rights reserved.

Ads by Google