Section 2.6 Rational Functions Part 1. What you should learn How to find the domains of rational...
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![Page 1: Section 2.6 Rational Functions Part 1. What you should learn How to find the domains of rational functions How to find the horizontal and vertical asymptotes.](https://reader036.fdocuments.in/reader036/viewer/2022082818/56649edf5503460f94bef548/html5/thumbnails/1.jpg)
Section 2.6 Rational FunctionsPart 1
![Page 2: Section 2.6 Rational Functions Part 1. What you should learn How to find the domains of rational functions How to find the horizontal and vertical asymptotes.](https://reader036.fdocuments.in/reader036/viewer/2022082818/56649edf5503460f94bef548/html5/thumbnails/2.jpg)
What you should learn
• How to find the domains of rational functions• How to find the horizontal and vertical
asymptotes of graphs of rational functions• How to analyze and sketch graphs of rationale
functions• How to sketch graphs of rational functions that
have slant asymptotes• How to use rational functions to model real-
life problems
![Page 3: Section 2.6 Rational Functions Part 1. What you should learn How to find the domains of rational functions How to find the horizontal and vertical asymptotes.](https://reader036.fdocuments.in/reader036/viewer/2022082818/56649edf5503460f94bef548/html5/thumbnails/3.jpg)
Rational Function
A rational function can be written in the form
Where N(x) and D(x) are polynomials.
D(x) is not the zero polynomial.
)(
)()(
xD
xNxf
![Page 4: Section 2.6 Rational Functions Part 1. What you should learn How to find the domains of rational functions How to find the horizontal and vertical asymptotes.](https://reader036.fdocuments.in/reader036/viewer/2022082818/56649edf5503460f94bef548/html5/thumbnails/4.jpg)
Find the domain of f(x)
158
1)(
2
xxxf
1580 2 xx)5)(3(0 xx
}5,3{ x
}5,3,|{ xxx
![Page 5: Section 2.6 Rational Functions Part 1. What you should learn How to find the domains of rational functions How to find the horizontal and vertical asymptotes.](https://reader036.fdocuments.in/reader036/viewer/2022082818/56649edf5503460f94bef548/html5/thumbnails/5.jpg)
Find the domain of g(x)
2
1)(
xxg
20 x}2{x
}2,|{ xxx
2
-2
5
Horizontal Asymptote at y = 0 when degree of numerator less than degree of the denominator.
Vertical Asymptote at the zero of the denominator.
![Page 6: Section 2.6 Rational Functions Part 1. What you should learn How to find the domains of rational functions How to find the horizontal and vertical asymptotes.](https://reader036.fdocuments.in/reader036/viewer/2022082818/56649edf5503460f94bef548/html5/thumbnails/6.jpg)
Asymptotes
• Vertical at zeros of the denominator
• HorizontalN < D asymptote at y = 0N = D asymptote at y = aN / aD
N > D no Horizontal asymptote
)(
)(
rdenominato degree
numerator degree
xD
xN
![Page 7: Section 2.6 Rational Functions Part 1. What you should learn How to find the domains of rational functions How to find the horizontal and vertical asymptotes.](https://reader036.fdocuments.in/reader036/viewer/2022082818/56649edf5503460f94bef548/html5/thumbnails/7.jpg)
N = D asymptote at y = aN / aD
6
4
2
-5 5
y=f(x)
2
2 53)(
x
xxf
1
3y0x
![Page 8: Section 2.6 Rational Functions Part 1. What you should learn How to find the domains of rational functions How to find the horizontal and vertical asymptotes.](https://reader036.fdocuments.in/reader036/viewer/2022082818/56649edf5503460f94bef548/html5/thumbnails/8.jpg)
Guidelines for Analyzing the Graphs of Rational Functions
1. Find and plot the y-intercept by evaluating f(0).
2. Find the zeros of the numerator and plot the points.
3. Find the zeros of the denominator and sketch the vertical asymptotes.
4. Find and sketch the horizontal asymptote if it exists.
![Page 9: Section 2.6 Rational Functions Part 1. What you should learn How to find the domains of rational functions How to find the horizontal and vertical asymptotes.](https://reader036.fdocuments.in/reader036/viewer/2022082818/56649edf5503460f94bef548/html5/thumbnails/9.jpg)
Guidelines for Analyzing the Graphs of Rational Functions
5. Test for symmetry
6. Plot at least one point between and one point beyond each x intercept and vertical asymptote.
7. Use smooth curves to complete the graph between and beyond the vertical asymptotes
![Page 10: Section 2.6 Rational Functions Part 1. What you should learn How to find the domains of rational functions How to find the horizontal and vertical asymptotes.](https://reader036.fdocuments.in/reader036/viewer/2022082818/56649edf5503460f94bef548/html5/thumbnails/10.jpg)
Graph
2
-2
-4
-5 5
2
2 4)(
x
xxf
1. y-intercept when x = 0
f(0) = undefined
x = 0 is a vertical asymptote
![Page 11: Section 2.6 Rational Functions Part 1. What you should learn How to find the domains of rational functions How to find the horizontal and vertical asymptotes.](https://reader036.fdocuments.in/reader036/viewer/2022082818/56649edf5503460f94bef548/html5/thumbnails/11.jpg)
Graph
2
-2
-4
-5 5
2
2 4)(
x
xxf
2. Find the zeros of the numerator
0 = x2 – 4
x = {-2, 2}
= (x + 2)(x – 2)
![Page 12: Section 2.6 Rational Functions Part 1. What you should learn How to find the domains of rational functions How to find the horizontal and vertical asymptotes.](https://reader036.fdocuments.in/reader036/viewer/2022082818/56649edf5503460f94bef548/html5/thumbnails/12.jpg)
Graph
2
-2
-4
-5 5
2
2 4)(
x
xxf
3. Find the zeros of the Denominator
0 = x2 x = {0}
We already found this in step #1
x = 0 is a vertical asymptote
![Page 13: Section 2.6 Rational Functions Part 1. What you should learn How to find the domains of rational functions How to find the horizontal and vertical asymptotes.](https://reader036.fdocuments.in/reader036/viewer/2022082818/56649edf5503460f94bef548/html5/thumbnails/13.jpg)
Graph
2
-2
-4
-5 5
2
2 4)(
x
xxf
4. Find and sketch the horizontal asymptote
Since the degree of the numerator and the denominator are the same the horizontal asymptote will be y = 1/1=1
y = 1
![Page 14: Section 2.6 Rational Functions Part 1. What you should learn How to find the domains of rational functions How to find the horizontal and vertical asymptotes.](https://reader036.fdocuments.in/reader036/viewer/2022082818/56649edf5503460f94bef548/html5/thumbnails/14.jpg)
Graph
2
-2
-4
-5 5
2
2 4)(
x
xxf
5. Test for Symmetry
Since f(x)= f(-x) we know this drawing will be symmetrical about the y-axis
![Page 15: Section 2.6 Rational Functions Part 1. What you should learn How to find the domains of rational functions How to find the horizontal and vertical asymptotes.](https://reader036.fdocuments.in/reader036/viewer/2022082818/56649edf5503460f94bef548/html5/thumbnails/15.jpg)
Graph
2
-2
-4
-5 5
2
2 4)(
x
xxf
6. Plot some points
f(1)= f(-1) = -3
f(3)= f(-3) = 5/9
![Page 16: Section 2.6 Rational Functions Part 1. What you should learn How to find the domains of rational functions How to find the horizontal and vertical asymptotes.](https://reader036.fdocuments.in/reader036/viewer/2022082818/56649edf5503460f94bef548/html5/thumbnails/16.jpg)
Graph
2
-2
-4
-5 5
2
2 4)(
x
xxf
7. Use smooth curves to complete the graph.
2
-2
-4
-5 5
y=f(x)
![Page 17: Section 2.6 Rational Functions Part 1. What you should learn How to find the domains of rational functions How to find the horizontal and vertical asymptotes.](https://reader036.fdocuments.in/reader036/viewer/2022082818/56649edf5503460f94bef548/html5/thumbnails/17.jpg)
2.6 Homework Part 1
• Rational Functions
• page 174
• 5 - 39 odd,