Warm Up Find a polynomial function with integer coefficient that has the given zero. Find the domain...
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![Page 1: Warm Up Find a polynomial function with integer coefficient that has the given zero. Find the domain of:](https://reader030.fdocuments.in/reader030/viewer/2022032415/56649efe5503460f94c122f4/html5/thumbnails/1.jpg)
Warm Up
Find a polynomial function with integer coefficient that has the given zero.
Find the domain of:
ii 4,4,2
f ( x) 2x 1
x2 4
![Page 2: Warm Up Find a polynomial function with integer coefficient that has the given zero. Find the domain of:](https://reader030.fdocuments.in/reader030/viewer/2022032415/56649efe5503460f94c122f4/html5/thumbnails/2.jpg)
Announcements
Assignment
◦p. 278 ◦# 3-12, 23, 26◦Study Guide
Notebook Quiz WednesdayReview Session Tuesday after school
![Page 3: Warm Up Find a polynomial function with integer coefficient that has the given zero. Find the domain of:](https://reader030.fdocuments.in/reader030/viewer/2022032415/56649efe5503460f94c122f4/html5/thumbnails/3.jpg)
2.7 Rational FunctionsHow to find the domains of rational functions
How to find horizontal and vertical asymptotes of graphs of rational functions
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Introduction to Rational Functions
A rational function is a function of the form f(x) = N(x)/D(x), where N and D are both polynomials. The domain of f is all x such that D(x) 0.◦Example:
1
12)(
x
xxf
![Page 5: Warm Up Find a polynomial function with integer coefficient that has the given zero. Find the domain of:](https://reader030.fdocuments.in/reader030/viewer/2022032415/56649efe5503460f94c122f4/html5/thumbnails/5.jpg)
Example 1.
Find the domain of
All Reals ≠ -2, 2
f ( x) 2x 1
x2 4x2 4 0
x 2 x 2 0
x 2 0 x 2
x 2 0 x 2
![Page 6: Warm Up Find a polynomial function with integer coefficient that has the given zero. Find the domain of:](https://reader030.fdocuments.in/reader030/viewer/2022032415/56649efe5503460f94c122f4/html5/thumbnails/6.jpg)
Horizontal and Vertical Asymptotes
1
12
x
xy
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Vertical Asymptote
X – values where there are no y – valuesFind vertical asymptotes by finding the
domain
1
01
x
x1
12
x
xy
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Horizontal asymptotes
The graph of f has one horizontal asymptote or no horizontal asymptote, depending on the degree of n and m.
a. If n < m, then y = 0 is the horizontal asymptote of the graph of f.
b. If n = m, then y = an/bm is the horizontal asymptote of the graph of f.
c. If n > m, then there is no horizontal asymptote of the graph of f.
...
...)(
mm
nn
xb
xaxf
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Hint, hint, note, note
Graphs CAN touch a horizontal asymptote
Graphs CAN’T touch a vertical asymptote
12 x
x
1
12
x
x
64
52
x
x
Example Example
Example
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Horizontal Asymptote
a. If n < m, then y = 0 is the horizontal asymptote of the graph of f.
xx
xy
11
0
m
n
x
xxf )(
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Horizontal Asymptote
b. If n = m, then y = an/bm is the horizontal asymptote of the graph of f.
1
12
x
xy
1
2
m
n
b
ay
...
...)(
mm
nn
xb
xaxf
![Page 12: Warm Up Find a polynomial function with integer coefficient that has the given zero. Find the domain of:](https://reader030.fdocuments.in/reader030/viewer/2022032415/56649efe5503460f94c122f4/html5/thumbnails/12.jpg)
Horizontal Asymptote
c. If n > m, then there is no horizontal asymptote of the graph of f.
53
22
3
x
xy
m
n
x
xxf )(
![Page 13: Warm Up Find a polynomial function with integer coefficient that has the given zero. Find the domain of:](https://reader030.fdocuments.in/reader030/viewer/2022032415/56649efe5503460f94c122f4/html5/thumbnails/13.jpg)
Find any horizontal and vertical asymptotes of the following.
The horizontal asymptote is at y= 1/2, and the vertical asymptote is at x = 3/2.
g( x) 2x 5
4x 6
What x-values will make the function undefined?
What is the relationship between the highest powers in the numerator and denominator?
![Page 14: Warm Up Find a polynomial function with integer coefficient that has the given zero. Find the domain of:](https://reader030.fdocuments.in/reader030/viewer/2022032415/56649efe5503460f94c122f4/html5/thumbnails/14.jpg)
Find any horizontal and vertical asymptotes of the following.
No horizontal asymptote and a vertical asymptote at x = -1
h(x) x2
x 1 What x-values will make the function undefined?
What is the relationship between the highest powers in the numerator and denominator?
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Domain of a rational function
To find the domain of a rational function of x, . .
set the denominator of the rational function equal to zero and solve for x. These values of x must be excluded from the domain of the function.
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Warm Up
Find the domain of the function and identify any horizontal and vertical asymptotes.
1)(
2
3
x
xxh
![Page 17: Warm Up Find a polynomial function with integer coefficient that has the given zero. Find the domain of:](https://reader030.fdocuments.in/reader030/viewer/2022032415/56649efe5503460f94c122f4/html5/thumbnails/17.jpg)
Announcements
Assignment
◦p. 281◦# 69 – 74◦Study Guide
Notebook Quiz tomorrowReview Session today after school
![Page 18: Warm Up Find a polynomial function with integer coefficient that has the given zero. Find the domain of:](https://reader030.fdocuments.in/reader030/viewer/2022032415/56649efe5503460f94c122f4/html5/thumbnails/18.jpg)
Objectives
How to analyze and sketch graphs of rational functions
How to sketch graphs of rational functions that have slant asymptotes
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Steps for finding the Graph of a Rational Functions
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1st Guideline for graphing rational functions
1. Find and plot the y-intercept (if any) by evaluating f(0)
2
3
20
3)0(
2
3)(
f
xxf
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2nd Guideline for graphing rational functions
1. Find the zeros of the numerator (if any) by setting the numerator = 0. Then plot them as x – intercepts
2
1
12
012
12)(
x
x
xx
xxf
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3rd Guideline for graphing rational functions
1. Find the zeros of the denominator (if any) by setting the denominator = 0. Then sketch the corresponding vertical asymptotes
2
022
3)(
x
xx
xf
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4th Guideline for graphing rational functions
1. Find and sketch the horizontal asymptote (if any) by using the rules for finding the horizontal asymptote
21
22
12)(
x
x
x
x
x
xxf
m
n
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5th Guideline for graphing rational functions
1. Plot at least one point between and at least one point beyond each x intercept and vertical asymptote
x
xxf
12)(
X Y
-1 3
(1/4) -2
1 1
X – int. = (1/2)
Vert. Asym. = 0
![Page 25: Warm Up Find a polynomial function with integer coefficient that has the given zero. Find the domain of:](https://reader030.fdocuments.in/reader030/viewer/2022032415/56649efe5503460f94c122f4/html5/thumbnails/25.jpg)
6th Guideline for graphing rational functions
1. Use smooth curves to complete the graph between and beyond the vertical asymptotes
x
xxf
12)(
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Example 1. Sketch the graph of the following function.
y-Intercept: Nonex-Intercept: (-1, 0)Vertical asymptote: x = 0Horizontal asymptote: y = 1Additional points: (-2, 0.5), (-1.5, 1/3),
(1, 2)
f ( x) x 1
x
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Sketch the graph of each of the following functions.
y-Intercept: (0, 0)x-Intercept: (0, 0)Vertical asymptote: noneHorizontal asymptote: y = 0Additional points: (-2,-0.4), (-1, -1/2), (1,
1/2)
h(x) x
x2 1
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Slant Asymptotes
y = -3x – 3Is our slant asymptote
If n is exactly one more than m, then the graph of f has a slant asymptote at y = q(x), where q(x) is the quotient from the division algorithm.
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Decide whether each of the following rational functions has a slant asymptote. If so, find the equation of the slant asymptote.
(a) Yes, y = x 3 (b) No
53
1)(
2
3
xx
xxf 52
23)(
3
x
xxf
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Example 2. Sketch the graph of
y-Intercept: (0, 0)x-Intercept: (0, 0)Vertical asymptote: x = 2Slant asymptote: y = x + 2
Additional points: (-1/2,-0.1), (1, -1), (3, 9)
y x 2
x 2x 2
4
x 2
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Slant Asymptotes
If n is exactly one more than m, then the graph of f has a slant asymptote at y = q(x), where q(x) is the quotient from the division algorithm.
2
125)(
x
xxf
.5
12..............
105.......
25.......
2
232
2
23)(
2
2
2
x
x
x
xx
xxx
x
xxxf
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Sketch the graph of each of the following functions.
y-Intercept: (0, -0.25)x-Intercept: (2, 0)Vertical asymptote: x = -2 and x = 4Horizontal asymptote: y = 0Additional points: (-4, -0.375), (0, 1/4),
(3, -1/5), (6, 1/4)
g( x) x 2
x2 2x 8
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Example 2. Find any horizontal and vertical asymptotes of the following.
The horizontal asymptote is y = 0. The only vertical asymptote is x = 1. There will be a hole in the graph at x = -1.
f ( x) x 1
x2 1
![Page 34: Warm Up Find a polynomial function with integer coefficient that has the given zero. Find the domain of:](https://reader030.fdocuments.in/reader030/viewer/2022032415/56649efe5503460f94c122f4/html5/thumbnails/34.jpg)
1st Guideline for graphing rational functions
2
3
20
3)0(
2
3)(
f
xxf
![Page 35: Warm Up Find a polynomial function with integer coefficient that has the given zero. Find the domain of:](https://reader030.fdocuments.in/reader030/viewer/2022032415/56649efe5503460f94c122f4/html5/thumbnails/35.jpg)
2nd Guideline for graphing rational functions
2
1
12
012
12)(
x
x
xx
xxf
![Page 36: Warm Up Find a polynomial function with integer coefficient that has the given zero. Find the domain of:](https://reader030.fdocuments.in/reader030/viewer/2022032415/56649efe5503460f94c122f4/html5/thumbnails/36.jpg)
3rd Guideline for graphing rational functions
2
022
3)(
x
xx
xf
![Page 37: Warm Up Find a polynomial function with integer coefficient that has the given zero. Find the domain of:](https://reader030.fdocuments.in/reader030/viewer/2022032415/56649efe5503460f94c122f4/html5/thumbnails/37.jpg)
4th Guideline for graphing rational functions
21
22
12)(
x
x
x
x
x
xxf
m
n
![Page 38: Warm Up Find a polynomial function with integer coefficient that has the given zero. Find the domain of:](https://reader030.fdocuments.in/reader030/viewer/2022032415/56649efe5503460f94c122f4/html5/thumbnails/38.jpg)
5th Guideline for graphing rational functions
x
xxf
12)(
X Y
-1 3
(1/4) -2
1 1
X – int. = (1/2)
Vert. Asym. = 0
![Page 39: Warm Up Find a polynomial function with integer coefficient that has the given zero. Find the domain of:](https://reader030.fdocuments.in/reader030/viewer/2022032415/56649efe5503460f94c122f4/html5/thumbnails/39.jpg)
6th Guideline for graphing rational functions
x
xxf
12)(