warm up find a polynomial function with integer coefficient that has the given zero. find the domain...
TRANSCRIPT
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
Announcements
Assignment
◦p. 278 ◦# 3-12, 23, 26◦Study Guide
Notebook Quiz WednesdayReview Session Tuesday after school
2.7 Rational FunctionsHow to find the domains of rational functions
How to find horizontal and vertical asymptotes of graphs of rational functions
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
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
Horizontal and Vertical Asymptotes
1
12
x
xy
Vertical Asymptote
X – values where there are no y – valuesFind vertical asymptotes by finding the
domain
1
01
x
x1
12
x
xy
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
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
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 )(
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
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 )(
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?
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?
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.
Warm Up
Find the domain of the function and identify any horizontal and vertical asymptotes.
1)(
2
3
x
xxh
Announcements
Assignment
◦p. 281◦# 69 – 74◦Study Guide
Notebook Quiz tomorrowReview Session today after school
Objectives
How to analyze and sketch graphs of rational functions
How to sketch graphs of rational functions that have slant asymptotes
Steps for finding the Graph of a Rational Functions
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
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
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
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
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
6th Guideline for graphing rational functions
1. Use smooth curves to complete the graph between and beyond the vertical asymptotes
x
xxf
12)(
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
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
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.
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
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
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
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
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
1st Guideline for graphing rational functions
2
3
20
3)0(
2
3)(
f
xxf
2nd Guideline for graphing rational functions
2
1
12
012
12)(
x
x
xx
xxf
3rd Guideline for graphing rational functions
2
022
3)(
x
xx
xf
4th Guideline for graphing rational functions
21
22
12)(
x
x
x
x
x
xxf
m
n
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
6th Guideline for graphing rational functions
x
xxf
12)(