key information starting last unit today –graphing –factoring –solving equations –common...
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Key Information• Starting Last Unit Today
– Graphing– Factoring– Solving Equations– Common Denominators– Domain and Range (Interval Notation)
• Factoring will be critical next week
• Retest is this Thursday PM and Friday AM
• Must have test corrections done by then
• Factoring Quiz Tuesday. Only 2 left!!
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Warm-upWhat is the degree of each of these?
4 23 3 7 1x x x 6 4 24 7 8x x x
8 10x
12
4
6
1
0
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Section 8-2 & 8-3
Graphing Rational Functions
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Objectives
• I can determine vertical asymptotes of a rational function and graph them
• I can determine horizontal asymptotes of a rational function and graph them
• I can find x and y intercepts to help graph
• I can graph rational functions using a calculator
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Rational Functions
• A rational function is any ratio of two polynomials, where denominator cannot be ZERO!
• Examples:
1)(
x
xxf
103
1)(
2
xx
xxf
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Asymptotes
• Asymptotes are the boundary lines that a rational function approaches, but never crosses.
• We draw these as Dashed Lines on our graphs.
• There are two types of asymptotes we will study in Alg-2: – Vertical– Horizontal
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Vertical Asymptotes
• Vertical Asymptotes exist where the denominator would be zero.
• They are graphed as Vertical Dashed Lines
• There can be more than one!
• To find them, set the denominator equal to zero and solve for “x”
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Example #1
• Find the vertical asymptotes for the following function:
1)(
x
xxf
•Set the denominator equal to zero
•x – 1 = 0, so x = 1
•This graph has a vertical asymptote at x = 1
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1 2 63 4 5 7 8 9 10
4
3
2
7
5
6
8
9
x-axis
y-axis
0
1-2-6 -3-4-5-7-8-910
-4
-3
-2
-1
-7
-5
-6
-8
-9
0
-1
Vertical Asymptote at
X = 1
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Other Examples:
• Find the vertical asymptotes for the following functions:
3
3)(
x
xg
)5)(2(
1)(
xx
xxg
3: xVA
5;2: xxVA
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Horizontal Asymptotes
• Horizontal Asymptotes are also Dashed Lines drawn horizontally to represent another boundary.
• To find the horizontal asymptote you compare the degree of the numerator with the degree of the denominator
• See next slide:
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Horizontal Asymptote
Given the Rational Function:
Compare DEGREE of Numerator to Denominator
If N < D , then y = 0 is the HA
If N = D, then the HA is
If N > D, then the graph has NO HA
Numerator( )
Denominatorf x
N
D
LCy
LC
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Example #1
• Find the horizontal asymptote for the following function:
1)(
x
xxf
•Since the degree of numerator is equal to degree of denominator (m = n)
•Then HA: y = 1/1 = 1
•This graph has a horizontal asymptote at y = 1
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1 2 63 4 5 7 8 9 10
4
3
2
7
5
6
8
9
x-axis
y-axis
0
1-2-6 -3-4-5-7-8-910
-4
-3
-2
-1
-7
-5
-6
-8
-9
0
-1
Horizontal Asymptote at
y = 1
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Other Examples:
• Find the horizontal asymptote for the following functions:
3
3)(
x
xg
13
13)(
2
2
xx
xxg
5
1)(
3
x
xxg
0: yHA
3: yHA
NoneHA :
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Intercepts
• x-intercepts (there can be more than one)
• Set Numerator = 0 and solve for “x”
• y-intercept (at most ONE y-intercept)
• Let all x’s =0 and solve
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Graphing Rational Expressions• Factor rational expression and reduce• Find VA (Denominator = 0)• Find HA (Compare degrees)• Find x-intercept(s) (Numerator = 0)• Find y-intercept (All x’s = 0)• Next type the function into the graphing calculator and
look up ordered pairs from the data table to graph the function.
• Remember that the graph will never cross the VA
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Calculator
ALWAYS use parenthesis!
(Numerator) (Denominator)y
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Key Data to Graph
)1)(3(
32)(
xx
xxf
1,3: xxVA
0: yHA
3int : ( ,0)
2x
)1,0(:int y
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Graph: f(x) =
• Vertical asymptote:• x – 2 = 0 so at x = 2• Dashed line at x = 2• m = 0, n = 1 so m<n• HA at y = 0• No x-int• y-int = (0, -1)• Put into graphing calc.• Pick ordered pairs
2
2
x
-7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7
-7
-6-5
-4
-3
-2-1
12
3
4
56
7
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f(x) = )3)(2(
6
xx
-7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7
-7
-6-5
-4
-3
-2-1
12
3
4
56
7Vertical Asymptotes:
x – 2 = 0 and x + 3 = 0
x = 2, x = -3
m = 0, n = 2 m < n
HA at y = 0
No x-int
y-int (0, -1)
Graph on right
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Calculator
(6) (( 2)( 3))y x x
6( )
( 2)( 3)f x
x x
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Homework
• WS 12-1
• Must know how to factor for next week!!!
• Factoring Quiz Tuesday