Rational Functions AWS
RATIONAL FUNCTIONS: PLAYLIST SUPPORT
Summit Public Schools
OBJECTIVE #1
Solving Rational Equations 2
RATIONAL EQUATIONS A rational equation is an equation that has one or more rational expressions on either or both sides of the equal sign. Example:
x2 4 3 x 1 x
Our goal is to eliminate the rational expressions by multiplying by the least common multiple of all of the denominators.
We also have to be careful about extraneous solutions. These are “extra” solutions that are introduced when we multiply both sides of an equation by a variable. These solutions often make the denominator 0, which we know cannot occur! So, we will check our solutions! 3
RECALL:Suppose WHYyouWEdivideCAN’T DIVIDE BY 0 two numbers, a / b, and the quotient is r. a r b This is the same as saying that:
rb a
Suppose you try to divide 4 by 0. This means there must be some number, r, that when multiplied by 0, gives you 4. But no such number exists because whenever we multiply a number by 0, we get 0!
4 r 0 4
A WORKED-OUT EXAMPLE
BASIC STEPS FOR SOLVING RATIONAL EQUATIONS: ***DETERMINE WHAT VALUES OF THE VARIABLE WILL MAKE ANY OF THE DENOMINATORS HAVE A VALUE OF ZERO. THESE VALUES CANNOT BE SOLUTIONS! 1) Determine the LCM of all of the denominators present in the equation. 2) Multiply both sides of the equation by the LCM. 3) Simplify and make sure you don’t have any rational expressions. 4) Solve the equation using necessary methods (isolating the variable if linear; setting everything equal to 0 and factoring if quadratic). 5) CHECK YOUR SOLUTION(S)!!! 5
EXAMPLE WHICH EQUATIONS LEAD TO EXTRANEOUS SOLUTIONS! 𝑦 2 − 𝑦 − 20 =0 5𝑦
𝑦 2 − 𝑦 − 20 =0 4𝑦
𝑦 2 − 𝑦 − 20 =0 𝑦+4
𝑦 2 − 𝑦 − 20 =0 𝑦−4
𝑦 2 − 𝑦 − 20 =0 𝑦+5
𝑦 2 − 𝑦 − 20 =0 𝑦−5
6
EXAMPLE SOLVE THE RATIONAL EQUATION. CHECK FOR EXTRANEOUS SOLUTIONS!
7 4 = 𝑥−1 𝑥−6
7
EXAMPLE SOLVE THE RATIONAL EQUATION. CHECK FOR EXTRANEOUS SOLUTIONS!
21 3 = 𝑥+2 𝑥−1
8
EXAMPLE SOLVE THE RATIONAL EQUATION. CHECK FOR EXTRANEOUS SOLUTIONS!
4 7 = 2𝑥 − 3 3𝑥 − 5
9
EXAMPLE SOLVE THE RATIONAL EQUATION. CHECK FOR EXTRANEOUS SOLUTIONS!
2 5 = 2𝑥 − 3 7𝑥 + 4
10
EXAMPLE SOLVE THE RATIONAL EQUATION. CHECK FOR EXTRANEOUS SOLUTIONS! 2
5𝑥 240 = 𝑥 − 4 3𝑥 − 12
11
EXAMPLE SOLVE THE RATIONAL EQUATION. CHECK FOR EXTRANEOUS SOLUTIONS!
4 𝑥+1= 𝑥+1
12
EXAMPLE SOLVE THE RATIONAL EQUATION. CHECK FOR EXTRANEOUS SOLUTIONS! 2
𝑥 +6 𝑥−2 + = 2𝑥 𝑥−1 𝑥−1
13
EXAMPLE SOLVE THE RATIONAL EQUATION. CHECK FOR EXTRANEOUS SOLUTIONS!
2 1 5 + = 2 𝑥 𝑥+1 𝑥 +𝑥
14
EXAMPLE SOLVE THE RATIONAL EQUATION. CHECK FOR EXTRANEOUS SOLUTIONS!
3 5 1 − 2 = 𝑥 + 3 𝑥 + 3𝑥 𝑥
15
EXAMPLE SOLVE THE RATIONAL EQUATION. CHECK FOR EXTRANEOUS SOLUTIONS!
𝑥+2 1 3𝑥 − 3 − = 2 3𝑥 − 1 𝑥 3𝑥 − 𝑥
16
EXAMPLE SOLVE THE RATIONAL EQUATION. CHECK FOR EXTRANEOUS SOLUTIONS!
7 8 2 − = 𝑥 + 6 𝑥 3𝑥
17
EXAMPLE SOLVE THE RATIONAL EQUATION. CHECK FOR EXTRANEOUS SOLUTIONS! 2
𝑥 4 −5𝑥 − = 2 𝑥+3 𝑥−2 𝑥 +𝑥−6
18
EXAMPLE SOLVE THE RATIONAL EQUATION. CHECK FOR EXTRANEOUS SOLUTIONS!
𝑥−2 1 1 − = 2 𝑥+3 𝑥−2 𝑥 +𝑥−6
19
EXAMPLE SOLVE THE RATIONAL EQUATION. CHECK FOR EXTRANEOUS SOLUTIONS!
𝑥 1 5 + = 2 𝑥 + 2 𝑥 + 1 𝑥 + 3𝑥 + 2
20
EXAMPLE SOLVE THE RATIONAL EQUATION. CHECK FOR EXTRANEOUS SOLUTIONS!
2𝑥 2 3𝑥 + = 2 𝑥 + 2 𝑥 − 4 𝑥 − 2𝑥 − 8
21
EXAMPLE SOLVE THE RATIONAL EQUATION. CHECK FOR EXTRANEOUS SOLUTIONS!
6 20 5 + 2 = 𝑥+3 𝑥 +𝑥−6 𝑥−2
22
EXAMPLE SOLVE THE RATIONAL EQUATION. CHECK FOR EXTRANEOUS SOLUTIONS! 2
2𝑥 3 −8𝑥 − = 2 𝑥 + 1 𝑥 + 5 𝑥 + 6𝑥 + 5
23
EXAMPLE SOLVE THE RATIONAL EQUATION. CHECK FOR EXTRANEOUS SOLUTIONS!
𝑥 1 11 − = 2 𝑥 − 1 𝑥 − 2 𝑥 − 3𝑥 + 2
24
OBJECTIVE #2
Determine 𝒙-intercepts (zeros) of Rational Functions 25
DEFINITION: ZERO The ZEROS of a function are the values of 𝑥 that, when substituted into the function, make the function have a value of zero. ZEROS are the same as 𝑥-intercepts. They are the values of 𝑥 where the graph of a function intersects the 𝑥-axis. Put quite simply: The ZERO (𝑥-intercept) of a function is the value of x when y is zero.
26
DEFINITION: ZERO OF A RATIONAL FUNCTION The ZEROS of rational functions are the values of 𝑥 that, when substituted into the numerator of the rational function, make the function have a value of zero. ZEROS are the same as 𝑥-intercepts. They are the values of 𝑥 where the graph of a function intersects the 𝑥-axis. Put quite simply: The ZERO (𝒙-intercept) OF A RATIONAL FNCTION is the value of x that makes the numerator (BUT NOT THE DENOMINATOR) equal zero. 27
28
WORK IT OUT
29
REMINDER: Y-INTERCEPTS
30
EXAMPLE IDENTIFY THE X-INTERCEPT(S) OF THE FUNCTION. 𝑥+7 𝑦= 𝑥−1
31
EXAMPLE IDENTIFY THE X-INTERCEPT(S) OF THE FUNCTION. 𝑥 − 14 𝑦= 𝑥 + 17
32
EXAMPLE IDENTIFY THE X-INTERCEPT(S) OF THE FUNCTION. 𝑥 𝑥−1 𝑥+2 𝑦= 𝑥−3 𝑥−4 𝑥+5
33
EXAMPLE IDENTIFY THE X-INTERCEPT(S) OF THE FUNCTION. 𝑥(𝑥 + 6)(𝑥 − 7) 𝑦= 2 𝑥 𝑥 − 7 (𝑥 − 8)
34
EXAMPLE IDENTIFY THE X-INTERCEPT(S) OF THE FUNCTION. 𝑥 2 − 5𝑥 𝑦= 2 𝑥 + 7𝑥 − 18
35
EXAMPLE IDENTIFY THE X-INTERCEPT(S) OF THE FUNCTION. 𝑥 3 − 4𝑥 𝑦= 2 𝑥 + 3𝑥 − 54
36
EXAMPLE IDENTIFY THE X-INTERCEPT(S) OF THE FUNCTION. 𝑥 2 − 5𝑥 − 14 𝑦= 2 𝑥 + 5𝑥 + 6
37
EXAMPLE IDENTIFY THE X-INTERCEPT(S) OF THE FUNCTION. 𝑥 2 − 3𝑥 − 28 𝑦= 2 𝑥 + 6𝑥 + 8
38
EXAMPLE IDENTIFY ALL INTERCEPTS (X AND Y) OF THE FUNCTION GRAPHED BELOW.
39
EXAMPLE IDENTIFY ALL INTERCEPTS (X AND Y) OF THE FUNCTION GRAPHED BELOW.
40
EXAMPLE IDENTIFY ALL INTERCEPTS (X AND Y) OF THE FUNCTION GRAPHED BELOW.
41
EXAMPLE IDENTIFY ALL INTERCEPTS (X AND Y) OF THE FUNCTION GRAPHED BELOW.
42
EXAMPLE IDENTIFY ALL INTERCEPTS (X AND Y) OF THE FUNCTION GRAPHED BELOW.
43
OBJECTIVE #3
Determine Horizontal Asymptotes of Rational Functions 44
HORIZONTAL ASYMPTOTES
45
WORKED OUT EXAMPLES
Degree of numerator = 1
Degree of denominator = 2
Because the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is 𝒚 = 𝟎.
Degree of numerator = 2
Degree of denominator = 2
Because the degree of the numerator is equal to the degree of the denominator, the horizontal asymptote is the ratio of the leading coefficients: 𝟏 𝒚 = = 𝟏. 𝟏
Degree of numerator = 2
Degree of denominator = 2
Because the degree of the numerator is equal to the degree of the denominator, the horizontal asymptote is the ratio of the leading coefficients: 𝟑 𝒚 = = 𝟑. 𝟏
46
HORIZONTAL ASYMPTOTES: SUMMARY
47
EXAMPLE IDENTIFY THE HORIZONTAL ASYMPTOTE OF THE FUNCTION. 3 𝑦= 𝑥−5
48
EXAMPLE IDENTIFY THE HORIZONTAL ASYMPTOTE OF THE FUNCTION. 𝑥−5 𝑦= 𝑥+4
49
EXAMPLE IDENTIFY THE HORIZONTAL ASYMPTOTE OF THE FUNCTION. 6𝑥 2 𝑦= 4 𝑥 −5
50
EXAMPLE IDENTIFY THE HORIZONTAL ASYMPTOTE OF THE FUNCTION. 7𝑥 2 − 5 𝑦= 2 3𝑥 + 4
51
EXAMPLE IDENTIFY THE HORIZONTAL ASYMPTOTE OF THE FUNCTION. 𝑥−5 𝑦= 3𝑥 + 4
52
EXAMPLE IDENTIFY THE HORIZONTAL ASYMPTOTE OF THE FUNCTION. 3𝑥 2 𝑦= 𝑥−5
53
EXAMPLE IDENTIFY THE HORIZONTAL ASYMPTOTE OF THE FUNCTION. (𝑥 − 1)(2𝑥 − 3) 𝑦= (3𝑥 − 4)(𝑥 + 5)
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EXAMPLE IDENTIFY THE HORIZONTAL ASYMPTOTE OF THE FUNCTION. 𝑥−3 𝑦= (𝑥 + 5)(𝑥 + 6)
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EXAMPLE IDENTIFY THE HORIZONTAL ASYMPTOTE OF THE FUNCTION. 3𝑥 3 + 5 𝑦= 𝑥+7 3
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EXAMPLE IDENTIFY THE HORIZONTAL ASYMPTOTE OF EACH FUNCTION. 49.
50.
51.
52.
53.
54.
55.
56.
57.
58.
59.
60.
61.
62.
63.
64. 57
OBJECTIVE #4
Determine Vertical Asymptotes of Rational Functions 58
VERTICAL ASYMPTOTES The graph of a rational function has vertical asymptotes at the values of 𝑥 that make the denominator (but not also the numerator) have a value of zero.
Vertical asymptote: 𝒙 = −𝟐
Vertical asymptote: 𝒙=𝟏
Vertical asymptotes: 𝒙 = −𝟐 and 𝒙 = 𝟏
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FINDING VERTICAL ASYMPTOTES To find the equations of vertical asymptotes of rational functions:
Make sure that the values you find for 𝑥 that make the denominator equal zero don’t also make the numerator equal zero.
60
WORK IT OUT!
61
EXAMPLE IDENTIFY THE VERTICAL ASYMPTOTE OF THE FUNCTION. 𝑥+4 𝑦= 𝑥+9
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EXAMPLE IDENTIFY THE VERTICAL ASYMPTOTE OF THE FUNCTION. 𝑥−7 𝑦= 𝑥−8
63
EXAMPLE IDENTIFY THE VERTICAL ASYMPTOTE OF THE FUNCTION. 𝑥 + 10 𝑦= 𝑥
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EXAMPLE IDENTIFY THE VERTICAL ASYMPTOTES OF THE FUNCTION. (𝑥 − 1)(𝑥 + 2) 𝑦= 𝑥 − 6 (𝑥 + 7)
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EXAMPLE IDENTIFY THE VERTICAL ASYMPTOTES OF THE FUNCTION. 𝑥2 + 𝑥 − 2 𝑦= 2 𝑥 − 3𝑥 − 54
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EXAMPLE IDENTIFY THE HORIZONTAL AND VERTICAL ASYMPTOTES OF THE FUNCTION.
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EXAMPLE IDENTIFY THE HORIZONTAL AND VERTICAL ASYMPTOTES OF THE FUNCTION.
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EXAMPLE IDENTIFY THE HORIZONTAL AND VERTICAL ASYMPTOTES OF THE FUNCTION.
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EXAMPLE CIRCLE THE EQUATION FOR THE FUNCTION GRAPHED BELOW.
𝒚=
𝟑𝒙 − 𝟗 𝒙+𝟐
𝟑𝒙 + 𝟗 𝒚= 𝒙+𝟐
𝒚=
𝒙+𝟐 𝟑𝒙 − 𝟗
𝒙+𝟐 𝒚= 𝟑𝒙 + 𝟗
𝒚=
𝟑𝒙 − 𝟗 𝒙−𝟐
𝒚=
𝒙−𝟐 𝟑𝒙 − 𝟗
𝒚=
𝟑𝒙 + 𝟗 𝒙−𝟐
𝒚=
𝒙−𝟐 𝟑𝒙 + 𝟗
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EXAMPLE CIRCLE THE EQUATION FOR THE FUNCTION GRAPHED BELOW.
𝒚=
𝒙 + 𝟓 𝟐𝒙 + 𝟕 (𝒙 − 𝟑)(𝒙 − 𝟒)
𝒚=
𝒙 − 𝟒 𝟐𝒙 + 𝟕 (𝒙 + 𝟓)(𝒙 − 𝟑)
𝒚=
𝒙+𝟓 𝒙−𝟒 (𝒙 − 𝟑)(𝟐𝒙 + 𝟕)
𝒚=
𝒙 − 𝟑 𝟐𝒙 + 𝟕 (𝒙 − 𝟓)(𝒙 + 𝟒)
𝒚=
(𝒙 + 𝟓)(𝒙 − 𝟒) 𝒙 − 𝟑 𝟐𝒙 + 𝟕
𝒚=
𝒙 + 𝟑 𝟐𝒙 − 𝟕 (𝒙 + 𝟓)(𝒙 − 𝟒)
𝒚=
𝒙 − 𝟑 𝟐𝒙 + 𝟕 (𝒙 + 𝟓)(𝒙 − 𝟒)
𝒚=
𝒙 + 𝟑 𝟐𝒙 − 𝟕 (𝒙 − 𝟓)(𝒙 + 𝟒)
71
EXAMPLE IDENTIFY THE VERTICAL ASYMPTOTES OF EACH FUNCTION. 49.
50.
51.
52.
53.
54.
55.
56.
57.
58.
59.
60.
61.
62.
63.
64. 72
OBJECTIVE #5
Determine Holes of Rational Functions 73
HOLES Graphs of rational functions have holes (also known as removable discontinuities) in them when there is a common factor in both the numerator and the denominator.
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STRATEGY FOR FINDING THE X-COORDINATE OF THE HOLE
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STRATEGY FOR FINDING THE Y-COORDINATE OF THE HOLE
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EXAMPLE IDENTIFY THE COORDINATES OF THE HOLE FOR THE RATIONAL FUNCTION. 𝒙𝟐 − 𝟐𝒙 − 𝟖 𝒚= 𝒙+𝟐
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EXAMPLE IDENTIFY THE COORDINATES OF THE HOLE FOR THE RATIONAL FUNCTION. 𝒙−𝟗 𝒚= 𝟐 𝒙 − 𝟕𝒙 − 𝟏𝟖
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EXAMPLE IDENTIFY THE COORDINATES OF THE HOLE FOR THE RATIONAL FUNCTION. 𝒙𝟐 + 𝒙 − 𝟔 𝒚= 𝟐 𝒙 − 𝟒𝒙 − 𝟐𝟏
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EXAMPLE IDENTIFY THE COORDINATES OF THE HOLE FOR THE RATIONAL FUNCTION. 𝒙𝟐 − 𝒙 − 𝟕𝟐 𝒚= 𝟐 𝒙 + 𝟓𝒙 − 𝟐𝟒
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Summit Public Schools
OBJECTIVE #1
Solving Rational Equations 2
RATIONAL EQUATIONS A rational equation is an equation that has one or more rational expressions on either or both sides of the equal sign. Example:
x2 4 3 x 1 x
Our goal is to eliminate the rational expressions by multiplying by the least common multiple of all of the denominators.
We also have to be careful about extraneous solutions. These are “extra” solutions that are introduced when we multiply both sides of an equation by a variable. These solutions often make the denominator 0, which we know cannot occur! So, we will check our solutions! 3
RECALL:Suppose WHYyouWEdivideCAN’T DIVIDE BY 0 two numbers, a / b, and the quotient is r. a r b This is the same as saying that:
rb a
Suppose you try to divide 4 by 0. This means there must be some number, r, that when multiplied by 0, gives you 4. But no such number exists because whenever we multiply a number by 0, we get 0!
4 r 0 4
A WORKED-OUT EXAMPLE
BASIC STEPS FOR SOLVING RATIONAL EQUATIONS: ***DETERMINE WHAT VALUES OF THE VARIABLE WILL MAKE ANY OF THE DENOMINATORS HAVE A VALUE OF ZERO. THESE VALUES CANNOT BE SOLUTIONS! 1) Determine the LCM of all of the denominators present in the equation. 2) Multiply both sides of the equation by the LCM. 3) Simplify and make sure you don’t have any rational expressions. 4) Solve the equation using necessary methods (isolating the variable if linear; setting everything equal to 0 and factoring if quadratic). 5) CHECK YOUR SOLUTION(S)!!! 5
EXAMPLE WHICH EQUATIONS LEAD TO EXTRANEOUS SOLUTIONS! 𝑦 2 − 𝑦 − 20 =0 5𝑦
𝑦 2 − 𝑦 − 20 =0 4𝑦
𝑦 2 − 𝑦 − 20 =0 𝑦+4
𝑦 2 − 𝑦 − 20 =0 𝑦−4
𝑦 2 − 𝑦 − 20 =0 𝑦+5
𝑦 2 − 𝑦 − 20 =0 𝑦−5
6
EXAMPLE SOLVE THE RATIONAL EQUATION. CHECK FOR EXTRANEOUS SOLUTIONS!
7 4 = 𝑥−1 𝑥−6
7
EXAMPLE SOLVE THE RATIONAL EQUATION. CHECK FOR EXTRANEOUS SOLUTIONS!
21 3 = 𝑥+2 𝑥−1
8
EXAMPLE SOLVE THE RATIONAL EQUATION. CHECK FOR EXTRANEOUS SOLUTIONS!
4 7 = 2𝑥 − 3 3𝑥 − 5
9
EXAMPLE SOLVE THE RATIONAL EQUATION. CHECK FOR EXTRANEOUS SOLUTIONS!
2 5 = 2𝑥 − 3 7𝑥 + 4
10
EXAMPLE SOLVE THE RATIONAL EQUATION. CHECK FOR EXTRANEOUS SOLUTIONS! 2
5𝑥 240 = 𝑥 − 4 3𝑥 − 12
11
EXAMPLE SOLVE THE RATIONAL EQUATION. CHECK FOR EXTRANEOUS SOLUTIONS!
4 𝑥+1= 𝑥+1
12
EXAMPLE SOLVE THE RATIONAL EQUATION. CHECK FOR EXTRANEOUS SOLUTIONS! 2
𝑥 +6 𝑥−2 + = 2𝑥 𝑥−1 𝑥−1
13
EXAMPLE SOLVE THE RATIONAL EQUATION. CHECK FOR EXTRANEOUS SOLUTIONS!
2 1 5 + = 2 𝑥 𝑥+1 𝑥 +𝑥
14
EXAMPLE SOLVE THE RATIONAL EQUATION. CHECK FOR EXTRANEOUS SOLUTIONS!
3 5 1 − 2 = 𝑥 + 3 𝑥 + 3𝑥 𝑥
15
EXAMPLE SOLVE THE RATIONAL EQUATION. CHECK FOR EXTRANEOUS SOLUTIONS!
𝑥+2 1 3𝑥 − 3 − = 2 3𝑥 − 1 𝑥 3𝑥 − 𝑥
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EXAMPLE SOLVE THE RATIONAL EQUATION. CHECK FOR EXTRANEOUS SOLUTIONS!
7 8 2 − = 𝑥 + 6 𝑥 3𝑥
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EXAMPLE SOLVE THE RATIONAL EQUATION. CHECK FOR EXTRANEOUS SOLUTIONS! 2
𝑥 4 −5𝑥 − = 2 𝑥+3 𝑥−2 𝑥 +𝑥−6
18
EXAMPLE SOLVE THE RATIONAL EQUATION. CHECK FOR EXTRANEOUS SOLUTIONS!
𝑥−2 1 1 − = 2 𝑥+3 𝑥−2 𝑥 +𝑥−6
19
EXAMPLE SOLVE THE RATIONAL EQUATION. CHECK FOR EXTRANEOUS SOLUTIONS!
𝑥 1 5 + = 2 𝑥 + 2 𝑥 + 1 𝑥 + 3𝑥 + 2
20
EXAMPLE SOLVE THE RATIONAL EQUATION. CHECK FOR EXTRANEOUS SOLUTIONS!
2𝑥 2 3𝑥 + = 2 𝑥 + 2 𝑥 − 4 𝑥 − 2𝑥 − 8
21
EXAMPLE SOLVE THE RATIONAL EQUATION. CHECK FOR EXTRANEOUS SOLUTIONS!
6 20 5 + 2 = 𝑥+3 𝑥 +𝑥−6 𝑥−2
22
EXAMPLE SOLVE THE RATIONAL EQUATION. CHECK FOR EXTRANEOUS SOLUTIONS! 2
2𝑥 3 −8𝑥 − = 2 𝑥 + 1 𝑥 + 5 𝑥 + 6𝑥 + 5
23
EXAMPLE SOLVE THE RATIONAL EQUATION. CHECK FOR EXTRANEOUS SOLUTIONS!
𝑥 1 11 − = 2 𝑥 − 1 𝑥 − 2 𝑥 − 3𝑥 + 2
24
OBJECTIVE #2
Determine 𝒙-intercepts (zeros) of Rational Functions 25
DEFINITION: ZERO The ZEROS of a function are the values of 𝑥 that, when substituted into the function, make the function have a value of zero. ZEROS are the same as 𝑥-intercepts. They are the values of 𝑥 where the graph of a function intersects the 𝑥-axis. Put quite simply: The ZERO (𝑥-intercept) of a function is the value of x when y is zero.
26
DEFINITION: ZERO OF A RATIONAL FUNCTION The ZEROS of rational functions are the values of 𝑥 that, when substituted into the numerator of the rational function, make the function have a value of zero. ZEROS are the same as 𝑥-intercepts. They are the values of 𝑥 where the graph of a function intersects the 𝑥-axis. Put quite simply: The ZERO (𝒙-intercept) OF A RATIONAL FNCTION is the value of x that makes the numerator (BUT NOT THE DENOMINATOR) equal zero. 27
28
WORK IT OUT
29
REMINDER: Y-INTERCEPTS
30
EXAMPLE IDENTIFY THE X-INTERCEPT(S) OF THE FUNCTION. 𝑥+7 𝑦= 𝑥−1
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EXAMPLE IDENTIFY THE X-INTERCEPT(S) OF THE FUNCTION. 𝑥 − 14 𝑦= 𝑥 + 17
32
EXAMPLE IDENTIFY THE X-INTERCEPT(S) OF THE FUNCTION. 𝑥 𝑥−1 𝑥+2 𝑦= 𝑥−3 𝑥−4 𝑥+5
33
EXAMPLE IDENTIFY THE X-INTERCEPT(S) OF THE FUNCTION. 𝑥(𝑥 + 6)(𝑥 − 7) 𝑦= 2 𝑥 𝑥 − 7 (𝑥 − 8)
34
EXAMPLE IDENTIFY THE X-INTERCEPT(S) OF THE FUNCTION. 𝑥 2 − 5𝑥 𝑦= 2 𝑥 + 7𝑥 − 18
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EXAMPLE IDENTIFY THE X-INTERCEPT(S) OF THE FUNCTION. 𝑥 3 − 4𝑥 𝑦= 2 𝑥 + 3𝑥 − 54
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EXAMPLE IDENTIFY THE X-INTERCEPT(S) OF THE FUNCTION. 𝑥 2 − 5𝑥 − 14 𝑦= 2 𝑥 + 5𝑥 + 6
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EXAMPLE IDENTIFY THE X-INTERCEPT(S) OF THE FUNCTION. 𝑥 2 − 3𝑥 − 28 𝑦= 2 𝑥 + 6𝑥 + 8
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EXAMPLE IDENTIFY ALL INTERCEPTS (X AND Y) OF THE FUNCTION GRAPHED BELOW.
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EXAMPLE IDENTIFY ALL INTERCEPTS (X AND Y) OF THE FUNCTION GRAPHED BELOW.
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EXAMPLE IDENTIFY ALL INTERCEPTS (X AND Y) OF THE FUNCTION GRAPHED BELOW.
41
EXAMPLE IDENTIFY ALL INTERCEPTS (X AND Y) OF THE FUNCTION GRAPHED BELOW.
42
EXAMPLE IDENTIFY ALL INTERCEPTS (X AND Y) OF THE FUNCTION GRAPHED BELOW.
43
OBJECTIVE #3
Determine Horizontal Asymptotes of Rational Functions 44
HORIZONTAL ASYMPTOTES
45
WORKED OUT EXAMPLES
Degree of numerator = 1
Degree of denominator = 2
Because the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is 𝒚 = 𝟎.
Degree of numerator = 2
Degree of denominator = 2
Because the degree of the numerator is equal to the degree of the denominator, the horizontal asymptote is the ratio of the leading coefficients: 𝟏 𝒚 = = 𝟏. 𝟏
Degree of numerator = 2
Degree of denominator = 2
Because the degree of the numerator is equal to the degree of the denominator, the horizontal asymptote is the ratio of the leading coefficients: 𝟑 𝒚 = = 𝟑. 𝟏
46
HORIZONTAL ASYMPTOTES: SUMMARY
47
EXAMPLE IDENTIFY THE HORIZONTAL ASYMPTOTE OF THE FUNCTION. 3 𝑦= 𝑥−5
48
EXAMPLE IDENTIFY THE HORIZONTAL ASYMPTOTE OF THE FUNCTION. 𝑥−5 𝑦= 𝑥+4
49
EXAMPLE IDENTIFY THE HORIZONTAL ASYMPTOTE OF THE FUNCTION. 6𝑥 2 𝑦= 4 𝑥 −5
50
EXAMPLE IDENTIFY THE HORIZONTAL ASYMPTOTE OF THE FUNCTION. 7𝑥 2 − 5 𝑦= 2 3𝑥 + 4
51
EXAMPLE IDENTIFY THE HORIZONTAL ASYMPTOTE OF THE FUNCTION. 𝑥−5 𝑦= 3𝑥 + 4
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EXAMPLE IDENTIFY THE HORIZONTAL ASYMPTOTE OF THE FUNCTION. 3𝑥 2 𝑦= 𝑥−5
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EXAMPLE IDENTIFY THE HORIZONTAL ASYMPTOTE OF THE FUNCTION. (𝑥 − 1)(2𝑥 − 3) 𝑦= (3𝑥 − 4)(𝑥 + 5)
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EXAMPLE IDENTIFY THE HORIZONTAL ASYMPTOTE OF THE FUNCTION. 𝑥−3 𝑦= (𝑥 + 5)(𝑥 + 6)
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EXAMPLE IDENTIFY THE HORIZONTAL ASYMPTOTE OF THE FUNCTION. 3𝑥 3 + 5 𝑦= 𝑥+7 3
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EXAMPLE IDENTIFY THE HORIZONTAL ASYMPTOTE OF EACH FUNCTION. 49.
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OBJECTIVE #4
Determine Vertical Asymptotes of Rational Functions 58
VERTICAL ASYMPTOTES The graph of a rational function has vertical asymptotes at the values of 𝑥 that make the denominator (but not also the numerator) have a value of zero.
Vertical asymptote: 𝒙 = −𝟐
Vertical asymptote: 𝒙=𝟏
Vertical asymptotes: 𝒙 = −𝟐 and 𝒙 = 𝟏
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FINDING VERTICAL ASYMPTOTES To find the equations of vertical asymptotes of rational functions:
Make sure that the values you find for 𝑥 that make the denominator equal zero don’t also make the numerator equal zero.
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WORK IT OUT!
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EXAMPLE IDENTIFY THE VERTICAL ASYMPTOTE OF THE FUNCTION. 𝑥+4 𝑦= 𝑥+9
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EXAMPLE IDENTIFY THE VERTICAL ASYMPTOTE OF THE FUNCTION. 𝑥−7 𝑦= 𝑥−8
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EXAMPLE IDENTIFY THE VERTICAL ASYMPTOTE OF THE FUNCTION. 𝑥 + 10 𝑦= 𝑥
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EXAMPLE IDENTIFY THE VERTICAL ASYMPTOTES OF THE FUNCTION. (𝑥 − 1)(𝑥 + 2) 𝑦= 𝑥 − 6 (𝑥 + 7)
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EXAMPLE IDENTIFY THE VERTICAL ASYMPTOTES OF THE FUNCTION. 𝑥2 + 𝑥 − 2 𝑦= 2 𝑥 − 3𝑥 − 54
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EXAMPLE IDENTIFY THE HORIZONTAL AND VERTICAL ASYMPTOTES OF THE FUNCTION.
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EXAMPLE IDENTIFY THE HORIZONTAL AND VERTICAL ASYMPTOTES OF THE FUNCTION.
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EXAMPLE IDENTIFY THE HORIZONTAL AND VERTICAL ASYMPTOTES OF THE FUNCTION.
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EXAMPLE CIRCLE THE EQUATION FOR THE FUNCTION GRAPHED BELOW.
𝒚=
𝟑𝒙 − 𝟗 𝒙+𝟐
𝟑𝒙 + 𝟗 𝒚= 𝒙+𝟐
𝒚=
𝒙+𝟐 𝟑𝒙 − 𝟗
𝒙+𝟐 𝒚= 𝟑𝒙 + 𝟗
𝒚=
𝟑𝒙 − 𝟗 𝒙−𝟐
𝒚=
𝒙−𝟐 𝟑𝒙 − 𝟗
𝒚=
𝟑𝒙 + 𝟗 𝒙−𝟐
𝒚=
𝒙−𝟐 𝟑𝒙 + 𝟗
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EXAMPLE CIRCLE THE EQUATION FOR THE FUNCTION GRAPHED BELOW.
𝒚=
𝒙 + 𝟓 𝟐𝒙 + 𝟕 (𝒙 − 𝟑)(𝒙 − 𝟒)
𝒚=
𝒙 − 𝟒 𝟐𝒙 + 𝟕 (𝒙 + 𝟓)(𝒙 − 𝟑)
𝒚=
𝒙+𝟓 𝒙−𝟒 (𝒙 − 𝟑)(𝟐𝒙 + 𝟕)
𝒚=
𝒙 − 𝟑 𝟐𝒙 + 𝟕 (𝒙 − 𝟓)(𝒙 + 𝟒)
𝒚=
(𝒙 + 𝟓)(𝒙 − 𝟒) 𝒙 − 𝟑 𝟐𝒙 + 𝟕
𝒚=
𝒙 + 𝟑 𝟐𝒙 − 𝟕 (𝒙 + 𝟓)(𝒙 − 𝟒)
𝒚=
𝒙 − 𝟑 𝟐𝒙 + 𝟕 (𝒙 + 𝟓)(𝒙 − 𝟒)
𝒚=
𝒙 + 𝟑 𝟐𝒙 − 𝟕 (𝒙 − 𝟓)(𝒙 + 𝟒)
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EXAMPLE IDENTIFY THE VERTICAL ASYMPTOTES OF EACH FUNCTION. 49.
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OBJECTIVE #5
Determine Holes of Rational Functions 73
HOLES Graphs of rational functions have holes (also known as removable discontinuities) in them when there is a common factor in both the numerator and the denominator.
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STRATEGY FOR FINDING THE X-COORDINATE OF THE HOLE
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STRATEGY FOR FINDING THE Y-COORDINATE OF THE HOLE
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EXAMPLE IDENTIFY THE COORDINATES OF THE HOLE FOR THE RATIONAL FUNCTION. 𝒙𝟐 − 𝟐𝒙 − 𝟖 𝒚= 𝒙+𝟐
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EXAMPLE IDENTIFY THE COORDINATES OF THE HOLE FOR THE RATIONAL FUNCTION. 𝒙−𝟗 𝒚= 𝟐 𝒙 − 𝟕𝒙 − 𝟏𝟖
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EXAMPLE IDENTIFY THE COORDINATES OF THE HOLE FOR THE RATIONAL FUNCTION. 𝒙𝟐 + 𝒙 − 𝟔 𝒚= 𝟐 𝒙 − 𝟒𝒙 − 𝟐𝟏
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EXAMPLE IDENTIFY THE COORDINATES OF THE HOLE FOR THE RATIONAL FUNCTION. 𝒙𝟐 − 𝒙 − 𝟕𝟐 𝒚= 𝟐 𝒙 + 𝟓𝒙 − 𝟐𝟒
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