H Algebra 2/Trig · Unit 4
Find the vertical asymptotes, the horizontal or oblique asymptote, and any holes of a rational function.
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Factor first
Factor the numerator and the denominator.
Cancel a common factor before you name the asymptotes.
Holes
A factor that cancels is a hole.
The x-value makes that factor 0. The y-value comes from the reduced fraction.
Vertical asymptotes
A vertical asymptote is a denominator zero that does not cancel.
The graph gets steeper and steeper near that vertical line.
The other asymptote
If the degrees match, the horizontal asymptote is the ratio of the leading coefficients.
If the numerator's degree is one more, the other asymptote is oblique.
Horizontal asymptote, no hole
Factor.
2(x − 2)(x + 2)(x − 3)(x + 3)
Nothing cancels, so there is no hole.
A vertical asymptote is a denominator zero that does not cancel.
x − 3 = 0
x = 3
x + 3 = 0
x = −3
The vertical asymptotes are x = −3 and x = 3.
The degrees are equal, so the horizontal asymptote is the ratio of the leading coefficients.
y = 2
The horizontal asymptote is y = 2.
Answer: no hole; vertical asymptotes x = −3 and x = 3; horizontal asymptote y = 2
Factor.
3x(x − 1)(x − 4)
Nothing cancels, so there is no hole.
A vertical asymptote is a denominator zero that does not cancel.
x − 1 = 0
x = 1
x − 4 = 0
x = 4
The vertical asymptotes are x = 1 and x = 4.
The degree of the numerator is less than the degree of the denominator.
y = 0
The horizontal asymptote is y = 0.
Answer: no hole; vertical asymptotes x = 1 and x = 4; horizontal asymptote y = 0
Factor.
3(x − 2)(x + 2)(x − 1)(x + 1)
Nothing cancels, so there is no hole.
A vertical asymptote is a denominator zero that does not cancel.
x − 1 = 0
x = 1
x + 1 = 0
x = −1
The vertical asymptotes are x = −1 and x = 1.
The degrees are equal, so the horizontal asymptote is the ratio of the leading coefficients.
y = 3
The horizontal asymptote is y = 3.
Answer: no hole; vertical asymptotes x = −1 and x = 1; horizontal asymptote y = 3
Factor.
4x(x − 3)(x + 3)
Nothing cancels, so there is no hole.
A vertical asymptote is a denominator zero that does not cancel.
x − 3 = 0
x = 3
x + 3 = 0
x = −3
The vertical asymptotes are x = −3 and x = 3.
The degree of the numerator is less than the degree of the denominator.
y = 0
The horizontal asymptote is y = 0.
Answer: no hole; vertical asymptotes x = −3 and x = 3; horizontal asymptote y = 0
Factor.
5(x − 2)(x + 2)
Nothing cancels, so there is no hole.
A vertical asymptote is a denominator zero that does not cancel.
x − 2 = 0
x = 2
x + 2 = 0
x = −2
The vertical asymptotes are x = −2 and x = 2.
The degree of the numerator is less than the degree of the denominator.
y = 0
The horizontal asymptote is y = 0.
Answer: no hole; vertical asymptotes x = −2 and x = 2; horizontal asymptote y = 0
Factor.
2x(x + 3)(x − 2)
Nothing cancels, so there is no hole.
A vertical asymptote is a denominator zero that does not cancel.
x + 3 = 0
x = −3
x − 2 = 0
x = 2
The vertical asymptotes are x = −3 and x = 2.
The degree of the numerator is less than the degree of the denominator.
y = 0
The horizontal asymptote is y = 0.
Answer: no hole; vertical asymptotes x = −3 and x = 2; horizontal asymptote y = 0
A hole and a horizontal asymptote
Factor.
(x − 2)(x + 1)(x − 2)(x + 2)
Cancel the common factor. A canceled factor is a hole, not a vertical asymptote.
(x + 1)(x + 2)
x = 2
Substitute that x-value into the reduced fraction.
y = (2 + 1)(2 + 2)
y = 34
The hole is (2, 34).
A vertical asymptote is a denominator zero that does not cancel.
x + 2 = 0
x = −2
The vertical asymptote is x = −2.
The degrees are equal, so the horizontal asymptote is the ratio of the leading coefficients.
y = 1
The horizontal asymptote is y = 1.
Answer: hole (2, 34); vertical asymptote x = −2; horizontal asymptote y = 1
Factor.
(x + 3)(x − 2)(x − 3)(x + 3)
Cancel the common factor. A canceled factor is a hole, not a vertical asymptote.
(x − 2)(x − 3)
x = −3
Substitute that x-value into the reduced fraction.
y = (−3 − 2)(−3 − 3)
y = −5−6
y = 56
The hole is (−3, 56).
A vertical asymptote is a denominator zero that does not cancel.
x − 3 = 0
x = 3
The vertical asymptote is x = 3.
The degrees are equal, so the horizontal asymptote is the ratio of the leading coefficients.
y = 1
The horizontal asymptote is y = 1.
Answer: hole (−3, 56); vertical asymptote x = 3; horizontal asymptote y = 1
Factor.
(x − 2)(x + 2)(x + 2)(x − 1)
Cancel the common factor. A canceled factor is a hole, not a vertical asymptote.
(x − 2)(x − 1)
x = −2
Substitute that x-value into the reduced fraction.
y = (−2 − 2)(−2 − 1)
y = −4−3
y = 43
The hole is (−2, 43).
A vertical asymptote is a denominator zero that does not cancel.
x − 1 = 0
x = 1
The vertical asymptote is x = 1.
The degrees are equal, so the horizontal asymptote is the ratio of the leading coefficients.
y = 1
The horizontal asymptote is y = 1.
Answer: hole (−2, 43); vertical asymptote x = 1; horizontal asymptote y = 1
Factor.
(x − 3)(x + 3)(x + 1)(x + 3)
Cancel the common factor. A canceled factor is a hole, not a vertical asymptote.
(x − 3)(x + 1)
x = −3
Substitute that x-value into the reduced fraction.
y = (−3 − 3)(−3 + 1)
y = −6−2
y = 3
The hole is (−3, 3).
A vertical asymptote is a denominator zero that does not cancel.
x + 1 = 0
x = −1
The vertical asymptote is x = −1.
The degrees are equal, so the horizontal asymptote is the ratio of the leading coefficients.
y = 1
The horizontal asymptote is y = 1.
Answer: hole (−3, 3); vertical asymptote x = −1; horizontal asymptote y = 1
Factor.
(x + 2)(x + 3)(x − 4)(x + 3)
Cancel the common factor. A canceled factor is a hole, not a vertical asymptote.
(x + 2)(x − 4)
x = −3
Substitute that x-value into the reduced fraction.
y = (−3 + 2)(−3 − 4)
y = −1−7
y = 17
The hole is (−3, 17).
A vertical asymptote is a denominator zero that does not cancel.
x − 4 = 0
x = 4
The vertical asymptote is x = 4.
The degrees are equal, so the horizontal asymptote is the ratio of the leading coefficients.
y = 1
The horizontal asymptote is y = 1.
Answer: hole (−3, 17); vertical asymptote x = 4; horizontal asymptote y = 1
Factor.
3x(x − 1)(x − 1)(x + 1)
Cancel the common factor. A canceled factor is a hole, not a vertical asymptote.
3x(x + 1)
x = 1
Substitute that x-value into the reduced fraction.
y = 3(1)(1 + 1)
y = 32
The hole is (1, 32).
A vertical asymptote is a denominator zero that does not cancel.
x + 1 = 0
x = −1
The vertical asymptote is x = −1.
The degrees are equal, so the horizontal asymptote is the ratio of the leading coefficients.
y = 3
The horizontal asymptote is y = 3.
Answer: hole (1, 32); vertical asymptote x = −1; horizontal asymptote y = 3
A hole and an oblique asymptote
Factor.
2x(x + 1)(x + 3)(x + 2)(x + 3)
Cancel the common factor. A canceled factor is a hole, not a vertical asymptote.
2x(x + 1)(x + 2)
x = −3
Substitute that x-value into the reduced fraction.
y = 2(−3)(−3 + 1)(−3 + 2)
y = 12−1
y = −12
The hole is (−3, −12).
A vertical asymptote is a denominator zero that does not cancel.
x + 2 = 0
x = −2
The vertical asymptote is x = −2.
The degree of the numerator is one more than the degree of the denominator, so the other asymptote is oblique.
(x + 2)(2x − 2) = 2x² + 2x − 4
2x² + 2x = (2x² + 2x − 4) + 4
y = 2x − 2
The oblique asymptote is y = 2x − 2.
Answer: hole (−3, −12); vertical asymptote x = −2; oblique asymptote y = 2x − 2
Factor.
x(x − 3)(x + 2)(x − 2)(x + 2)
Cancel the common factor. A canceled factor is a hole, not a vertical asymptote.
x(x − 3)(x − 2)
x = −2
Substitute that x-value into the reduced fraction.
y = (−2)(−2 − 3)(−2 − 2)
y = 10−4
y = −52
The hole is (−2, −52).
A vertical asymptote is a denominator zero that does not cancel.
x − 2 = 0
x = 2
The vertical asymptote is x = 2.
The degree of the numerator is one more than the degree of the denominator, so the other asymptote is oblique.
(x − 2)(x − 1) = x² − 3x + 2
x² − 3x = (x² − 3x + 2) − 2
y = x − 1
The oblique asymptote is y = x − 1.
Answer: hole (−2, −52); vertical asymptote x = 2; oblique asymptote y = x − 1
Factor.
x(x + 4)(x − 2)(x + 3)(x − 2)
Cancel the common factor. A canceled factor is a hole, not a vertical asymptote.
x(x + 4)(x + 3)
x = 2
Substitute that x-value into the reduced fraction.
y = (2)(2 + 4)(2 + 3)
y = 125
The hole is (2, 125).
A vertical asymptote is a denominator zero that does not cancel.
x + 3 = 0
x = −3
The vertical asymptote is x = −3.
The degree of the numerator is one more than the degree of the denominator, so the other asymptote is oblique.
(x + 3)(x + 1) = x² + 4x + 3
x² + 4x = (x² + 4x + 3) − 3
y = x + 1
The oblique asymptote is y = x + 1.
Answer: hole (2, 125); vertical asymptote x = −3; oblique asymptote y = x + 1
Factor.
x(x + 1)(x + 3)(x − 1)(x + 1)
Cancel the common factor. A canceled factor is a hole, not a vertical asymptote.
x(x + 3)(x − 1)
x = −1
Substitute that x-value into the reduced fraction.
y = (−1)(−1 + 3)(−1 − 1)
y = −2−2
y = 1
The hole is (−1, 1).
A vertical asymptote is a denominator zero that does not cancel.
x − 1 = 0
x = 1
The vertical asymptote is x = 1.
The degree of the numerator is one more than the degree of the denominator, so the other asymptote is oblique.
(x − 1)(x + 4) = x² + 3x − 4
x² + 3x = (x² + 3x − 4) + 4
y = x + 4
The oblique asymptote is y = x + 4.
Answer: hole (−1, 1); vertical asymptote x = 1; oblique asymptote y = x + 4
Factor.
2x(x + 2)(x − 1)(x − 2)(x + 2)
Cancel the common factor. A canceled factor is a hole, not a vertical asymptote.
2x(x − 1)(x − 2)
x = −2
Substitute that x-value into the reduced fraction.
y = 2(−2)(−2 − 1)(−2 − 2)
y = 12−4
y = −3
The hole is (−2, −3).
A vertical asymptote is a denominator zero that does not cancel.
x − 2 = 0
x = 2
The vertical asymptote is x = 2.
The degree of the numerator is one more than the degree of the denominator, so the other asymptote is oblique.
(x − 2)(2x + 2) = 2x² − 2x − 4
2x² − 2x = (2x² − 2x − 4) + 4
y = 2x + 2
The oblique asymptote is y = 2x + 2.
Answer: hole (−2, −3); vertical asymptote x = 2; oblique asymptote y = 2x + 2
Factor.
x(x − 3)(x + 1)(x − 3)(x + 3)
Cancel the common factor. A canceled factor is a hole, not a vertical asymptote.
x(x + 1)(x + 3)
x = 3
Substitute that x-value into the reduced fraction.
y = (3)(3 + 1)(3 + 3)
y = 126
y = 2
The hole is (3, 2).
A vertical asymptote is a denominator zero that does not cancel.
x + 3 = 0
x = −3
The vertical asymptote is x = −3.
The degree of the numerator is one more than the degree of the denominator, so the other asymptote is oblique.
(x + 3)(x − 2) = x² + x − 6
x² + x = (x² + x − 6) + 6
y = x − 2
The oblique asymptote is y = x − 2.
Answer: hole (3, 2); vertical asymptote x = −3; oblique asymptote y = x − 2
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