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Rational Functions: Asymptotes & Holes — Easy Worksheet 2
Grade 12 · MathSheets Name: ____________________ Date: __________
Answer each question. Write vertical asymptotes like x = 3 (two of them: x = -1 or x = 3) and horizontal or slant asymptotes like y = 2/3 or y = x + 1.
How to solve
Factor and cancel first: a cancelled factor gives a hole, a leftover zero of the denominator gives a vertical asymptote. For the horizontal asymptote compare degrees: lower on top gives y = 0, equal degrees give y = (ratio of leading coefficients). If the top is one degree higher, divide to get a slant asymptote.
Example
Find the vertical asymptote of f(x) = 5x - 6x + 1.
Answer: x = -1
Not quite. The answer is
💡 The denominator x + 3 is 0 when x = −3. The numerator there is 5(−3) − 5 = −15 − 5 = −20, not 0, so nothing cancels. The vertical asymptote is x = −3.
✓ Correct!
Not quite. The answer is
💡 The denominator x + 3 is 0 when x = −3. The numerator there is 5(−3) − 5 = −15 − 5 = −20, not 0, so nothing cancels. The vertical asymptote is x = −3.
💡 How to solve: Rational Functions: Asymptotes & Holes
Factor and cancel first: a cancelled factor gives a hole, a leftover zero of the denominator gives a vertical asymptote. For the horizontal asymptote compare degrees: lower on top gives y = 0, equal degrees give y = (ratio of leading coefficients). If the top is one degree higher, divide to get a slant asymptote.
Try a similar one
Find the vertical asymptote of f(x) = 3x + 1x - 1.
✓ You've got it! Press Next to keep going. Not yet — the answer is . Read the steps above once more.
Not quite. The answer is
💡 The top and bottom both have degree 1, so the horizontal asymptote is y = (leading coefficient of the top) ÷ (leading coefficient of the bottom) = −86 = −43. The answer is y = −43.
✓ Correct!
Not quite. The answer is
💡 The top and bottom both have degree 1, so the horizontal asymptote is y = (leading coefficient of the top) ÷ (leading coefficient of the bottom) = −86 = −43. The answer is y = −43.
💡 How to solve: Rational Functions: Asymptotes & Holes
Factor and cancel first: a cancelled factor gives a hole, a leftover zero of the denominator gives a vertical asymptote. For the horizontal asymptote compare degrees: lower on top gives y = 0, equal degrees give y = (ratio of leading coefficients). If the top is one degree higher, divide to get a slant asymptote.
Try a similar one
Find the vertical asymptote of f(x) = x - 8x - 5.
✓ You've got it! Press Next to keep going. Not yet — the answer is . Read the steps above once more.
Not quite. The answer is
💡 The top has degree 1 and the bottom has degree 2. When the bottom has the higher degree, f(x) gets closer and closer to 0 as x grows, so the horizontal asymptote is y = 0.
✓ Correct!
Not quite. The answer is
💡 The top has degree 1 and the bottom has degree 2. When the bottom has the higher degree, f(x) gets closer and closer to 0 as x grows, so the horizontal asymptote is y = 0.
💡 How to solve: Rational Functions: Asymptotes & Holes
Factor and cancel first: a cancelled factor gives a hole, a leftover zero of the denominator gives a vertical asymptote. For the horizontal asymptote compare degrees: lower on top gives y = 0, equal degrees give y = (ratio of leading coefficients). If the top is one degree higher, divide to get a slant asymptote.
Try a similar one
Find the vertical asymptote of f(x) = 2x - 4x - 5.
✓ You've got it! Press Next to keep going. Not yet — the answer is . Read the steps above once more.
Not quite. The answer is
💡 A fraction is 0 when its numerator is 0. Solve 3x + 15 = 0. Subtract 15 from both sides: 3x = −15. Divide both sides by 3: x = −5. The denominator there is (−5) + 2 = −5 + 2 = −3, not 0, so the x-intercept is x = −5.
✓ Correct!
Not quite. The answer is
💡 A fraction is 0 when its numerator is 0. Solve 3x + 15 = 0. Subtract 15 from both sides: 3x = −15. Divide both sides by 3: x = −5. The denominator there is (−5) + 2 = −5 + 2 = −3, not 0, so the x-intercept is x = −5.
💡 How to solve: Rational Functions: Asymptotes & Holes
Factor and cancel first: a cancelled factor gives a hole, a leftover zero of the denominator gives a vertical asymptote. For the horizontal asymptote compare degrees: lower on top gives y = 0, equal degrees give y = (ratio of leading coefficients). If the top is one degree higher, divide to get a slant asymptote.
Try a similar one
Find the vertical asymptote of f(x) = 2x + 2x + 2.
✓ You've got it! Press Next to keep going. Not yet — the answer is . Read the steps above once more.
Not quite. The answer is
💡 The top has degree 1 and the bottom has degree 2. When the bottom has the higher degree, f(x) gets closer and closer to 0 as x grows, so the horizontal asymptote is y = 0.
✓ Correct!
Not quite. The answer is
💡 The top has degree 1 and the bottom has degree 2. When the bottom has the higher degree, f(x) gets closer and closer to 0 as x grows, so the horizontal asymptote is y = 0.
💡 How to solve: Rational Functions: Asymptotes & Holes
Factor and cancel first: a cancelled factor gives a hole, a leftover zero of the denominator gives a vertical asymptote. For the horizontal asymptote compare degrees: lower on top gives y = 0, equal degrees give y = (ratio of leading coefficients). If the top is one degree higher, divide to get a slant asymptote.
Try a similar one
Find the vertical asymptote of f(x) = x + 6x - 4.
✓ You've got it! Press Next to keep going. Not yet — the answer is . Read the steps above once more.
Not quite. The answer is
💡 The top and bottom both have degree 1, so the horizontal asymptote is y = (leading coefficient of the top) ÷ (leading coefficient of the bottom) = 31 = 3. The answer is y = 3.
✓ Correct!
Not quite. The answer is
💡 The top and bottom both have degree 1, so the horizontal asymptote is y = (leading coefficient of the top) ÷ (leading coefficient of the bottom) = 31 = 3. The answer is y = 3.
💡 How to solve: Rational Functions: Asymptotes & Holes
Factor and cancel first: a cancelled factor gives a hole, a leftover zero of the denominator gives a vertical asymptote. For the horizontal asymptote compare degrees: lower on top gives y = 0, equal degrees give y = (ratio of leading coefficients). If the top is one degree higher, divide to get a slant asymptote.
Try a similar one
Find the vertical asymptote of f(x) = 3x + 8x - 3.
✓ You've got it! Press Next to keep going. Not yet — the answer is . Read the steps above once more.
Not quite. The answer is
💡 The top and bottom both have degree 1, so the horizontal asymptote is y = (leading coefficient of the top) ÷ (leading coefficient of the bottom) = −72. The answer is y = −72.
✓ Correct!
Not quite. The answer is
💡 The top and bottom both have degree 1, so the horizontal asymptote is y = (leading coefficient of the top) ÷ (leading coefficient of the bottom) = −72. The answer is y = −72.
💡 How to solve: Rational Functions: Asymptotes & Holes
Factor and cancel first: a cancelled factor gives a hole, a leftover zero of the denominator gives a vertical asymptote. For the horizontal asymptote compare degrees: lower on top gives y = 0, equal degrees give y = (ratio of leading coefficients). If the top is one degree higher, divide to get a slant asymptote.
Try a similar one
Find the vertical asymptote of f(x) = 4x + 8x - 1.
✓ You've got it! Press Next to keep going. Not yet — the answer is . Read the steps above once more.
Not quite. The answer is
💡 A fraction is 0 when its numerator is 0. Solve x + 4 = 0. Subtract 4 from both sides: x = −4. The denominator there is (−4) + 2 = −4 + 2 = −2, not 0, so the x-intercept is x = −4.
✓ Correct!
Not quite. The answer is
💡 A fraction is 0 when its numerator is 0. Solve x + 4 = 0. Subtract 4 from both sides: x = −4. The denominator there is (−4) + 2 = −4 + 2 = −2, not 0, so the x-intercept is x = −4.
💡 How to solve: Rational Functions: Asymptotes & Holes
Factor and cancel first: a cancelled factor gives a hole, a leftover zero of the denominator gives a vertical asymptote. For the horizontal asymptote compare degrees: lower on top gives y = 0, equal degrees give y = (ratio of leading coefficients). If the top is one degree higher, divide to get a slant asymptote.
Try a similar one
Find the vertical asymptote of f(x) = 2x - 9x + 4.
✓ You've got it! Press Next to keep going. Not yet — the answer is . Read the steps above once more.
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Rational Functions: Asymptotes & Holes — Easy Worksheet 2
Answer each question. Write vertical asymptotes like x = 3 (two of them: x = -1 or x = 3) and horizontal or slant asymptotes like y = 2/3 or y = x + 1.