Questions and Answers
What occurs when the denominator of a rational function is equal to zero?
How do you find horizontal asymptotes of rational functions?
What is a hole in a rational function?
How do you find y-intercepts of rational functions?
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What is the condition for a horizontal asymptote to occur in a rational function?
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What is the purpose of finding asymptotes in rational functions?
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What is the vertical asymptote of the function f(x) = 3 / (x + 4)?
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For the function f(x) = 4x^2 / (2x^2 + 5), what is the horizontal asymptote?
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What happens to the rational function f(x) = (x^2 - 9) / (x - 3)?
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What is the x-intercept of the function f(x) = (2x - 6) / (x + 1)?
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Which of the following statements is true regarding horizontal asymptotes?
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For the function f(x) = (x^2 + 2x) / (x^2 - 1), where is the hole located?
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What is the y-intercept of the function f(x) = 5x^2 / (x - 4)?
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Given f(x) = (x^3 - 1) / (x^2 + 3), how do you find the intercepts?
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Study Notes
Asymptotes of Rational Equations
Vertical Asymptotes
- A vertical asymptote occurs when the denominator of the rational function is equal to zero.
- To find vertical asymptotes, set the denominator equal to zero and solve for x.
- Vertical asymptotes are not part of the graph, but rather a boundary beyond which the function cannot be defined.
Horizontal Asymptotes
- A horizontal asymptote occurs when the degree of the numerator is less than or equal to the degree of the denominator.
- To find horizontal asymptotes, divide the numerator by the denominator and look for the quotient.
- If the quotient is a constant, that constant is the horizontal asymptote.
- If the quotient is a polynomial, the horizontal asymptote is the ratio of the leading coefficients.
Holes
- A hole occurs when there is a common factor in the numerator and denominator that cancels out.
- To find holes, factor the numerator and denominator and cancel out any common factors.
- Holes are points on the graph where the function is not defined, but the graph appears to have a "hole" at that point.
Intercepts
- Intercepts occur when the graph crosses the x-axis or y-axis.
- To find x-intercepts, set the numerator equal to zero and solve for x.
- To find y-intercepts, set x equal to zero and solve for y.
- Intercepts are points on the graph where the function crosses the axis.
Asymptotes of Rational Equations
Vertical Asymptotes
- Occur when the denominator of a rational function is equal to zero.
- Found by setting the denominator equal to zero and solving for x.
- Represent a boundary beyond which the function cannot be defined and are not part of the graph.
Horizontal Asymptotes
- Occur when the degree of the numerator is less than or equal to the degree of the denominator.
- Found by dividing the numerator by the denominator and looking for the quotient.
- If the quotient is a constant, it is the horizontal asymptote.
- If the quotient is a polynomial, the horizontal asymptote is the ratio of the leading coefficients.
Holes
- Occur when there is a common factor in the numerator and denominator that cancels out.
- Found by factoring the numerator and denominator and cancelling out common factors.
- Represent points on the graph where the function is not defined, but the graph appears to have a "hole" at that point.
Intercepts
- Occur when the graph crosses the x-axis or y-axis.
- x-intercepts found by setting the numerator equal to zero and solving for x.
- y-intercepts found by setting x equal to zero and solving for y.
- Represent points on the graph where the function crosses the axis.
Asymptotes of Rational Equations
Vertical Asymptotes
- Occur when the denominator of a rational function is equal to zero
- Found by setting the denominator equal to zero and solving for x
- Solutions to the equation are the vertical asymptotes
- Represented by a vertical line on the graph
Horizontal Asymptotes
- Occur when the degree of the numerator is less than or equal to the degree of the denominator
- Found by dividing the leading coefficients of the numerator and denominator
- If the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is y = 0
- Represented by a horizontal line on the graph
Holes
- Occur when there is a common factor in the numerator and denominator that cancels out
- Found by factoring the numerator and denominator and canceling out any common factors
- Resulting function will have a hole at the point where the common factor is equal to zero
Intercepts
- x-intercepts occur when y = 0
- Found by setting the numerator equal to zero and solving for x
- y-intercepts occur when x = 0
- Found by plugging in x = 0 into the function
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Description
Learn about asymptotes in rational functions, including vertical and horizontal asymptotes, and how to find them. Understood the concept of boundaries beyond which the function cannot be defined.
