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Questions and Answers
What occurs when the denominator of a rational function is equal to zero?
What occurs when the denominator of a rational function is equal to zero?
- A horizontal asymptote
- A vertical asymptote (correct)
- An x-intercept
- A hole in the graph
How do you find horizontal asymptotes of rational functions?
How do you find horizontal asymptotes of rational functions?
- Set the denominator equal to zero and solve for x
- Set the numerator equal to zero and solve for x
- Add the numerator and denominator and solve for x
- Divide the numerator by the denominator and look for the quotient (correct)
What is a hole in a rational function?
What is a hole in a rational function?
- A point on the graph where the function is defined
- A vertical asymptote
- A horizontal asymptote
- A point on the graph where the function is not defined (correct)
How do you find y-intercepts of rational functions?
How do you find y-intercepts of rational functions?
What is the condition for a horizontal asymptote to occur in a rational function?
What is the condition for a horizontal asymptote to occur in a rational function?
What is the purpose of finding asymptotes in rational functions?
What is the purpose of finding asymptotes in rational functions?
What is the vertical asymptote of the function f(x) = 3 / (x + 4)?
What is the vertical asymptote of the function f(x) = 3 / (x + 4)?
For the function f(x) = 4x^2 / (2x^2 + 5), what is the horizontal asymptote?
For the function f(x) = 4x^2 / (2x^2 + 5), what is the horizontal asymptote?
What happens to the rational function f(x) = (x^2 - 9) / (x - 3)?
What happens to the rational function f(x) = (x^2 - 9) / (x - 3)?
What is the x-intercept of the function f(x) = (2x - 6) / (x + 1)?
What is the x-intercept of the function f(x) = (2x - 6) / (x + 1)?
Which of the following statements is true regarding horizontal asymptotes?
Which of the following statements is true regarding horizontal asymptotes?
For the function f(x) = (x^2 + 2x) / (x^2 - 1), where is the hole located?
For the function f(x) = (x^2 + 2x) / (x^2 - 1), where is the hole located?
What is the y-intercept of the function f(x) = 5x^2 / (x - 4)?
What is the y-intercept of the function f(x) = 5x^2 / (x - 4)?
Given f(x) = (x^3 - 1) / (x^2 + 3), how do you find the intercepts?
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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