Rational Function Graphs & Characteristics

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What is a rational function?

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A rational function is a function that can be expressed as the quotient of two polynomials, typically in the form f(x) = P(x)/Q(x), where P and Q are polynomials.

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What is an asymptote?

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An asymptote is a line that a graph approaches but never touches. It can be vertical, horizontal, or oblique.

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What is a horizontal asymptote?

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A horizontal asymptote is a horizontal line that the graph of a function approaches as x approaches positive or negative infinity.

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How do you find the horizontal asymptote of a rational function?

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To find the horizontal asymptote of a rational function, compare the degrees of the numerator and denominator. If the degree of the numerator is less, the asymptote is y = 0. If they are equal, the asymptote is y = leading coefficient of P / leading coefficient of Q.

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What is a vertical asymptote?

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A vertical asymptote is a vertical line that the graph of a function approaches as the input approaches a certain value, typically where the denominator of a rational function equals zero.

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How do you find the vertical asymptote of a rational function?

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To find the vertical asymptote, set the denominator of the rational function equal to zero and solve for x.

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What is the significance of the vertical asymptote in a rational function?

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The vertical asymptote indicates the values of x where the function is undefined and where the graph will approach infinity.

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What does the term 'domain' refer to in the context of rational functions?

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The domain of a rational function is the set of all possible input values (x) for which the function is defined, excluding values that make the denominator zero.

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What is the range of a rational function?

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The range of a rational function is the set of all possible output values (y) that the function can produce.

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What is the relationship between the degrees of the numerator and denominator in determining horizontal asymptotes?

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If the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is y = 0. If they are equal, the horizontal asymptote is y = leading coefficient of numerator / leading coefficient of denominator.