Pre-Calc P1 HW:12.05.24: Graphs of Rational Functions

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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, i.e., \( f(x) = \frac{P(x)}{Q(x)} \) where \( P(x) \) and \( Q(x) \) are polynomials.

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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 infinity or negative infinity. It indicates the behavior of the function at extreme values of \( x \).

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How do you find horizontal asymptotes for rational functions?

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To find horizontal asymptotes of a rational function \( f(x) = \frac{P(x)}{Q(x)} \), compare the degrees of the polynomials \( P \) and \( Q \): 1. If degree of \( P < \) degree of \( Q \), then \( y = 0 \) is the horizontal asymptote. 2. If degree of \( P = \) degree of \( Q \), then \( y = \frac{a}{b} \) where \( a \) and \( b \) are the leading coefficients of \( P \) and \( Q \). 3. If degree of \( P > \) degree of \( Q \), there is no horizontal asymptote.

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What is a vertical asymptote?

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A vertical asymptote is a vertical line \( x = a \) where the function approaches infinity or negative infinity as \( x \) approaches \( a \). It occurs where the denominator of a rational function is zero.

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How do you find vertical asymptotes for rational functions?

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To find vertical asymptotes of a rational function \( f(x) = \frac{P(x)}{Q(x)} \), set the denominator \( Q(x) = 0 \) and solve for \( x \).

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What is the domain of a rational function?

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The domain of a rational function is all real numbers except where the denominator is zero. For example, for \( f(x) = \frac{1}{x-3} \), the domain is \( x \neq 3 \).

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How do you find the x-intercepts of a rational function?

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To find the x-intercepts of a rational function \( f(x) = \frac{P(x)}{Q(x)} \), set the numerator \( P(x) = 0 \) and solve for \( x \).

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How do you find the y-intercept of a rational function?

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To find the y-intercept of a rational function \( f(x) = \frac{P(x)}{Q(x)} \), evaluate the function at \( x = 0 \), i.e., find \( f(0) \).

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What is the significance of the degree of the numerator and denominator in rational functions?

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The degree of the numerator and denominator determines the end behavior of the rational function and the existence of horizontal asymptotes.

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What is the relationship between the x-intercepts and vertical asymptotes?

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X-intercepts occur where the function crosses the x-axis (numerator = 0), while vertical asymptotes occur where the function is undefined (denominator = 0). They are not the same.