Unit 4 (Rationals) Test Review

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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 and Q are polynomials and Q(x) ≠ 0.

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What is the horizontal asymptote of a rational function?

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The horizontal asymptote of a rational function is a horizontal line that the graph approaches as x approaches infinity. It can be determined by comparing the degrees of the numerator and denominator.

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

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Vertical asymptotes occur at the values of x that make the denominator zero, provided that these values do not also make the numerator zero.

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What is an x-intercept?

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An x-intercept is a point where the graph of a function crosses the x-axis, which occurs when \( f(x) = 0 \).

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

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To find the x-intercept of a rational function, set the numerator equal to zero and solve for x.

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

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The degree of the numerator and denominator helps determine the behavior of the function, including the presence and location of asymptotes.

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What is the formula for adding two rational functions?

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To add two rational functions \( \frac{a}{b} + \frac{c}{d} \), find a common denominator: \( \frac{a \cdot d + c \cdot b}{b \cdot d} \).

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What is the formula for dividing two rational functions?

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To divide two rational functions \( \frac{a}{b} \div \frac{c}{d} \), multiply by the reciprocal: \( \frac{a}{b} \cdot \frac{d}{c} = \frac{a \cdot d}{b \cdot c} \).

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What is the process for simplifying a rational function?

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To simplify a rational function, factor both the numerator and denominator and then cancel any common factors.

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What does it mean for a rational function to have no x-intercept?

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A rational function has no x-intercept if the numerator is never zero, meaning the function does not cross the x-axis.