Rational Function Graphs

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What is a rational function?

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What are vertical asymptotes in rational functions?

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Vertical asymptotes are lines that the graph of a rational function approaches but never touches or crosses. They occur at values of x that make the denominator Q(x) equal to zero.

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What are horizontal asymptotes in rational functions?

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Horizontal asymptotes describe the behavior of a rational function as x approaches infinity or negative infinity. They indicate the value that f(x) approaches as x becomes very large or very small.

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How do you find vertical asymptotes of a rational function?

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To find vertical asymptotes, set the denominator Q(x) equal to zero and solve for x. The solutions are the x-values where vertical asymptotes occur.

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How do you find horizontal asymptotes of a rational function?

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To find horizontal asymptotes, compare the degrees of the numerator and denominator: 1) If degree of P < degree of Q, y = 0 is the horizontal asymptote. 2) If degree of P = degree of Q, y = \frac{leading coefficient of P}{leading coefficient of Q}. 3) If degree of P > degree of Q, there is no horizontal asymptote.

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What is the significance of holes in the graph of a rational function?

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Holes occur in the graph of a rational function at x-values that make both the numerator and denominator equal to zero. They indicate points where the function is undefined but can be simplified.

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What is the end behavior of rational functions?

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The end behavior of a rational function describes how the function behaves as x approaches positive or negative infinity, often determined by the leading terms of the numerator and denominator.

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What is the domain of a rational function?

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The domain of a rational function consists of all real numbers except for the x-values that make the denominator equal to zero.

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How do you graph a rational function?

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To graph a rational function, identify vertical and horizontal asymptotes, find intercepts, determine the domain, and plot points to understand the function's behavior.

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What is the x-intercept of a rational function?

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The x-intercept of a rational function is found by setting the numerator P(x) equal to zero and solving for x.