Sketch the graphs of the rational functions in Exercises 53–60.

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y = ------------------
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Identify the rational function: \( f(x) = \frac{x^2}{x^2 - 4} \). This function is a ratio of two polynomials.

Determine the domain of the function. The denominator \( x^2 - 4 \) cannot be zero, so solve \( x^2 - 4 = 0 \) to find the values of \( x \) that are not in the domain.

Find the vertical asymptotes by setting the denominator equal to zero: \( x^2 - 4 = 0 \). Solve for \( x \) to find the vertical asymptotes at \( x = 2 \) and \( x = -2 \).

Determine the horizontal asymptote by comparing the degrees of the numerator and the denominator. Since both are degree 2, the horizontal asymptote is \( y = \frac{1}{1} = 1 \).

Analyze the behavior of the function around the asymptotes and intercepts. Check the sign of the function in the intervals determined by the vertical asymptotes and plot key points to sketch the graph.

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주요 개념

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Rational Functions

A rational function is a ratio of two polynomials, where the numerator and the denominator are polynomials. In the given function y = x² / (x² - 4), x² is the numerator and x² - 4 is the denominator. Understanding the behavior of rational functions involves analyzing their asymptotes, intercepts, and domain.

Vertical Asymptotes

Vertical asymptotes occur in rational functions where the denominator equals zero, causing the function to approach infinity. For y = x² / (x² - 4), setting the denominator x² - 4 = 0 gives x = ±2. These values are where the vertical asymptotes occur, indicating the function's undefined points and guiding the graph's behavior near these lines.

Horizontal Asymptotes

Horizontal asymptotes describe the behavior of a function as x approaches infinity. For rational functions, if the degrees of the numerator and denominator are equal, the horizontal asymptote is the ratio of their leading coefficients. In y = x² / (x² - 4), both the numerator and denominator have the same degree, so the horizontal asymptote is y = 1, indicating the function's end behavior.

Sketch the graphs of the rational functions in Exercises 53–60.𝓍²y = ------------------𝓍² ― 4