Follow the seven steps to graph each rational function. f(x)=2x²/(x²+4)

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Identify the domain of the function by finding all values of \(x\) for which the denominator is not zero. For \(f(x) = \frac{2x^{2}}{x^{2} + 4}\), set the denominator equal to zero: \(x^{2} + 4 = 0\) and solve for \(x\).

Find the intercepts: To find the \(y\)-intercept, evaluate \(f(0)\). To find the \(x\)-intercepts, set the numerator equal to zero and solve for \(x\).

Determine any vertical asymptotes by analyzing where the denominator is zero and the numerator is not zero. Since the denominator is \(x^{2} + 4\), check if it has real roots.

Find the horizontal asymptote by comparing the degrees of the numerator and denominator polynomials. Since both numerator and denominator are degree 2, divide the leading coefficients to find the horizontal asymptote.

Analyze the behavior of the function near the asymptotes and at large values of \(x\) to understand the end behavior, then sketch the graph using all the information gathered.

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주요 개념

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Rational Functions

A rational function is a ratio of two polynomials, expressed as f(x) = P(x)/Q(x). Understanding the domain, where the denominator Q(x) ≠ 0, is essential to avoid undefined values and to analyze the function's behavior.

Graphing Steps for Rational Functions

Graphing rational functions involves seven key steps: finding the domain, intercepts, asymptotes (vertical, horizontal, or oblique), analyzing end behavior, plotting points, and sketching the curve. These steps help visualize the function accurately.

Asymptotes and End Behavior

Asymptotes are lines that the graph approaches but never touches. Vertical asymptotes occur where the denominator is zero, and horizontal or oblique asymptotes describe the function's behavior as x approaches infinity or negative infinity.

Follow the seven steps to graph each rational function. f(x)=2x2/(x2+4)