In Exercises 57–80, follow the seven steps to graph each rational function. f(x)=2x²/(x²+4)

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1. Identify the domain of the function $f\left(x\right)=\frac{2x^2}{x^2+4}$. Since the denominator is $x^2+\; 4$, which is always positive (never zero), the domain is all real numbers $x\in \; ℝ$.
2. Find the intercepts: For the y-intercept, evaluate $f(0)$. For the x-intercepts, set the numerator equal to zero and solve for $x$, i.e., solve $2x^2\; =\; 0$.
3. Determine the vertical and horizontal asymptotes: Since the denominator never equals zero, there are no vertical asymptotes. For the horizontal asymptote, compare the degrees of numerator and denominator (both degree 2). The horizontal asymptote is the ratio of the leading coefficients, which is $y\; =\; \backslash frac\{2\}\{1\}\; =\; 2$.
4. Analyze the behavior of the function near the asymptotes and at extreme values of $x$. Consider limits as $x\; \to \; \pm \infty$ to confirm the horizontal asymptote and check values near zero to understand the shape of the graph.
5. Sketch the graph using the information gathered: plot intercepts, draw the horizontal asymptote $y=2$, and use the behavior analysis to shape the curve accordingly.

Key Concepts

Rational Functions

A rational function is a ratio of two polynomials, expressed as f(x) = P(x)/Q(x). Understanding its domain, where the denominator is not zero, and behavior is essential for graphing. In this case, f(x) = 2x²/(x² + 4) is a rational function with no real zeros in the denominator.

Domain and Asymptotes

The domain of a rational function excludes values that make the denominator zero. Vertical asymptotes occur at these values, while horizontal or oblique asymptotes describe end behavior as x approaches infinity. For f(x) = 2x²/(x² + 4), the denominator never zeroes, so no vertical asymptotes exist, but horizontal asymptotes can be found by comparing degrees.

Graphing Steps for Rational Functions

Graphing rational functions involves seven steps: finding the domain, intercepts, asymptotes, analyzing end behavior, plotting points, and sketching the curve. This systematic approach helps visualize the function accurately, especially for functions like f(x) = 2x²/(x² + 4), where symmetry and limits guide the shape.