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

Verified step by step guidance

1. Identify the domain of the function by finding the values of \(x\) that make the denominator zero. Solve the equation \(x^{2} - 4 = 0\) to find these values.
2. Determine the vertical asymptotes by setting the denominator equal to zero and solving for \(x\). These are the values excluded from the domain where the function tends to infinity.
3. Find the horizontal asymptote by comparing the degrees of the numerator and denominator. Since the numerator is degree 1 and the denominator is degree 2, the horizontal asymptote is \(y = 0\).
4. Calculate the \(x\)-intercepts by setting the numerator equal to zero and solving for \(x\). This gives the points where the graph crosses the \(x\)-axis.
5. Calculate the \(y\)-intercept by evaluating \(f(0)\), substituting \(x=0\) into the function to find where the graph crosses the \(y\)-axis.

Key Concepts

Domain of a Rational Function

The domain of a rational function includes all real numbers except where the denominator equals zero. For f(x) = 2x/(x²−4), set the denominator x²−4 = 0 to find excluded values, which are x = ±2. Understanding the domain helps identify vertical asymptotes and restrictions on the graph.

Asymptotes of Rational Functions

Asymptotes are lines that the graph approaches but never touches. Vertical asymptotes occur where the denominator is zero (and numerator is nonzero), while horizontal asymptotes describe end behavior based on degrees of numerator and denominator. For f(x) = 2x/(x²−4), vertical asymptotes are at x = ±2, and the horizontal asymptote is y = 0.

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. Each step builds understanding of the function’s behavior, ensuring an accurate and complete graph of f(x) = 2x/(x²−4).