Follow the seven steps to graph each rational function. f(x)=− 1/(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. Set the denominator equal to zero: \(x^{2} - 4 = 0\).
2. Solve the equation \(x^{2} - 4 = 0\) by factoring it as \((x - 2)(x + 2) = 0\), which gives the values \(x = 2\) and \(x = -2\). These values are excluded from the domain because they make the denominator zero, causing vertical asymptotes.
3. Determine the vertical asymptotes by noting that the function is undefined at \(x = 2\) and \(x = -2\). So, draw vertical dashed lines at these \(x\)-values.
4. Find the horizontal asymptote by analyzing the degrees of the numerator and denominator. The numerator is a constant (\(-1\)), and the denominator is a quadratic (\(x^{2} - 4\)). Since the degree of the denominator is greater than the numerator, the horizontal asymptote is \(y = 0\).
5. Create a table of values by choosing \(x\)-values around the vertical asymptotes (for example, values less than \(-2\), between \(-2\) and \$2\(, and greater than \)2\(), then calculate the corresponding \)f(x)$ values to understand the behavior of the graph in each interval.

Key Concepts

Rational Functions

A rational function is a ratio of two polynomials, expressed as f(x) = P(x)/Q(x). Understanding its domain, zeros, and behavior is essential, especially where the denominator equals zero, causing vertical asymptotes or undefined points.

Asymptotes of Rational Functions

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

Graphing Steps for Rational Functions

Graphing involves identifying domain restrictions, intercepts, asymptotes, and behavior near asymptotes, then plotting points to sketch the curve. Following a systematic approach ensures an accurate representation of the function.