Graph each rational function. See Examples 5–9. ƒ(x)=[(x-5)(x-2)]/(x^2+9)

Verified step by step guidance

1. Identify the numerator and the denominator of the rational function: The numerator is \((x-5)(x-2)\) and the denominator is \(x^2 + 9\).
2. Determine the x-intercepts by setting the numerator equal to zero: Solve \((x-5)(x-2) = 0\) to find the x-values where the function crosses the x-axis.
3. Identify the vertical asymptotes by setting the denominator equal to zero: Solve \(x^2 + 9 = 0\). Note that this equation has no real solutions, indicating no vertical asymptotes.
4. Determine the horizontal asymptote by comparing the degrees of the numerator and the denominator: Since the degree of the numerator (2) is equal to the degree of the denominator (2), the horizontal asymptote is \(y = \frac{1}{1} = 1\).
5. Plot the x-intercepts, horizontal asymptote, and analyze the behavior of the function as \(x\) approaches positive and negative infinity to sketch the graph of the rational function.

Key Concepts

Rational Functions

A rational function is a function that can be expressed as the ratio of two polynomials. The general form is f(x) = P(x)/Q(x), where P(x) and Q(x) are polynomials. Understanding the behavior of rational functions involves analyzing their asymptotes, intercepts, and overall shape, which are influenced by the degrees and coefficients of the polynomials.

Graphing Techniques

Graphing rational functions requires specific techniques, such as identifying vertical and horizontal asymptotes, finding x- and y-intercepts, and determining the function's end behavior. Vertical asymptotes occur where the denominator is zero, while horizontal asymptotes describe the behavior as x approaches infinity. These features help sketch the graph accurately.

Asymptotes

Asymptotes are lines that the graph of a function approaches but never touches. For rational functions, vertical asymptotes are found at values of x that make the denominator zero, indicating points of discontinuity. Horizontal asymptotes describe the function's behavior as x approaches infinity, providing insight into the function's long-term trends and limits.