Graph each rational function. See Examples 5–9. ƒ(x)=(x^2+8x+16)/(x^2+4x-5)

Verified step by step guidance

1. Identify the vertical asymptotes by setting the denominator equal to zero and solving for \(x\).
2. Determine the horizontal asymptote by comparing the degrees of the numerator and the denominator.
3. Find the x-intercepts by setting the numerator equal to zero and solving for \(x\).
4. Calculate the y-intercept by evaluating \(f(0)\).
5. Sketch the graph using the asymptotes, intercepts, and any additional points as needed.

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 rational functions involves analyzing their behavior, including asymptotes, intercepts, and domain restrictions, which arise from the values that make the denominator zero.

Graphing Techniques

Graphing rational functions requires specific techniques to accurately represent their behavior. Key steps include finding the x-intercepts by setting the numerator to zero, the y-intercept by evaluating f(0), and vertical asymptotes by identifying values that make the denominator zero. Horizontal asymptotes can also be determined by comparing the degrees of the numerator and denominator.

Asymptotes

Asymptotes are lines that the graph of a function approaches but never touches. For rational functions, vertical asymptotes occur at values of x that make the denominator zero, while horizontal asymptotes describe the end behavior of the function as x approaches infinity. Understanding asymptotes is crucial for sketching accurate graphs and predicting the function's behavior.