Graph each rational function. See Examples 5–9. ƒ(x)=[(x+3)(x-5)]/[(x+1)(x-4)]

Verified step by step guidance

1. Identify the vertical asymptotes by setting the denominator equal to zero: \((x+1)(x-4) = 0\). Solve for \(x\) to find the values where the function is undefined.
2. Determine the horizontal asymptote by comparing the degrees of the numerator and the denominator. Since both are quadratic (degree 2), the horizontal asymptote is \(y = \frac{a}{b}\), where \(a\) and \(b\) are the leading coefficients of the numerator and denominator, respectively.
3. Find the x-intercepts by setting the numerator equal to zero: \((x+3)(x-5) = 0\). Solve for \(x\) to find the points where the graph crosses the x-axis.
4. Calculate the y-intercept by evaluating \(f(0)\). Substitute \(x = 0\) into the function and simplify to find the y-coordinate where the graph crosses the y-axis.
5. Sketch the graph using the information from the previous steps: plot the intercepts, draw the asymptotes as dashed lines, and sketch the curve, ensuring it approaches the asymptotes appropriately.

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 discontinuities, which are critical for graphing them accurately.

Asymptotes

Asymptotes are lines that a graph approaches but never touches. There are vertical asymptotes, which occur where the denominator of a rational function is zero, and horizontal asymptotes, which describe the behavior of the function as x approaches infinity. Identifying these asymptotes is essential for understanding the overall shape and limits of the graph.

Intercepts

Intercepts are points where the graph of a function crosses the axes. The x-intercepts occur when f(x) = 0, which happens when the numerator of the rational function is zero. The y-intercept is found by evaluating f(0). Knowing the intercepts helps in sketching the graph and understanding the function's behavior at specific points.