Problem 41 - Precalculus : Building Concepts and Connections

Question

Sketch a graph of the rational function. Indicate any vertical and horizontal asymptote(s) and all intercepts. $$g(x)=\frac{x+5}{x-2}$$

Step-by-Step Solution

Answer

The graph of the function \(g(x)=\frac{x+5}{x-2}\) has a vertical asymptote at \(x=2\), a horizontal asymptote at \(y=1\), an x-intercept at \(x=-5\) and a y-intercept at \(y=-2.5\).

Step 1: Determine the Vertical Asymptote

The vertical asymptote of a rational function is found by setting the denominator equal to zero and solving for x. Thus, \(x-2=0\), which gives \(x=2\). This is the vertical asymptote.

Step 2: Determine the Horizontal Asymptote

For the given function \(g(x)=\frac{x+5}{x-2}\), we can observe that the degrees of the numerator and the denominator are equal. In such a situation, the horizontal asymptote is the ratio of the leading coefficients, i.e., \(y = \frac{1}{1} = 1\). Therefore, the horizontal asymptote is at \(y=1\).

Step 3: Find the x-intercept and the y-intercept

To find the x-intercept, set \(y=0\), which means solving \(\frac{x+5}{x-2} = 0\). This gives \(x = -5\) for the x-intercept. To find the y-intercept, set \(x=0\), which gives us \(\frac{0+5}{0-2} = -2.5\) for the y-intercept.

Step 4: Graph the Function

Plot the vertical asymptote at \(x=2\) and the horizontal asymptote at \(y=1\). Next, mark the points of the intercepts at \(x=-5\) and \(y=-2.5\). Finally, sketch the graph of the function, keeping in mind these intercepts and asymptotes.