Question

sketch the graph of the rational function.
$f(x) = \frac{x + 2}{x - 2}$

Explanation:

Step1: Find Vertical Asymptote

Set denominator \(x - 2 = 0\), so \(x = 2\) is vertical asymptote.

Step2: Find Horizontal Asymptote

Degree of numerator (\(x + 2\)) and denominator (\(x - 2\)) are equal (both 1). So horizontal asymptote is ratio of leading coefficients: \(y=\frac{1}{1}=1\).

Step3: Find Intercepts

• x - intercept: Set \(f(x)=0\), \(\frac{x + 2}{x - 2}=0\) ⇒ \(x=-2\).
• y - intercept: Set \(x = 0\), \(f(0)=\frac{0 + 2}{0 - 2}=-1\).

Step4: Analyze Behavior

• For \(x\to2^+\), \(f(x)\to+\infty\); for \(x\to2^-\), \(f(x)\to-\infty\).
• For \(x\to\pm\infty\), \(f(x)\to1\) (approaches horizontal asymptote).
• Plot points: \((-2,0)\), \((0, - 1)\), and use asymptotes to sketch two branches: one left of \(x = 2\) (passing through \((-2,0)\) and \((0,-1)\), approaching \(x = 2\) from left and \(y = 1\) as \(x\to-\infty\)), one right of \(x = 2\) (approaching \(x = 2\) from right and \(y = 1\) as \(x\to+\infty\)).

Answer:

To sketch \(f(x)=\frac{x + 2}{x - 2}\), follow these key steps (as explained above) to draw the graph with vertical asymptote \(x = 2\), horizontal asymptote \(y = 1\), x - intercept at \((-2,0)\), y - intercept at \((0, - 1)\), and two branches approaching the asymptotes. (The actual sketch would be drawn on the provided coordinate plane using the analyzed behavior and intercepts.)