Question

cation
all intercepts and asymptotes of the following rational function. if you get a fraction or decimal then plot as close to the true location as possible.
$f(x) = 2 + \frac{3}{x + 6}$
plot rational function
vertical asymptote horizontal asymptote
x-intercept y-intercept
click on a feature then drag it into place.

Explanation:

Step1: Find Vertical Asymptote

The vertical asymptote of a rational function occurs where the denominator is zero (and the numerator is not zero there). For \( f(x) = 2+\frac{3}{x + 6}\), the denominator is \(x + 6\). Set \(x+6=0\), so \(x=-6\).

Step2: Find Horizontal Asymptote

For a rational function, if the degrees of the numerator and denominator of the fractional part are equal (or the fractional part approaches zero as \(|x|\to\infty\)), we analyze the limit as \(x\to\pm\infty\). As \(x\to\pm\infty\), \(\frac{3}{x + 6}\to0\), so \(f(x)\to2\). Thus, the horizontal asymptote is \(y = 2\).

Step3: Find x - intercept

Set \(f(x)=0\): \(0=2+\frac{3}{x + 6}\). Subtract 2: \(-2=\frac{3}{x + 6}\). Multiply both sides by \(x + 6\) (assuming \(x
eq - 6\)): \(-2(x + 6)=3\). Expand: \(-2x-12 = 3\). Add 12: \(-2x=15\). Divide by - 2: \(x=-\frac{15}{2}=-7.5\).

Step4: Find y - intercept

Set \(x = 0\) in \(f(x)\): \(f(0)=2+\frac{3}{0 + 6}=2+\frac{3}{6}=2 + 0.5=2.5=\frac{5}{2}\).

Answer:

• Vertical Asymptote: \(x=-6\)
• Horizontal Asymptote: \(y = 2\)
• x - intercept: \(x=-\frac{15}{2}\) (or \(x=-7.5\))
• y - intercept: \(y=\frac{5}{2}\) (or \(y = 2.5\))