Question

quiz 2 lt2
fill in the blanks
$f(x)=\frac{5}{x + 6}-6$
vertical asymptote: $x =$
horizontal asymptote: $y =$
x - intercept: $x =$
y - intercept: $y =$

Explanation:

Step1: Find vertical asymptote

Set denominator \(x + 6=0\), so \(x=-6\).

Step2: Find horizontal asymptote

For \(y=\frac{5}{x + 6}-6\), as \(x\to\pm\infty\), \(\frac{5}{x + 6}\to0\), so \(y=-6\).

Step3: Find \(x\)-intercept

Set \(y = 0\), \(\frac{5}{x + 6}-6=0\), \(\frac{5}{x + 6}=6\), \(6(x + 6)=5\), \(6x+36 = 5\), \(6x=-31\), \(x=-\frac{31}{6}\).

Step4: Find \(y\)-intercept

Set \(x = 0\), \(y=\frac{5}{0 + 6}-6=\frac{5}{6}-6=\frac{5 - 36}{6}=-\frac{31}{6}\).

Answer:

Vertical asymptote: \(x = -6\)
Horizontal asymptote: \(y = -6\)
\(x\)-intercept: \(x = -\frac{41}{6}\)
\(y\)-intercept: \(y = -\frac{31}{6}\)