Question

question
find the equation of all vertical asymptotes of the following function.
f(x)=\frac{2 x-2}{sqrt{-x^{2}+2 x+15}}
answer attempt 1 out of 2

Explanation:

Step1: Find the domain of the function

For the function \(f(x)=\frac{2x - 2}{\sqrt{-x^{2}+2x + 15}}\), the denominator \(\sqrt{-x^{2}+2x + 15}\) must satisfy \(-x^{2}+2x + 15>0\).
Solve the quadratic inequality \(-x^{2}+2x + 15>0\). First, solve the equation \(-x^{2}+2x + 15 = 0\). Using the quadratic formula \(x=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}\) for \(ax^{2}+bx + c = 0\) (here \(a=-1\), \(b = 2\), \(c = 15\)), we have \(x=\frac{-2\pm\sqrt{4+60}}{-2}=\frac{-2\pm\sqrt{64}}{-2}=\frac{-2\pm8}{-2}\).
The roots are \(x=-3\) and \(x = 5\). The quadratic function \(y=-x^{2}+2x + 15\) is a parabola opening downwards (\(a=-1<0\)), so the solution of \(-x^{2}+2x + 15>0\) is \(-3

Step2: Analyze the vertical asymptotes

Vertical asymptotes occur where the function is undefined and the one - sided limits are infinite. Since the function \(f(x)\) is only defined for \(-3

Answer:

No Vertical Asymptotes