Question

f(x) = \frac{-x^2 + 4}{x + 2}
answer attempt 1 out of 2
horizontal asymptote: y = \square no horizontal asymptote
vertical asymptote: x = \square no vertical asymptote
x-intercept: (\square, 0) no x-intercept
y-intercept: (0, \square) no y-intercept
hole: (\square, \square) no hole

Explanation:

Step1: Simplify the function

First, factor the numerator: \(-x^2 + 4 = -(x^2 - 4) = -(x - 2)(x + 2)\). So the function \(f(x)=\frac{-(x - 2)(x + 2)}{x + 2}\). We can cancel out the common factor \(x + 2\) (for \(x
eq - 2\)), so \(f(x)=-(x - 2)=-x + 2\) (with a hole at \(x=-2\) because the original function is undefined at \(x = - 2\) but the simplified function is defined there).

Step2: Find Horizontal Asymptote

For a rational function \(y=\frac{N(x)}{D(x)}\), if the degree of \(N(x)\) is greater than the degree of \(D(x)\), there is no horizontal asymptote. Here, after simplifying, the function is linear (\(f(x)=-x + 2\)), but originally the numerator degree (2) is greater than denominator degree (1), so no horizontal asymptote.

Step3: Find Vertical Asymptote

Vertical asymptotes occur where the denominator is zero and the numerator is non - zero. But we canceled \(x + 2\), so there is no vertical asymptote (because the factor that made the denominator zero was canceled, and the remaining function is linear, which has no vertical asymptote).

Step4: Find x - Intercept

Set \(y = 0\) in the simplified function \(f(x)=-x + 2\). So \(0=-x + 2\), solving for \(x\) gives \(x = 2\). So the x - intercept is \((2,0)\).

Step5: Find y - Intercept

Set \(x = 0\) in the simplified function \(f(x)=-x + 2\). Then \(f(0)=-0 + 2=2\). So the y - intercept is \((0,2)\).

Step6: Find Hole

The hole occurs at the \(x\) - value where we canceled the factor, i.e., \(x=-2\). To find the \(y\) - value of the hole, plug \(x = - 2\) into the simplified function \(f(x)=-x + 2\). So \(f(-2)=-(-2)+2=2 + 2 = 4\). So the hole is at \((-2,4)\).

Answer:

Horizontal Asymptote: No horizontal asymptote
Vertical Asymptote: No vertical asymptote
x - Intercept: \((2,0)\)
y - Intercept: \((0,2)\)
Hole: \((-2,4)\)