To analyze the function $f(x) = \frac{x - 4}{2x + 2}$, we need to find the zeros, vertical asymptotes, and the oblique asymptote.

Step 1: Find the Zero

To find the zero of the function, set the numerator equal to zero:

$x - 4 = 0$

Solving for $x$, we get:

$x = 4$

Thus, the zero of the function is at $x = 4$.

Step 2: Find the Vertical Asymptote

A vertical asymptote occurs where the denominator is zero, provided it does not cancel with the numerator. Set the denominator equal to zero:

$2x + 2 = 0$

Solving for $x$, we get:

$x = -1$

Therefore, there is a vertical asymptote at $x = -1$.

Step 3: Find the Oblique Asymptote

Since the degree of the numerator (1) is equal to the degree of the denominator (1), there is a slant, or oblique, asymptote. To find it, we divide the numerator by the denominator:

$\frac{x - 4}{2x + 2} = \frac{1}{2} - \frac{3}{2x + 2}$

As $x$ approaches infinity, the second term $\frac{3}{2x + 2}$ approaches zero, so the oblique asymptote is:

[ f(x) = \frac{1}{2} x - \frac{1