The function you are asked to graph is a rational function:

$f(x) = \frac{2x + 2}{x + 4}$

Step-by-Step Guide for Graphing:

1. Find Vertical Asymptote: The vertical asymptote occurs where the denominator is zero. Set $x + 4 = 0$, giving us $x = -4$. So, the vertical asymptote is the vertical line $x = -4$.

2. Find Horizontal Asymptote: To determine the horizontal asymptote, compare the degrees of the numerator and denominator. Both the numerator and denominator have a degree of 1. The horizontal asymptote is found by dividing the leading coefficients of the terms: $\frac{2}{1} = 2$ So, the horizontal asymptote is $y = 2$.

3. Plot Points: Select points on either side of the vertical asymptote to understand the curve's shape.

   • For $x = -5$:
     $f(-5) = \frac{2(-5) + 2}{-5 + 4} = \frac{-10 + 2}{-1} = \frac{-8}{-1} = 8$ So, the point $(-5, 8)$ lies on the curve.

   • For $x = 0$:
     $f(0) = \frac{2(0) + 2}{0 + 4} = \frac{2}{4} = \frac{1}{2}$ So, the point $(0, 0.5)$ lies on the curve.

4. Sketch the Graph: Now that you have the vertical asymptote $x = -4$, horizontal asymptote $y = 2$, and some points, sketch the curve approaching these asymptotes.

Would you like any additional details or clarifications?

Here are 5 related questions to deepen your understanding:

1. How do you find slant asymptotes for rational functions?
2. Can a rational function have more than one vertical asymptote?
3. What happens to the function near the vertical asymptote?
4. How do you determine if a rational function crosses its horizontal asymptote?
5. How can holes occur in the graph of a rational function?

Tip: Always check if a rational function can be simplified further by factoring both the numerator and denominator before analyzing asymptotes.