Math Snap

STEP 1

What is this asking?
We need to graph the function $f(x) = \frac{3x+3}{x+2}$, including asymptotes and two points on each piece of the graph.
Watch out!
Don't forget to consider both sides of the vertical asymptote when plotting points!

STEP 2

1. Find the vertical asymptote
2. Find the horizontal asymptote
3. Find points to the left of the vertical asymptote
4. Find points to the right of the vertical asymptote

STEP 3

A vertical asymptote occurs when the denominator of a rational function is equal to zero, and the numerator is not zero.
Let's set the denominator equal to zero and solve for $x$:
$$x + 2 = 0$$ $$x = -2$$Since the numerator is $3(-2) + 3 = -3 \ne 0$ when $x = -2$, we have a vertical asymptote at $x = -2$.

STEP 4

To find the horizontal asymptote, we look at the degrees of the numerator and denominator.
Both the numerator, $3x + 3$, and the denominator, $x + 2$, have a degree of 1.
When the degrees are the same, the horizontal asymptote is the ratio of the leading coefficients.

STEP 5

The leading coefficient of the numerator is 3, and the leading coefficient of the denominator is 1.
So, our horizontal asymptote is:
$$y = \frac{3}{1} = 3$$

STEP 6

Let's choose $x = -3$ and $x = -4$, which are to the left of our vertical asymptote at $x = -2$.

STEP 7

For $x = -3$, we have:
$$f(-3) = \frac{3(-3) + 3}{-3 + 2} = \frac{-9 + 3}{-1} = \frac{-6}{-1} = 6$$ So, we have the point $(-3, 6)$.

STEP 8

For $x = -4$, we have:
$$f(-4) = \frac{3(-4) + 3}{-4 + 2} = \frac{-12 + 3}{-2} = \frac{-9}{-2} = \frac{9}{2} = 4.5$$ So, we have the point $(-4, 4.5)$.

STEP 9

Now let's choose $x = -1$ and $x = 0$, which are to the right of our vertical asymptote at $x = -2$.

STEP 10

For $x = -1$, we have:
$$f(-1) = \frac{3(-1) + 3}{-1 + 2} = \frac{-3 + 3}{1} = \frac{0}{1} = 0$$ So, we have the point $(-1, 0)$.

STEP 11

For $x = 0$, we have:
$$f(0) = \frac{3(0) + 3}{0 + 2} = \frac{0 + 3}{2} = \frac{3}{2} = 1.5$$ So, we have the point $(0, 1.5)$.

We found a vertical asymptote at $x = -2$ and a horizontal asymptote at $y = 3$.
We also found two points on each side of the vertical asymptote: $(-3, 6)$ and $(-4, 4.5)$ to the left, and $(-1, 0)$ and $(0, 1.5)$ to the right.
Graph the asymptotes and plot these points to get the graph of $f(x)$!