Let's analyze the given function and determine the requested characteristics.

The function provided is:

$f(x) = \frac{3x + 3}{2x - 4}$

We are asked to identify the vertical asymptote, horizontal asymptote, and intercepts of this function.

Step 1: Vertical Asymptote

Vertical asymptotes occur where the denominator equals zero, as long as the numerator is not zero at the same point.

Set the denominator equal to zero:

$2x - 4 = 0$

Solve for $x$:

$2x = 4 \quad \Rightarrow \quad x = 2$

So, the vertical asymptote is at $x = 2$.

Step 2: Horizontal Asymptote

To determine the horizontal asymptote, we compare the degrees of the numerator and denominator.

• The degree of the numerator is 1 (since the highest power of $x$ is $x^1$).
• The degree of the denominator is also 1 (since the highest power of $x$ is $x^1$).

When the degrees of the numerator and denominator are equal, the horizontal asymptote is given by the ratio of the leading coefficients. The leading coefficient of the numerator is 3, and the leading coefficient of the denominator is 2. Therefore, the horizontal asymptote is:

$y = \frac{3}{2}$

Step 3: Y-intercept

The y-intercept occurs when $x = 0$. Substitute $x = 0$ into the function:

$f(0) = \frac{3(0) + 3}{2(0) - 4} = \frac{3}{-4} = -\frac{3}{4}$

So, the y-intercept is at $(0, -\frac{3}{4})$.

Step 4: X-intercept

The x-intercept occurs when $f(x) = 0$. Set the numerator equal to zero:

$3x + 3 = 0$

Solve for $x$:

$3x = -3 \quad \Rightarrow \quad x = -1$

So, the x-intercept is at $(-1, 0)$.

Final Answer:

• Vertical Asymptote: $x = 2$
• Horizontal Asymptote: $y = \frac{3}{2}$
• Y-intercept: $(0, -\frac{3}{4})$
• X-intercept: $(-1, 0)$

Would you like a graph to visualize this function or any additional explanation on how to graph it?