We are given the rational function:

$f(x) = \frac{3x^2 - 9x - 12}{4x^2 + 8x + 4}$

Let's determine each feature of the graph:

1. Horizontal Asymptote:

For rational functions, the horizontal asymptote depends on the degrees of the numerator and denominator:

• If the degree of the numerator is equal to the degree of the denominator, the horizontal asymptote is the ratio of the leading coefficients.
• Here, both the numerator and the denominator have degree 2. The leading coefficients are $3$ (numerator) and $4$ (denominator).

Thus, the horizontal asymptote is: $y = \frac{3}{4}$

2. Vertical Asymptote:

Vertical asymptotes occur where the denominator is zero (but the numerator is non-zero at that point).

• Set the denominator equal to zero and solve: $4x^2 + 8x + 4 = 0$ This is a quadratic equation. Factoring: $4(x^2 + 2x + 1) = 4(x + 1)^2 = 0$ So, $x = -1$.

Thus, there is a vertical asymptote at: $x = -1$

3. x-Intercept:

To find the x-intercepts, set the numerator equal to zero and solve for $x$: $3x^2 - 9x - 12 = 0$ Dividing by 3: $x^2 - 3x - 4 = 0$ Factoring: $(x - 4)(x + 1) = 0$ So, $x = 4$ and $x = -1$.

However, since $x = -1$ is a vertical asymptote, it is not a valid x-intercept. Therefore, the only x-intercept is: $(4, 0)$

4. y-Intercept:

To find the y-intercept, set $x = 0$ and evaluate $f(0)$: $f(0) = \frac{3(0)^2 - 9(0) - 12}{4(0)^2 + 8(0) + 4} = \frac{-12}{4} = -3$ Thus, the y-intercept is: $(0, -3)$

5. Hole:

A hole occurs when both the numerator and denominator are zero at the same point. From the vertical asymptote and x-intercept calculations, we see that there is no common factor between the numerator and denominator. Therefore, there is no hole in this function.

Summary:

• Horizontal Asymptote: $y = \frac{3}{4}$
• Vertical Asymptote: $x = -1$
• x-Intercept: $(4, 0)$
• y-Intercept: $(0, -3)$
• Hole: None

Let me know if you'd like further details or if you have any questions! Here are some related questions:

1. How are vertical and horizontal asymptotes determined for other types of functions?
2. How do holes differ from vertical asymptotes?
3. What happens if both numerator and denominator have higher degrees?
4. Can a function have more than one vertical asymptote?
5. What effect does the asymptote have on the overall graph of the function?

Tip: Always simplify the numerator and denominator when possible to check for removable discontinuities like holes!