The given function is:

$f(x) = \frac{2x^2 - 4}{x^2 + 7x - 8}$

To find the asymptotes of this function, we will consider two types of asymptotes: vertical and horizontal.

1. Vertical Asymptotes

Vertical asymptotes occur where the denominator equals zero, as long as the numerator doesn't also become zero at the same points.

First, let's factor the denominator:

$x^2 + 7x - 8 = (x + 8)(x - 1)$

Now, set the denominator equal to zero:

$(x + 8)(x - 1) = 0$

Thus, the vertical asymptotes occur at $x = -8$ and $x = 1$, since these values make the denominator zero.

2. Horizontal Asymptotes

To find the horizontal asymptote, we look at the degrees of the polynomial in the numerator and denominator:

• The degree of the numerator is 2 (since the highest power of $x$ is $x^2$).
• The degree of the denominator is also 2.

When the degrees of the numerator and denominator are equal, the horizontal asymptote is given by the ratio of the leading coefficients. In this case, the leading coefficient of the numerator is 2, and the leading coefficient of the denominator is 1. Therefore, the horizontal asymptote is:

$y = \frac{2}{1} = 2$

Summary of Asymptotes:

• Vertical asymptotes: $x = -8$ and $x = 1$
• Horizontal asymptote: $y = 2$

Would you like a deeper explanation or any further clarifications?

Here are 5 related questions you might find interesting:

1. How do you determine if a function has a slant asymptote?
2. Can vertical asymptotes ever be crossed by a function?
3. What happens to a function near its vertical asymptotes?
4. How does the degree of polynomials influence horizontal asymptotes?
5. Can a function have both a horizontal and a slant asymptote?

Tip: Always check for holes in rational functions by factoring the numerator and denominator to see if they share any common factors.