The image you've uploaded explains the process of finding the vertical asymptotes of a rational function. The vertical asymptotes occur when the denominator of the function is equal to zero. The function given is:

$f(x) = \frac{x^2 + 2}{8x^2 - 31x - 4}$

Process of Finding Vertical Asymptotes:

1. Step 1: Set the denominator equal to zero: $8x^2 - 31x - 4 = 0$

2. Step 2: Factor the quadratic expression: $8x^2 - 31x - 4 = (8x + 1)(x - 4)$ This gives two factors: $8x + 1$ and $x - 4$.

3. Step 3: Solve each factor for $x$: $8x + 1 = 0 \quad \Rightarrow \quad x = -\frac{1}{8}$ $x - 4 = 0 \quad \Rightarrow \quad x = 4$

Thus, the vertical asymptotes are at $x = -\frac{1}{8}$ and $x = 4$.

Summary:

• The vertical asymptotes are located at $x = -\frac{1}{8}$ and $x = 4$, which correspond to the points where the denominator equals zero and the function becomes undefined (approaches infinity).

Would you like a more detailed explanation of any part of this process?

Here are 5 related questions:

1. How do we know that vertical asymptotes occur when the denominator equals zero?
2. Why is factoring the quadratic expression necessary in this process?
3. Can a rational function have more than two vertical asymptotes?
4. What happens to the graph of the function near the vertical asymptotes?
5. How would the presence of horizontal or slant asymptotes affect this function?

Tip: Always check if the numerator and denominator have common factors before concluding the vertical asymptotes. If there's a common factor, it might indicate a hole instead of an asymptote.