To find the vertical and horizontal asymptotes of the rational function $f(x) = \frac{2x^2 + 6x - 8}{x^2 + 3x + 2}$, let’s go through the process step-by-step.

Step 1: Find the Vertical Asymptotes

Vertical asymptotes occur where the denominator is equal to zero (assuming these points don't also make the numerator zero).

1. Set the denominator equal to zero: $x^2 + 3x + 2 = 0$

2. Factor the quadratic expression: $(x + 1)(x + 2) = 0$

3. Solve for $x$: $x = -1 \quad \text{and} \quad x = -2$

Thus, the vertical asymptotes are at $x = -1$ and $x = -2$.

Step 2: Find the Horizontal Asymptote

For horizontal asymptotes, we compare the degrees of the numerator and the denominator.

• The degrees of both the numerator and denominator are the same (both are degree 2).
• When the degrees are the same, 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

• Vertical Asymptotes: $x = -1$ and $x = -2$
• Horizontal Asymptote: $y = 2$

Plotting on the Graph

To graph these asymptotes on the provided grid:

1. Draw vertical dashed lines at $x = -1$ and $x = -2$.
2. Draw a horizontal dashed line at $y = 2$.

Would you like further details on this solution, or do you have any questions?

Here are five related questions that can deepen your understanding:

1. How would the asymptotes change if the numerator had a higher degree than the denominator?
2. What happens to the function near the asymptotes $x = -1$ and $x = -2$?
3. How do you determine slant (oblique) asymptotes for a rational function?
4. Can a rational function have more than one horizontal asymptote? Why or why not?
5. How does the behavior of the graph differ as it approaches a vertical asymptote compared to a horizontal one?

Tip: Remember that vertical asymptotes are points where the function is undefined due to a zero in the denominator, while horizontal asymptotes show the function's behavior as $x$ approaches infinity.