To find the vertical and horizontal asymptotes of the rational function $f(x) = \frac{x^2 + 3x + 2}{2x^2 + 5}$, let’s go through each step.

1. Horizontal Asymptote

The horizontal asymptote of a rational function depends on the degrees of the numerator and denominator polynomials:

• If the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is $y = 0$.
• If the degrees are equal, the horizontal asymptote is the ratio of the leading coefficients.
• If the degree of the numerator is greater than the degree of the denominator, there is no horizontal asymptote (there might be an oblique asymptote).

In this function:

• The degree of the numerator, $x^2 + 3x + 2$, is 2.
• The degree of the denominator, $2x^2 + 5$, is also 2.

Since the degrees are equal, the horizontal asymptote is the ratio of the leading coefficients, which are 1 (from $x^2$ in the numerator) and 2 (from $2x^2$ in the denominator).

So, the horizontal asymptote is: $y = \frac{1}{2}$

2. Vertical Asymptote

The vertical asymptotes occur where the denominator equals zero, as long as these points do not cancel out with factors in the numerator.

Set the denominator equal to zero: $2x^2 + 5 = 0$

Solving for $x$: $x^2 = -\frac{5}{2}$

Since $x^2 = -\frac{5}{2}$ has no real solutions (it results in a negative value under the square root), there are no vertical asymptotes for this function.

Summary of Asymptotes

• Horizontal Asymptote: $y = \frac{1}{2}$
• Vertical Asymptote: None

Graphing Instructions

On the graph provided:

1. Draw a dashed horizontal line at $y = \frac{1}{2}$ to represent the horizontal asymptote.
2. Since there are no vertical asymptotes, no additional dashed lines for vertical asymptotes are needed.

Would you like further details on any of these steps, or have questions about the concept of asymptotes?

Related Questions

1. What are the conditions for an oblique asymptote in rational functions?
2. How do you find horizontal asymptotes when the numerator's degree is greater than the denominator’s?
3. What is the significance of the leading coefficients in determining horizontal asymptotes?
4. How do you identify holes in the graph of a rational function?
5. Can a rational function have both a horizontal and an oblique asymptote?

Tip

When finding asymptotes, always check for factors that might cancel between the numerator and denominator, as they can affect the presence of vertical asymptotes or holes.