To analyze the given rational function $y = \frac{4x^2 - 16}{2x + 9}$, we need to determine its domain and asymptotes.

1. Domain:

The domain of a rational function is all real numbers except where the denominator equals zero because division by zero is undefined.

The denominator of the function is $2x + 9$.

Set the denominator equal to zero and solve for $x$:

$2x + 9 = 0$

$2x = -9$

$x = -\frac{9}{2}$

So, the domain is all real numbers except $x = -\frac{9}{2}$.

Domain: $x \in \mathbb{R}, x \neq -\frac{9}{2}$

2. Vertical Asymptote:

Vertical asymptotes occur where the denominator equals zero and the numerator does not simultaneously equal zero.

From the domain calculation, the vertical asymptote is at $x = -\frac{9}{2}$.

Vertical Asymptote: $x = -\frac{9}{2}$

3. Horizontal Asymptote:

To find the horizontal asymptote, we compare the degrees of the numerator and the denominator:

• The degree of the numerator $4x^2 - 16$ is 2.
• The degree of the denominator $2x + 9$ is 1.

Since the degree of the numerator is greater than the degree of the denominator (2 > 1), there is no horizontal asymptote.

Horizontal Asymptote: None

4. Oblique Asymptote:

When the degree of the numerator is one more than the degree of the denominator, there is an oblique (or slant) asymptote. Here, the numerator's degree is 2 and the denominator's degree is 1, so an oblique asymptote exists.

To find the oblique asymptote, perform polynomial long division of the numerator by the denominator:

$\frac{4x^2 - 16}{2x + 9}$

Performing the division:

• Divide $4x^2$ by $2x$ to get $2x$.
• Multiply $2x$ by $2x + 9$, resulting in $4x^2 + 18x$.
• Subtract $4x^2 + 18x$ from $4x^2 - 16$, yielding $-18x - 16$.
• Divide $-18x$ by $2x$ to get $-9$.
• Multiply $-9$ by $2x + 9$, resulting in $-18x - 81$.
• Subtract $-18x - 81$ from $-18x - 16$, giving a remainder of $65$.

So, the oblique asymptote is:

$y = 2x - 9$

Oblique Asymptote: $y = 2x - 9$

Summary:

• Domain: $x \in \mathbb{R}, x \neq -\frac{9}{2}$
• Vertical Asymptote: $x = -\frac{9}{2}$
• Horizontal Asymptote: None
• Oblique Asymptote: $y = 2x - 9$

Would you like a detailed explanation of any part of the solution? Here are some related questions you might consider:

1. What is the significance of the vertical asymptote in graphing the function?
2. How does the oblique asymptote affect the shape of the graph as $x$ approaches infinity?
3. Can a function have both a horizontal and an oblique asymptote?
4. How would you graph this function, given the asymptotes and the domain?
5. What happens to the function as $x$ approaches the vertical asymptote?

Tip: When dealing with rational functions, always check for common factors between the numerator and denominator to simplify the function if possible, which can reveal removable discontinuities.