Complete the following for the function 

​f(x)equals=StartFraction x squared minus 3 Over x plus 4 EndFractionx2−3x+4.

**a.** Find the slant asymptote of f. 

**b.** Find the vertical​ asymptote(s) of f​ (if any).

**c.** Graph f and all of its asymptotes with a graphing utility.

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Part 1

**a.** The slant asymptote is 

enter your response here.

​(Type your answer in​ slope-intercept form.)

Let's tackle each part of the problem for the function \( f(x) = \frac{x^2 - 3}{x + 4} \).

### **Part a: Finding the Slant Asymptote**

A slant (or oblique) asymptote occurs when the degree of the numerator is exactly one more than the degree of the denominator. Here, the numerator \( x^2 - 3 \) is a quadratic (degree 2), and the denominator \( x + 4 \) is linear (degree 1), so a slant asymptote will exist.

To find the slant asymptote, we perform polynomial division:

#### Polynomial Division:
1. **Divide the leading term of the numerator by the leading term of the denominator:**
   \[
   \frac{x^2}{x} = x
   \]
2. **Multiply the entire denominator by this result:**
   \[
   x(x + 4) = x^2 + 4x
   \]
3. **Subtract this product from the numerator:**
   \[
   (x^2 - 3) - (x^2 + 4x) = -4x - 3
   \]
4. **The quotient so far is \( x \).** Now, continue dividing:
   \[
   \frac{-4x}{x} = -4
   \]
5. **Multiply the entire denominator by this result:**
   \[
   -4(x + 4) = -4x - 16
   \]
6. **Subtract this from the previous remainder:**
   \[
   (-4x - 3) - (-4x - 16) = 13
   \]

The result of the division is:
\[
f(x) = x - 4 + \frac{13}{x + 4}
\]
As \( x \) approaches \( \infty \) or \( -\infty \), the term \( \frac{13}{x + 4} \) approaches 0. Therefore, the slant asymptote is the line:
\[
\boxed{y = x - 4}
\]

### **Part b: Finding the Vertical Asymptote(s)**

Vertical asymptotes occur where the denominator is zero and the numerator is not zero at that point. Set the denominator equal to zero and solve for \( x \):

\[
x + 4 = 0 \implies x = -4
\]

So, there is a vertical asymptote at:
\[
\boxed{x = -4}
\]

### **Part c: Graphing the Function and Its Asymptotes**

To graph the function \( f(x) = \frac{x^2 - 3}{x + 4} \) along with its slant and vertical asymptotes:

1. **Graph the slant asymptote** \( y = x - 4 \) as a dashed line.
2. **Graph the vertical asymptote** \( x = -4 \) as a dashed vertical line.
3. **Graph the function** using a graphing utility, plotting key points, and making sure the graph approaches the asymptotes appropriately.

You can use graphing software, or graphing calculators, such as Desmos or GeoGebra, to visualize the function and its asymptotes.

Would you like help with the graph or any further questions? Here are some related questions to consider:

1. How do you determine when a function has a horizontal asymptote instead of a slant asymptote?
2. Can a function have both a slant and horizontal asymptote?
3. What is the behavior of \( f(x) \) as \( x \) approaches the vertical asymptote \( x = -4 \)?
4. How would the slant asymptote change if the numerator were a cubic polynomial?
5. How do you interpret the remainder in polynomial division when finding slant asymptotes?

**Tip:** The slant asymptote can be thought of as the line that the function resembles as \( x \) becomes very large in magnitude.

Learn how to find the slant asymptote and vertical asymptotes of the function f(x) = (x^2 - 3) / (x + 4). This problem covers polynomial division and asymptotic behavior in functions, including graphical representation using graphing utilities. Suitable for advanced high school students in Grades 11-12.

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