https://file.aiquickdraw.com/mathbot/file/1724682882890bpv1xk48.jpg

Use the graph to answer the question.
Find the asymptotes of the function. Select all that apply.
choices:
a. x=-1
b. y=-2
c. x=3
d. x=0
e. y=0
f. y=-3

To determine the asymptotes of the function depicted in the graph, let's analyze the behavior of the function:

1. **Vertical Asymptotes**: These occur where the function approaches infinity or negative infinity as \( x \) approaches a particular value.
   - The graph shows vertical asymptotes at \( x = -3 \) and \( x = 3 \), as the function approaches \(\infty\) as \(x\) approaches these values from the left and right. 

2. **Horizontal Asymptotes**: These occur when the function approaches a particular \( y \)-value as \( x \) approaches infinity or negative infinity.
   - The graph approaches \( y = 0 \) as \( x \) moves towards positive or negative infinity. Therefore, there is a horizontal asymptote at \( y = 0 \).

Given these observations, the correct selections for the asymptotes are:

- **\(x = 3\)** (vertical asymptote)
- **\(y = 0\)** (horizontal asymptote)

The correct choices from your list are:

- c. \(x = 3\)
- e. \(y = 0\)

Would you like further details or have any questions?

Here are 5 related questions to expand your understanding:

1. What are the equations of the asymptotes for a hyperbola centered at the origin?
2. How do you determine the vertical asymptotes of a rational function?
3. What is the significance of horizontal asymptotes in the behavior of functions as \(x\) approaches infinity?
4. Can a function have more than one horizontal asymptote?
5. How do you find oblique (slant) asymptotes for a rational function?

**Tip:** When identifying asymptotes, always look for where the function behaves abnormally, such as rising or falling steeply, or flattening out at the extremes of the graph.

Learn how to identify asymptotes of a function using graphical analysis. This problem explores vertical and horizontal asymptotes, providing step-by-step solutions and graphical examples. Suitable for high school students studying functions and calculus.

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