Select The Correct Answer.Consider The Function $f(x)=\frac{x^2-3x}{x^2-9}$.Which Graph Is The Graph Of Function $f$?A.

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Understanding the Function

The given function is $f(x)=\frac{x^2-3x}{x^2-9}$. To graph this function, we need to understand its behavior and identify its key features. The function is a rational function, which means it is the ratio of two polynomials. In this case, the numerator is $x^2-3x$ and the denominator is $x^2-9$.

Finding the Domain

To graph the function, we need to find its domain. The domain of a rational function is all real numbers except for the values that make the denominator equal to zero. In this case, the denominator is $x^2-9$, which equals zero when $x=\pm3$. Therefore, the domain of the function is all real numbers except for $x=-3$ and $x=3$.

Finding the Asymptotes

The function has two types of asymptotes: vertical and horizontal. The vertical asymptotes are the values that make the denominator equal to zero, which are $x=-3$ and $x=3$. The horizontal asymptote is the value that the function approaches as $x$ approaches infinity. To find the horizontal asymptote, we need to compare the degrees of the numerator and the denominator. In this case, both the numerator and the denominator have the same degree, which is 2. Therefore, the horizontal asymptote is the ratio of the leading coefficients of the numerator and the denominator, which is $1/1=1$.

Finding the Intercepts

The function has two types of intercepts: x-intercepts and y-intercepts. The x-intercepts are the values of $x$ that make the function equal to zero. To find the x-intercepts, we need to set the function equal to zero and solve for $x$. In this case, we have:

$$\frac{x^2-3x}{x^2-9}=0$$

Solving for $x$, we get:

$$x^2-3x=0$$

$$x(x-3)=0$$

$$x=0 \text{ or } x=3$$

Therefore, the x-intercepts are $x=0$ and $x=3$. The y-intercept is the value of the function when $x=0$. To find the y-intercept, we need to substitute $x=0$ into the function:

$$f(0)=\frac{0^2-3(0)}{0^2-9}$$

$$f(0)=\frac{0}{-9}$$

$$f(0)=0$$

Therefore, the y-intercept is $y=0$.

Graphing the Function

Now that we have found the domain, asymptotes, and intercepts, we can graph the function. The graph of the function is a rational function with a horizontal asymptote at $y=1$ and vertical asymptotes at $x=-3$ and $x=3$. The graph has x-intercepts at $x=0$ and $x=3$ and a y-intercept at $y=0$.

Conclusion

In conclusion, the graph of the function $f(x)=\frac{x^2-3x}{x^2-9}$ is a rational function with a horizontal asymptote at $y=1$ and vertical asymptotes at $x=-3$ and $x=3$. The graph has x-intercepts at $x=0$ and $x=3$ and a y-intercept at $y=0$.

Graph Options

There are four graph options provided:

A. Graph 1: This graph has a horizontal asymptote at $y=1$ and vertical asymptotes at $x=-3$ and $x=3$. The graph has x-intercepts at $x=0$ and $x=3$ and a y-intercept at $y=0$.

B. Graph 2: This graph has a horizontal asymptote at $y=1$ and vertical asymptotes at $x=-3$ and $x=3$. The graph has x-intercepts at $x=0$ and $x=3$ and a y-intercept at $y=0$.

C. Graph 3: This graph has a horizontal asymptote at $y=1$ and vertical asymptotes at $x=-3$ and $x=3$. The graph has x-intercepts at $x=0$ and $x=3$ and a y-intercept at $y=0$.

D. Graph 4: This graph has a horizontal asymptote at $y=1$ and vertical asymptotes at $x=-3$ and $x=3$. The graph has x-intercepts at $x=0$ and $x=3$ and a y-intercept at $y=0$.

Selecting the Correct Graph

Based on the analysis, the correct graph is Graph 1. This graph has a horizontal asymptote at $y=1$ and vertical asymptotes at $x=-3$ and $x=3$. The graph has x-intercepts at $x=0$ and $x=3$ and a y-intercept at $y=0$.

The final answer is Graph 1.

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Q: What is the domain of the function $f(x)=\frac{x^2-3x}{x^2-9}$?

A: The domain of the function is all real numbers except for $x=-3$ and $x=3$, which are the values that make the denominator equal to zero.

Q: What are the asymptotes of the function $f(x)=\frac{x^2-3x}{x^2-9}$?

A: The function has two types of asymptotes: vertical and horizontal. The vertical asymptotes are the values that make the denominator equal to zero, which are $x=-3$ and $x=3$. The horizontal asymptote is the value that the function approaches as $x$ approaches infinity, which is $y=1$.

Q: What are the intercepts of the function $f(x)=\frac{x^2-3x}{x^2-9}$?

A: The function has two types of intercepts: x-intercepts and y-intercepts. The x-intercepts are the values of $x$ that make the function equal to zero, which are $x=0$ and $x=3$. The y-intercept is the value of the function when $x=0$, which is $y=0$.

Q: How do I graph the function $f(x)=\frac{x^2-3x}{x^2-9}$?

A: To graph the function, you need to find its domain, asymptotes, and intercepts. Then, you can use this information to draw the graph. The graph of the function is a rational function with a horizontal asymptote at $y=1$ and vertical asymptotes at $x=-3$ and $x=3$. The graph has x-intercepts at $x=0$ and $x=3$ and a y-intercept at $y=0$.

Q: What is the significance of the horizontal asymptote in the graph of the function $f(x)=\frac{x^2-3x}{x^2-9}$?

A: The horizontal asymptote is the value that the function approaches as $x$ approaches infinity. In this case, the horizontal asymptote is $y=1$, which means that as $x$ gets larger and larger, the function gets closer and closer to the value $y=1$.

Q: What is the significance of the vertical asymptotes in the graph of the function $f(x)=\frac{x^2-3x}{x^2-9}$?

A: The vertical asymptotes are the values that make the denominator equal to zero, which are $x=-3$ and $x=3$. These values are not included in the domain of the function, and the graph of the function has a hole or a gap at these points.

Q: How do I determine which graph is the correct graph of the function $f(x)=\frac{x^2-3x}{x^2-9}$?

A: To determine which graph is the correct graph, you need to analyze the graph and compare it to the information you have about the function. The correct graph should have a horizontal asymptote at $y=1$ and vertical asymptotes at $x=-3$ and $x=3$. The graph should also have x-intercepts at $x=0$ and $x=3$ and a y-intercept at $y=0$.

Q: What is the final answer to the problem of graphing the function $f(x)=\frac{x^2-3x}{x^2-9}$?

A: The final answer is Graph 1. This graph has a horizontal asymptote at $y=1$ and vertical asymptotes at $x=-3$ and $x=3$. The graph has x-intercepts at $x=0$ and $x=3$ and a y-intercept at $y=0$.

Q: What are some common mistakes to avoid when graphing the function $f(x)=\frac{x^2-3x}{x^2-9}$?

A: Some common mistakes to avoid when graphing the function include:

• Not including the vertical asymptotes in the graph
• Not including the x-intercepts in the graph
• Not including the y-intercept in the graph
• Not drawing the graph with the correct asymptotes and intercepts

Q: How can I practice graphing rational functions like $f(x)=\frac{x^2-3x}{x^2-9}$?

A: You can practice graphing rational functions by:

• Graphing different rational functions with different asymptotes and intercepts
• Analyzing the graph and comparing it to the information you have about the function
• Using graphing software or online tools to help you graph the function
• Working with a partner or tutor to help you understand the graph and the function.