2. Use Desmos To Graph The Function $f(x)=x \sqrt{6-x}$ And Use The Graph To Identify The Function's Extrema, Domain, And Intercepts.a. This Function Has A (circle Your Choice) - Local Minimum - Local Maximum Value Of $y

Introduction

Graphing functions is an essential skill in mathematics, and with the help of technology, it has become easier and more accessible. Desmos is a popular online graphing calculator that allows users to create and explore mathematical functions in a visually engaging way. In this article, we will use Desmos to graph the function $f(x)=x \sqrt{6-x}$ and identify its extrema, domain, and intercepts.

Graphing the Function

To graph the function $f(x)=x \sqrt{6-x}$ using Desmos, follow these steps:

1. Open Desmos and create a new graph.
2. Enter the function $f(x)=x \sqrt{6-x}$ in the input field.
3. Press the "Graph" button to visualize the function.

Analyzing the Graph

The graph of the function $f(x)=x \sqrt{6-x}$ is a curve that opens downwards, indicating that the function is decreasing as $x$ increases. To identify the function's extrema, domain, and intercepts, we need to analyze the graph carefully.

Extrema

The graph of the function $f(x)=x \sqrt{6-x}$ has a local maximum at $x=3$. This is because the function changes from increasing to decreasing at $x=3$. To find the value of the local maximum, we need to substitute $x=3$ into the function:

$$f(3)=3 \sqrt{6-3}=3 \sqrt{3}$$

Therefore, the local maximum value of the function is $3 \sqrt{3}$.

Domain

The domain of the function $f(x)=x \sqrt{6-x}$ is the set of all possible input values for $x$. From the graph, we can see that the function is defined for $x \in [0, 6]$. This is because the square root function is only defined for non-negative values, and the expression under the square root, $6-x$, must be non-negative.

Intercepts

The graph of the function $f(x)=x \sqrt{6-x}$ has two $x$-intercepts at $x=0$ and $x=6$. To find the $y$-intercept, we need to substitute $x=0$ into the function:

$$f(0)=0 \sqrt{6-0}=0$$

Therefore, the $y$-intercept is $(0, 0)$.

Conclusion

In this article, we used Desmos to graph the function $f(x)=x \sqrt{6-x}$ and identified its extrema, domain, and intercepts. We found that the function has a local maximum at $x=3$, a domain of $x \in [0, 6]$, and two $x$-intercepts at $x=0$ and $x=6$. The $y$-intercept is $(0, 0)$. This exercise demonstrates the importance of graphing functions in mathematics and the role of technology in making it easier and more accessible.

Discussion Questions

1. What is the local maximum value of the function $f(x)=x \sqrt{6-x}$?
2. What is the domain of the function $f(x)=x \sqrt{6-x}$?
3. How many $x$-intercepts does the graph of the function $f(x)=x \sqrt{6-x}$ have?
4. What is the $y$-intercept of the graph of the function $f(x)=x \sqrt{6-x}$?

Answer Key

1. The local maximum value of the function $f(x)=x \sqrt{6-x}$ is $3 \sqrt{3}$.
2. The domain of the function $f(x)=x \sqrt{6-x}$ is $x \in [0, 6]$.
3. The graph of the function $f(x)=x \sqrt{6-x}$ has two $x$-intercepts at $x=0$ and $x=6$.
4. The $y$-intercept of the graph of the function $f(x)=x \sqrt{6-x}$ is $(0, 0)$.
   Q&A: Graphing Functions with Desmos =====================================

Introduction

In our previous article, we used Desmos to graph the function $f(x)=x \sqrt{6-x}$ and identified its extrema, domain, and intercepts. In this article, we will answer some frequently asked questions about graphing functions with Desmos.

Q: What is Desmos?

A: Desmos is a popular online graphing calculator that allows users to create and explore mathematical functions in a visually engaging way.

Q: How do I use Desmos to graph a function?

A: To use Desmos to graph a function, follow these steps:

1. Open Desmos and create a new graph.
2. Enter the function in the input field.
3. Press the "Graph" button to visualize the function.

Q: What is the difference between a local maximum and a local minimum?

A: A local maximum is a point on the graph where the function changes from increasing to decreasing, while a local minimum is a point on the graph where the function changes from decreasing to increasing.

Q: How do I find the local maximum and local minimum of a function?

A: To find the local maximum and local minimum of a function, you need to analyze the graph carefully. Look for points where the function changes from increasing to decreasing (local maximum) or decreasing to increasing (local minimum).

Q: What is the domain of a function?

A: The domain of a function is the set of all possible input values for the function.

Q: How do I find the domain of a function?

A: To find the domain of a function, look for the values of x that make the function undefined. For example, if the function has a square root, the expression under the square root must be non-negative.

Q: What are the x-intercepts of a function?

A: The x-intercepts of a function are the points where the graph of the function crosses the x-axis.

Q: How do I find the x-intercepts of a function?

A: To find the x-intercepts of a function, look for the points where the graph of the function crosses the x-axis.

Q: What is the y-intercept of a function?

A: The y-intercept of a function is the point where the graph of the function crosses the y-axis.

Q: How do I find the y-intercept of a function?

A: To find the y-intercept of a function, substitute x=0 into the function.

Conclusion

In this article, we answered some frequently asked questions about graphing functions with Desmos. We hope that this article has been helpful in clarifying any doubts you may have had about graphing functions.

Discussion Questions

1. What is the difference between a local maximum and a local minimum?
2. How do you find the local maximum and local minimum of a function?
3. What is the domain of a function?
4. How do you find the domain of a function?
5. What are the x-intercepts of a function?
6. How do you find the x-intercepts of a function?
7. What is the y-intercept of a function?
8. How do you find the y-intercept of a function?

Answer Key

1. A local maximum is a point on the graph where the function changes from increasing to decreasing, while a local minimum is a point on the graph where the function changes from decreasing to increasing.
2. To find the local maximum and local minimum of a function, you need to analyze the graph carefully. Look for points where the function changes from increasing to decreasing (local maximum) or decreasing to increasing (local minimum).
3. The domain of a function is the set of all possible input values for the function.
4. To find the domain of a function, look for the values of x that make the function undefined. For example, if the function has a square root, the expression under the square root must be non-negative.
5. The x-intercepts of a function are the points where the graph of the function crosses the x-axis.
6. To find the x-intercepts of a function, look for the points where the graph of the function crosses the x-axis.
7. The y-intercept of a function is the point where the graph of the function crosses the y-axis.
8. To find the y-intercept of a function, substitute x=0 into the function.