Write The Function Whose Graph Is The Graph Of Y = 2 X 3 Y=2 \sqrt[3]{x} Y = 2 3 X ​ But Is Reflected About The X X X -axis.The Function Is Y = − 2 X 3 Y=-2 \sqrt[3]{x} Y = − 2 3 X ​ . (Type An Exact Answer, Using Radicals As Needed.)

Introduction

In mathematics, reflecting a function across the x-axis involves changing the sign of the function's output. This means that for every input value of x, the output value of the reflected function will be the negative of the original function's output. In this article, we will explore how to reflect a function across the x-axis and provide a step-by-step guide on how to do it.

What is Reflection in Mathematics?

Reflection in mathematics is a transformation that involves flipping a function or a shape over a line or a point. In the case of reflecting a function across the x-axis, the line of reflection is the x-axis itself. When a function is reflected across the x-axis, the y-coordinates of the function's graph are negated, resulting in a new function that is a mirror image of the original function.

How to Reflect a Function Across the x-axis

To reflect a function across the x-axis, we need to follow these steps:

1. Identify the original function: The first step is to identify the original function that we want to reflect. In this case, the original function is $y=2 \sqrt[3]{x}$.
2. Negate the output: The next step is to negate the output of the original function. This means that for every input value of x, the output value of the reflected function will be the negative of the original function's output.
3. Write the reflected function: Once we have negated the output of the original function, we can write the reflected function. In this case, the reflected function is $y=-2 \sqrt[3]{x}$.

Example: Reflecting the Function $y=2 \sqrt[3]{x}$ Across the x-axis

Let's take the function $y=2 \sqrt[3]{x}$ as an example. To reflect this function across the x-axis, we need to negate the output of the function. This means that for every input value of x, the output value of the reflected function will be the negative of the original function's output.

Step 1: Identify the original function

The original function is $y=2 \sqrt[3]{x}$.

Step 2: Negate the output

To negate the output of the original function, we need to multiply the output by -1. This means that the reflected function will have the same input values as the original function, but the output values will be negated.

Step 3: Write the reflected function

Once we have negated the output of the original function, we can write the reflected function. In this case, the reflected function is $y=-2 \sqrt[3]{x}$.

Conclusion

In conclusion, reflecting a function across the x-axis involves changing the sign of the function's output. This means that for every input value of x, the output value of the reflected function will be the negative of the original function's output. By following the steps outlined in this article, we can reflect a function across the x-axis and write the reflected function.

Reflection of a Function Across the x-axis: Key Takeaways

• Reflection in mathematics is a transformation that involves flipping a function or a shape over a line or a point.
• To reflect a function across the x-axis, we need to negate the output of the original function.
• The reflected function will have the same input values as the original function, but the output values will be negated.
• The reflected function can be written by multiplying the output of the original function by -1.

Reflection of a Function Across the x-axis: Real-World Applications

Reflection of a function across the x-axis has many real-world applications in mathematics and science. Some of the key applications include:

• Graphing functions: Reflection of a function across the x-axis is used to graph functions that are not easily graphed using traditional methods.
• Solving equations: Reflection of a function across the x-axis is used to solve equations that involve functions that are not easily solved using traditional methods.
• Optimization: Reflection of a function across the x-axis is used to optimize functions that involve multiple variables.

Reflection of a Function Across the x-axis: Final Thoughts

Introduction

In our previous article, we discussed how to reflect a function across the x-axis. In this article, we will answer some of the most frequently asked questions about reflecting a function across the x-axis.

Q: What is the purpose of reflecting a function across the x-axis?

A: The purpose of reflecting a function across the x-axis is to change the sign of the function's output. This means that for every input value of x, the output value of the reflected function will be the negative of the original function's output.

Q: How do I reflect a function across the x-axis?

A: To reflect a function across the x-axis, you need to follow these steps:

1. Identify the original function: The first step is to identify the original function that you want to reflect.
2. Negate the output: The next step is to negate the output of the original function. This means that for every input value of x, the output value of the reflected function will be the negative of the original function's output.
3. Write the reflected function: Once you have negated the output of the original function, you can write the reflected function.

Q: What is the difference between reflecting a function across the x-axis and reflecting a function across the y-axis?

A: Reflecting a function across the x-axis involves changing the sign of the function's output, while reflecting a function across the y-axis involves changing the sign of the function's input.

Q: Can I reflect a function across the x-axis using a graphing calculator?

A: Yes, you can reflect a function across the x-axis using a graphing calculator. Most graphing calculators have a function that allows you to reflect a graph across the x-axis.

Q: How do I reflect a function across the x-axis using a graphing calculator?

A: To reflect a function across the x-axis using a graphing calculator, follow these steps:

1. Enter the original function: Enter the original function into the graphing calculator.
2. Use the reflection function: Use the reflection function on the graphing calculator to reflect the graph across the x-axis.
3. Graph the reflected function: Graph the reflected function to see the result.

Q: Can I reflect a function across the x-axis using a computer algebra system (CAS)?

A: Yes, you can reflect a function across the x-axis using a computer algebra system (CAS). Most CAS software has a function that allows you to reflect a function across the x-axis.

Q: How do I reflect a function across the x-axis using a CAS?

A: To reflect a function across the x-axis using a CAS, follow these steps:

1. Enter the original function: Enter the original function into the CAS software.
2. Use the reflection function: Use the reflection function on the CAS software to reflect the function across the x-axis.
3. View the reflected function: View the reflected function to see the result.

Q: What are some real-world applications of reflecting a function across the x-axis?

A: Some real-world applications of reflecting a function across the x-axis include:

• Graphing functions: Reflecting a function across the x-axis is used to graph functions that are not easily graphed using traditional methods.
• Solving equations: Reflecting a function across the x-axis is used to solve equations that involve functions that are not easily solved using traditional methods.
• Optimization: Reflecting a function across the x-axis is used to optimize functions that involve multiple variables.

Conclusion

In conclusion, reflecting a function across the x-axis is an important concept in mathematics that has many real-world applications. By understanding how to reflect a function across the x-axis, you can solve equations, graph functions, and optimize functions that involve multiple variables.