Which Function Represents A Reflection Of F ( X ) = 3 7 ( 2 ) X F(x)=\frac{3}{7}(2)^x F ( X ) = 7 3 ​ ( 2 ) X Over The X X X -axis?A. G ( X ) = − 3 7 ( 2 ) X G(x)=-\frac{3}{7}(2)^x G ( X ) = − 7 3 ​ ( 2 ) X B. G ( X ) = 3 7 ( − 2 ) X G(x)=\frac{3}{7}(-2)^x G ( X ) = 7 3 ​ ( − 2 ) X C. G ( X ) = 3 7 ( 1 2 ) − X G(x)=\frac{3}{7}\left(\frac{1}{2}\right)^{-x} G ( X ) = 7 3 ​ ( 2 1 ​ ) − X D.

Understanding Reflections in Mathematics

Reflections are an essential concept in mathematics, particularly in the study of functions. A reflection of a function over the x-axis is a transformation that flips the function upside down, resulting in a new function that is a mirror image of the original function. In this article, we will explore the concept of reflecting functions over the x-axis and determine which function represents a reflection of the given function $f(x)=\frac{3}{7}(2)^x$.

What is a Reflection Over the x-axis?

A reflection over the x-axis is a transformation that flips a function upside down, resulting in a new function that is a mirror image of the original function. This transformation involves multiplying the function by -1, which changes the sign of the function's values. In other words, if a function $f(x)$ has a value of $y$ at a given point $x$, the reflected function $g(x)$ will have a value of $-y$ at the same point $x$.

Reflection of the Given Function

The given function is $f(x)=\frac{3}{7}(2)^x$. To reflect this function over the x-axis, we need to multiply the function by -1. This will change the sign of the function's values, resulting in a new function that is a mirror image of the original function.

Determining the Reflected Function

To determine the reflected function, we can multiply the given function by -1:

$$g(x)=-\frac{3}{7}(2)^x$$

This is the reflected function of $f(x)=\frac{3}{7}(2)^x$ over the x-axis.

Comparing the Reflected Function with the Options

Now that we have determined the reflected function, we can compare it with the options provided:

A. $g(x)=-\frac{3}{7}(2)^x$ B. $g(x)=\frac{3}{7}(-2)^x$ C. $g(x)=\frac{3}{7}\left(\frac{1}{2}\right)^{-x}$

The reflected function we determined is $g(x)=-\frac{3}{7}(2)^x$, which matches option A.

Conclusion

In conclusion, the function that represents a reflection of $f(x)=\frac{3}{7}(2)^x$ over the x-axis is $g(x)=-\frac{3}{7}(2)^x$. This function is a mirror image of the original function, with the same shape but flipped upside down.

Reflections in Mathematics: A Deeper Look

Reflections are an essential concept in mathematics, particularly in the study of functions. A reflection of a function over the x-axis is a transformation that flips the function upside down, resulting in a new function that is a mirror image of the original function. In this article, we explored the concept of reflecting functions over the x-axis and determined which function represents a reflection of the given function $f(x)=\frac{3}{7}(2)^x$.

Types of Reflections

There are two types of reflections in mathematics: reflection over the x-axis and reflection over the y-axis. A reflection over the x-axis involves multiplying the function by -1, while a reflection over the y-axis involves replacing $x$ with $-x$.

Reflection Over the y-axis

A reflection over the y-axis involves replacing $x$ with $-x$. This transformation changes the sign of the function's values, resulting in a new function that is a mirror image of the original function.

Example: Reflection of $f(x)=\frac{3}{7}(2)^x$ Over the y-axis

To reflect the function $f(x)=\frac{3}{7}(2)^x$ over the y-axis, we need to replace $x$ with $-x$:

$$g(x)=\frac{3}{7}(2)^{-x}$$

This is the reflected function of $f(x)=\frac{3}{7}(2)^x$ over the y-axis.

Reflections in Real-World Applications

Reflections have numerous real-world applications, particularly in the fields of physics, engineering, and computer science. For example, reflections are used in the study of optics, where they are used to describe the behavior of light as it reflects off a surface.

Conclusion

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Frequently Asked Questions About Reflections

Reflections are an essential concept in mathematics, particularly in the study of functions. In this article, we will answer some frequently asked questions about reflections, including what a reflection is, how to reflect a function over the x-axis, and how to reflect a function over the y-axis.

Q: What is a reflection in mathematics?

A: A reflection in mathematics is a transformation that flips a function upside down, resulting in a new function that is a mirror image of the original function.

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

A: To reflect a function over the x-axis, you need to multiply the function by -1. This will change the sign of the function's values, resulting in a new function that is a mirror image of the original function.

Q: How do I reflect a function over the y-axis?

A: To reflect a function over the y-axis, you need to replace x with -x. This will change the sign of the function's values, resulting in a new function that is a mirror image of the original function.

Q: What is the difference between a reflection over the x-axis and a reflection over the y-axis?

A: A reflection over the x-axis involves multiplying the function by -1, while a reflection over the y-axis involves replacing x with -x. This means that a reflection over the x-axis will flip the function upside down, while a reflection over the y-axis will flip the function left to right.

Q: Can I reflect a function over both the x-axis and the y-axis?

A: Yes, you can reflect a function over both the x-axis and the y-axis. To do this, you need to multiply the function by -1 and replace x with -x.

Q: How do I determine the reflected function?

A: To determine the reflected function, you need to follow the steps outlined above. If you are reflecting a function over the x-axis, you need to multiply the function by -1. If you are reflecting a function over the y-axis, you need to replace x with -x.

Q: Can I use reflections to solve real-world problems?

A: Yes, you can use reflections to solve real-world problems. Reflections are used in numerous fields, including physics, engineering, and computer science. For example, reflections are used in the study of optics, where they are used to describe the behavior of light as it reflects off a surface.

Q: What are some common applications of reflections in mathematics?

A: Some common applications of reflections in mathematics include:

• Reflections in geometry: Reflections are used to describe the behavior of shapes and figures in geometry.
• Reflections in algebra: Reflections are used to solve equations and inequalities in algebra.
• Reflections in calculus: Reflections are used to describe the behavior of functions and their derivatives in calculus.

Conclusion

In conclusion, reflections are an essential concept in mathematics, particularly in the study of functions. In this article, we answered some frequently asked questions about reflections, including what a reflection is, how to reflect a function over the x-axis, and how to reflect a function over the y-axis. We also discussed some common applications of reflections in mathematics and how they can be used to solve real-world problems.

Reflections in Mathematics: A Final Note

Reflections are a powerful tool in mathematics, and they have numerous applications in various fields. By understanding reflections, you can solve complex problems and describe the behavior of functions and shapes in mathematics.