Reflection of the Given Function

Oct 13, 2025 by windailys.com 49 views

Introduction

In mathematics, reflecting a function over the y-axis is a fundamental concept that helps us understand the symmetry and properties of functions. When a function is reflected over the y-axis, its x-values are negated, resulting in a new function that is symmetric to the original function with respect to the y-axis. In this article, we will explore the reflection of the function ${$ f(x) = 5\left(\frac{1}{5}\right)^x $}$ over the y-axis and determine the equations that represent the reflected function.

Reflection of Functions Over the Y-Axis

When a function is reflected over the y-axis, its x-values are negated, resulting in a new function that is symmetric to the original function with respect to the y-axis. This can be achieved by replacing x with -x in the original function. In mathematical terms, if we have a function ${$ f(x) $}$, its reflection over the y-axis is given by ${$ f(-x) $}$.

Reflection of the Given Function

The given function is ${$ f(x) = 5\left(\frac{1}{5}\right)^x $}$. To reflect this function over the y-axis, we need to replace x with -x in the original function. This results in the following equation:

${$ f(-x) = 5\left(\frac{1}{5}\right)^{-x} $}$

Simplifying the Reflected Function

To simplify the reflected function, we can use the property of exponents that states ${$ a^{-x} = \frac{1}{a^x} $}$. Applying this property to the reflected function, we get:

${$ f(-x) = 5\left(\frac{1}{5}\right)^{-x} = 5\left(\frac{5^x}{1}\right) = 5^{x+1} $}$

However, this is not the only possible equation that represents the reflected function. We can also express the reflected function in terms of the original function as follows:

${$ f(-x) = 5\left(\frac{1}{5}\right)^{-x} = \frac{5}{\left(\frac{1}{5}\right)^x} = \frac{5}{5^{-x}} = 5^{1-x} $}$

Comparing the Options

Now that we have simplified the reflected function, we can compare it with the given options to determine which ones represent the reflected function.

Option A: ${$ f(x) = \frac{1}{5}(5)^{-x} $}$

This option is not correct because it does not represent the reflected function. The reflected function is ${$ f(-x) = 5\left(\frac{1}{5}\right)^{-x} $}$, which is different from the given option.

Option B: ${$ f(x) = 5^{1-x} $}$

This option is correct because it represents the reflected function. The reflected function is ${$ f(-x) = 5\left(\frac{1}{5}\right)^{-x} $}$, which is equivalent to ${$ f(x) = 5^{1-x} $}$.

Conclusion

In conclusion, the reflection of the function ${$ f(x) = 5\left(\frac{1}{5}\right)^x $}$ over the y-axis is given by ${$ f(-x) = 5\left(\frac{1}{5}\right)^{-x} $}$. This can be simplified to ${$ f(-x) = 5^{x+1} $}$ or ${$ f(-x) = 5^{1-x} $}$. The correct option that represents the reflected function is ${$ f(x) = 5^{1-x} $}$.

References

[1] "Reflection of Functions Over the Y-Axis". Math Open Reference. Retrieved 2023-02-20.
[2] "Exponents and Logarithms". Khan Academy. Retrieved 2023-02-20.

Introduction

In our previous article, we explored the reflection of the function ${$ f(x) = 5\left(\frac{1}{5}\right)^x $}$ over the y-axis and determined the equations that represent the reflected function. In this article, we will answer some frequently asked questions related to the reflection of functions over the y-axis.

Q&A

Q: What is the reflection of a function over the y-axis?

A: The reflection of a function over the y-axis is achieved by replacing x with -x in the original function. This results in a new function that is symmetric to the original function with respect to the y-axis.

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 in the original function. This can be achieved by using the following formula:

${$ f(-x) = f(x) $}$

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

A: Reflecting a function over the x-axis involves replacing y with -y in the original function, while reflecting it over the y-axis involves replacing x with -x in the original function.

Q: Can I reflect a function over the y-axis using a calculator?

A: Yes, you can reflect a function over the y-axis using a calculator. Most graphing calculators have a built-in function to reflect a graph over the y-axis.

Q: How do I determine the equation of the reflected function?

A: To determine the equation of the reflected function, you need to replace x with -x in the original function. This will result in a new function that is symmetric to the original function with respect to the y-axis.

Q: Can I reflect a function over the y-axis using a graphing software?

A: Yes, you can reflect a function over the y-axis using a graphing software. Most graphing software packages have a built-in function to reflect a graph over the y-axis.

Q: What is the importance of reflecting functions over the y-axis?

A: Reflecting functions over the y-axis is an important concept in mathematics that helps us understand the symmetry and properties of functions. It is used in various fields such as physics, engineering, and computer science.

Q: Can I reflect a function over the y-axis using a programming language?

A: Yes, you can reflect a function over the y-axis using a programming language. Most programming languages have a built-in function to reflect a graph over the y-axis.

Conclusion

In conclusion, reflecting a function over the y-axis is an important concept in mathematics that helps us understand the symmetry and properties of functions. By replacing x with -x in the original function, we can determine the equation of the reflected function. We hope that this Q&A guide has helped you understand the concept of reflecting functions over the y-axis.

References

[1] "Reflection of Functions Over the Y-Axis". Math Open Reference. Retrieved 2023-02-20.
[2] "Exponents and Logarithms". Khan Academy. Retrieved 2023-02-20.

Common Mistakes to Avoid

Not replacing x with -x in the original function when reflecting it over the y-axis.
Not using the correct formula to reflect a function over the y-axis.
Not understanding the concept of symmetry and properties of functions.

Tips and Tricks

Use a graphing calculator or software to visualize the reflection of a function over the y-axis.
Use a programming language to reflect a function over the y-axis.
Understand the concept of symmetry and properties of functions before reflecting a function over the y-axis.