Function Reflection Formula

Oct 13, 2025 by miodeya.com 112 views

Understanding Reflection Across the Y-Axis

In mathematics, a reflection of a function across the y-axis is a transformation that flips the function over the y-axis. This means that the x-values of the original function are negated, resulting in a new function that is a mirror image of the original function across the y-axis.

Function Reflection Formula

The formula for reflecting a function f(x) across the y-axis is given by:

g(x) = f(-x)

This formula indicates that we need to replace x with -x in the original function f(x) to obtain the reflected function g(x).

Original Function

The original function is given by:

f(x) = \frac{1}{2}(3)^x

Reflection Across the Y-Axis

To reflect the original function f(x) across the y-axis, we need to replace x with -x in the original function. This results in:

g(x) = f(-x) = \frac{1}{2}(3)^{-x}

Simplifying the Reflected Function

We can simplify the reflected function g(x) by using the property of exponents that states:

a^(-x) = \frac{1}{a^x}

Applying this property to the reflected function g(x), we get:

g(x) = \frac{1}{2}(3)^{-x} = \frac{1}{2}\left(\frac{1}{3}\right)^x

Comparing with the Options

Now, let's compare the reflected function g(x) with the options given:

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

The reflected function g(x) matches option C.

Conclusion

In conclusion, the function that represents g(x), a reflection of f(x) = \frac{1}{2}(3)^x across the y-axis, is:

g(x) = \frac{1}{2}(3)^{-x}

This function is a mirror image of the original function f(x) across the y-axis.

Reflection of Functions Across the Y-Axis: Key Takeaways

The formula for reflecting a function f(x) across the y-axis is g(x) = f(-x).
To reflect a function across the y-axis, replace x with -x in the original function.
The reflected function g(x) is a mirror image of the original function f(x) across the y-axis.
The reflected function g(x) can be simplified using the property of exponents a^(-x) = \frac{1}{a^x}.
Reflection of Functions Across the Y-Axis: Q&A =============================================

Q: What is the formula for reflecting a function f(x) across the y-axis?

A: The formula for reflecting a function f(x) across the y-axis is g(x) = f(-x).

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

A: To reflect a function across the y-axis, replace x with -x in the original function.

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

A: Reflecting a function across the x-axis involves negating the y-values of the function, while reflecting it across the y-axis involves negating the x-values of the function.

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

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

Q: How do I simplify a reflected function?

A: To simplify a reflected function, use the property of exponents a^(-x) = \frac{1}{a^x}.

Q: What is the relationship between the original function and the reflected function?

A: The reflected function is a mirror image of the original function across the y-axis.

Q: Can I reflect a function across the y-axis multiple times?

A: Yes, you can reflect a function across the y-axis multiple times. Each reflection will result in a new function that is a mirror image of the previous function across the y-axis.

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

A: To determine the equation of a reflected function, replace x with -x in the original function.

Q: Can I reflect a function across the y-axis if it is a polynomial function?

A: Yes, you can reflect a polynomial function across the y-axis by replacing x with -x in the original function.

Q: Can I reflect a function across the y-axis if it is a rational function?

A: Yes, you can reflect a rational function across the y-axis by replacing x with -x in the original function.

Q: Can I reflect a function across the y-axis if it is a trigonometric function?

A: Yes, you can reflect a trigonometric function across the y-axis by replacing x with -x in the original function.

Q: Can I reflect a function across the y-axis if it is an exponential function?

A: Yes, you can reflect an exponential function across the y-axis by replacing x with -x in the original function.

Q: Can I reflect a function across the y-axis if it is a logarithmic function?

A: Yes, you can reflect a logarithmic function across the y-axis by replacing x with -x in the original function.

Reflection of Functions Across the Y-Axis: Key Takeaways

The formula for reflecting a function f(x) across the y-axis is g(x) = f(-x).
To reflect a function across the y-axis, replace x with -x in the original function.
The reflected function g(x) is a mirror image of the original function f(x) across the y-axis.
The reflected function g(x) can be simplified using the property of exponents a^(-x) = \frac{1}{a^x}.
You can reflect a function across the y-axis multiple times.
You can reflect a function across the y-axis if it is a polynomial function, a rational function, a trigonometric function, an exponential function, or a logarithmic function.