Applying Translations to the Function

Oct 13, 2025 by magicalceylontours.com 76 views

When working with functions, it's essential to understand how translations affect the original function. A translation is a transformation that moves the graph of a function to a new position on the coordinate plane. In this article, we'll explore how to translate the function $f(x) = (1.6)^x$ 5 units up and 9 units to the right.

What is a Translation?

A translation is a transformation that moves the graph of a function to a new position on the coordinate plane. It's a horizontal or vertical shift of the graph. When a function is translated horizontally, the x-values change, and when it's translated vertically, the y-values change.

Horizontal Translation

A horizontal translation moves the graph of a function to the left or right. If the graph is translated 9 units to the right, the x-values change from $x$ to $x - 9$ . This is because the graph is shifted 9 units to the right, so the new x-value is 9 units less than the original x-value.

Vertical Translation

A vertical translation moves the graph of a function up or down. If the graph is translated 5 units up, the y-values change from $f(x)$ to $f(x) + 5$ . This is because the graph is shifted 5 units up, so the new y-value is 5 units greater than the original y-value.

Applying Translations to the Function

Now that we understand how translations work, let's apply them to the function $f(x) = (1.6)^x$ . We want to translate the function 5 units up and 9 units to the right. To do this, we need to replace $x$ with $x - 9$ to account for the horizontal translation, and add 5 to the function to account for the vertical translation.

The Correct Equation

The correct equation for the translated function is:

g(x) = (1.6)^{x-9} + 5

This equation represents the function $f(x) = (1.6)^x$ after it has been translated 5 units up and 9 units to the right.

Comparing the Options

Let's compare the correct equation with the options provided:

A. $g(x) = (1.6)^{x+5} - 9$ B. $g(x) = (1.6)^{x+5} + 9$ C. $g(x) = (1.6)^{x-9} + 5$ D. $g(x) = (1.6)^{x-9} - 5$

Only option C matches the correct equation.

Conclusion

In this article, we explored how to translate the function $f(x) = (1.6)^x$ 5 units up and 9 units to the right. We learned how to apply horizontal and vertical translations to a function and how to write the correct equation for the translated function. By understanding function translations, we can better analyze and work with functions in mathematics.

Key Takeaways

A translation is a transformation that moves the graph of a function to a new position on the coordinate plane.
A horizontal translation moves the graph of a function to the left or right.
A vertical translation moves the graph of a function up or down.
To apply a translation to a function, replace $x$ with $x - h$ to account for the horizontal translation, and add $k$ to the function to account for the vertical translation.
The correct equation for the translated function is $g(x) = (1.6)^{x-9} + 5$ .
Function Translations Q&A ==========================

In our previous article, we explored how to translate the function $f(x) = (1.6)^x$ 5 units up and 9 units to the right. We learned how to apply horizontal and vertical translations to a function and how to write the correct equation for the translated function. In this article, we'll answer some frequently asked questions about function translations.

Q: What is the difference between a horizontal and vertical translation?

A: A horizontal translation moves the graph of a function to the left or right, while a vertical translation moves the graph of a function up or down.

Q: How do I apply a horizontal translation to a function?

A: To apply a horizontal translation to a function, replace $x$ with $x - h$ , where $h$ is the number of units the graph is translated to the right or left.

Q: How do I apply a vertical translation to a function?

A: To apply a vertical translation to a function, add $k$ to the function, where $k$ is the number of units the graph is translated up or down.

Q: What is the correct equation for the translated function?

A: The correct equation for the translated function is $g(x) = (1.6)^{x-9} + 5$ .

Q: Why is the correct equation $g(x) = (1.6)^{x-9} + 5$ and not $g(x) = (1.6)^{x+5} + 9$ ?

A: The correct equation is $g(x) = (1.6)^{x-9} + 5$ because we need to replace $x$ with $x - 9$ to account for the horizontal translation, and add 5 to the function to account for the vertical translation.

Q: Can I apply multiple translations to a function?

A: Yes, you can apply multiple translations to a function. For example, if you want to translate the function $f(x) = (1.6)^x$ 3 units up and 2 units to the right, you would replace $x$ with $x - 2$ and add 3 to the function.

Q: How do I determine the correct equation for a translated function?

A: To determine the correct equation for a translated function, you need to replace $x$ with $x - h$ to account for the horizontal translation, and add $k$ to the function to account for the vertical translation.

Q: What are some common mistakes to avoid when applying translations to a function?

A: Some common mistakes to avoid when applying translations to a function include:

Not replacing $x$ with $x - h$ to account for the horizontal translation
Not adding $k$ to the function to account for the vertical translation
Not using the correct equation for the translated function

Conclusion

In this article, we answered some frequently asked questions about function translations. We learned how to apply horizontal and vertical translations to a function and how to write the correct equation for the translated function. By understanding function translations, we can better analyze and work with functions in mathematics.

Key Takeaways

A horizontal translation moves the graph of a function to the left or right.
A vertical translation moves the graph of a function up or down.
To apply a horizontal translation to a function, replace $x$ with $x - h$ .
To apply a vertical translation to a function, add $k$ to the function.
The correct equation for the translated function is $g(x) = (1.6)^{x-9} + 5$ .
You can apply multiple translations to a function.
To determine the correct equation for a translated function, replace $x$ with $x - h$ and add $k$ to the function.