Applying Vertical Translations to the Given Function

Oct 13, 2025 by windailys.com 86 views

In mathematics, function translations are a crucial concept in understanding how functions change when they are shifted or transformed in some way. In this article, we will explore the concept of function translations and how to apply them to a given function.

What is a Function Translation?

A function translation is a transformation that changes the position of a function's graph. This can be done in various ways, including shifting the graph horizontally, vertically, or a combination of both. In this article, we will focus on vertical translations, which involve shifting the graph up or down.

Vertical Translations

When a function is translated vertically, its graph is shifted up or down by a certain number of units. This can be represented by the equation:

f(x) → f(x) + k

where k is the number of units the graph is shifted.

Applying Vertical Translations to the Given Function

The given function is f(x) = 3(x+9)^2 - 2. We are asked to translate this function 8 units down. To do this, we need to apply the vertical translation formula:

f(x) → f(x) - k

where k is the number of units the graph is shifted. In this case, k = 8.

Calculating the Transformed Function

To calculate the transformed function, we need to substitute the original function into the vertical translation formula:

g(x) = f(x) - 8 g(x) = 3(x+9)^2 - 2 - 8 g(x) = 3(x+9)^2 - 10

Comparing the Transformed Function to the Options

Now that we have calculated the transformed function, we can compare it to the options provided:

A. g(x) = 3(x+17)^2 - 2 B. g(x) = 3(x+9)^2 + 8 C. g(x) = 3(x+9)^2 - 10 D. g(x) = 3(x+1)^2 - 2

From the calculation above, we can see that the correct transformed function is:

C. g(x) = 3(x+9)^2 - 10

Conclusion

In this article, we explored the concept of function translations and applied it to a given function. We calculated the transformed function by applying the vertical translation formula and compared it to the options provided. The correct transformed function is g(x) = 3(x+9)^2 - 10.

Key Takeaways

Function translations involve shifting the graph of a function up or down by a certain number of units.
The vertical translation formula is f(x) → f(x) + k or f(x) → f(x) - k.
To apply a vertical translation to a function, substitute the original function into the vertical translation formula.
Compare the calculated transformed function to the options provided to determine the correct answer.
Function Translations Q&A ==========================

In the previous article, we explored the concept of function translations and applied it to a given function. In this article, we will answer some frequently asked questions about function translations.

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

A: A horizontal translation involves shifting the graph of a function left or right by a certain number of units, while a vertical translation involves shifting the graph up or down by a certain number of units.

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

A: To apply a horizontal translation to a function, you need to change the value inside the parentheses of the function. For example, if you want to shift the graph of f(x) = 3(x+9)^2 - 2 3 units to the left, you would change the value inside the parentheses to f(x) = 3(x+6)^2 - 2.

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

A: To apply a vertical translation to a function, you need to add or subtract a value from the function. For example, if you want to shift the graph of f(x) = 3(x+9)^2 - 2 8 units down, you would subtract 8 from the function: f(x) = 3(x+9)^2 - 2 - 8.

Q: What is the effect of a negative value in a function translation?

A: A negative value in a function translation will shift the graph of the function in the opposite direction. For example, if you want to shift the graph of f(x) = 3(x+9)^2 - 2 8 units down, you would subtract 8 from the function: f(x) = 3(x+9)^2 - 2 - 8. This will shift the graph down by 8 units.

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 shift the graph of f(x) = 3(x+9)^2 - 2 8 units down and 3 units to the left, you would first apply the horizontal translation: f(x) = 3(x+6)^2 - 2, and then apply the vertical translation: f(x) = 3(x+6)^2 - 2 - 8.

Q: How do I determine the correct order of translations?

A: When applying multiple translations to a function, you need to determine the correct order of translations. The general rule is to apply the horizontal translations first, followed by the vertical translations. For example, if you want to shift the graph of f(x) = 3(x+9)^2 - 2 8 units down and 3 units to the left, you would first apply the horizontal translation: f(x) = 3(x+6)^2 - 2, and then apply the vertical translation: f(x) = 3(x+6)^2 - 2 - 8.

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

A: Yes, function translations can be used to solve real-world problems. For example, if you are designing a roller coaster and you want to shift the graph of the roller coaster's path 8 units down and 3 units to the left, you can use function translations to calculate the new path of the roller coaster.

Conclusion

In this article, we answered some frequently asked questions about function translations. We covered topics such as horizontal and vertical translations, applying multiple translations, and determining the correct order of translations. We also discussed how function translations can be used to solve real-world problems.