What Is The Equation Of The Translated Function, { G(x) $}$, If { F(x) = X^2 $}$?A. { G(x) = (x+5)^2 + 2 $}$B. { G(x) = (x+2)^2 + 5 $}$C. { G(x) = (x-2)^2 + 5 $}$D. { G(x) = (x-5)^2 + 2 $}$

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Introduction

In mathematics, a translated function is a function that has been shifted or moved from its original position. This can be achieved by adding or subtracting a constant value to the input variable of the function. In this article, we will explore the concept of translated functions and how to find the equation of a translated function given the original function.

What is a Translated Function?

A translated function is a function that has been shifted or moved from its original position. This can be achieved by adding or subtracting a constant value to the input variable of the function. For example, if we have a function f(x) = x^2, we can create a translated function by adding or subtracting a constant value to x.

Types of Translations

There are two types of translations: horizontal and vertical. A horizontal translation involves shifting the function to the left or right, while a vertical translation involves shifting the function up or down.

  • Horizontal Translation: A horizontal translation involves shifting the function to the left or right. This can be achieved by adding or subtracting a constant value to x. For example, if we have a function f(x) = x^2, we can create a horizontal translation by adding 5 to x, resulting in the function g(x) = (x+5)^2.
  • Vertical Translation: A vertical translation involves shifting the function up or down. This can be achieved by adding or subtracting a constant value to the function. For example, if we have a function f(x) = x^2, we can create a vertical translation by adding 2 to the function, resulting in the function g(x) = x^2 + 2.

Finding the Equation of a Translated Function

To find the equation of a translated function, we need to identify the type of translation and the constant value used. Let's consider the following example:

Given the function f(x) = x^2, find the equation of the translated function g(x) if it is shifted 2 units to the right and 5 units up.

To find the equation of the translated function, we need to add 2 to x and add 5 to the function. This results in the equation g(x) = (x+2)^2 + 5.

Example Solutions

Let's consider the following example solutions:

  • Option A: g(x) = (x+5)^2 + 2. This is a horizontal translation of 5 units to the left and a vertical translation of 2 units up.
  • Option B: g(x) = (x+2)^2 + 5. This is a horizontal translation of 2 units to the right and a vertical translation of 5 units up.
  • Option C: g(x) = (x-2)^2 + 5. This is a horizontal translation of 2 units to the left and a vertical translation of 5 units up.
  • Option D: g(x) = (x-5)^2 + 2. This is a horizontal translation of 5 units to the left and a vertical translation of 2 units up.

Conclusion

In conclusion, a translated function is a function that has been shifted or moved from its original position. This can be achieved by adding or subtracting a constant value to the input variable of the function. To find the equation of a translated function, we need to identify the type of translation and the constant value used. By following the steps outlined in this article, we can find the equation of a translated function given the original function.

Final Answer

Frequently Asked Questions

In this article, we will answer some of the most frequently asked questions about translated functions.

Q: What is a translated function?

A: A translated function is a function that has been shifted or moved from its original position. This can be achieved by adding or subtracting a constant value to the input variable of the function.

Q: What are the two types of translations?

A: There are two types of translations: horizontal and vertical. A horizontal translation involves shifting the function to the left or right, while a vertical translation involves shifting the function up or down.

Q: How do I find the equation of a translated function?

A: To find the equation of a translated function, you need to identify the type of translation and the constant value used. For example, if you have a function f(x) = x^2 and you want to shift it 2 units to the right and 5 units up, you would add 2 to x and add 5 to the function, resulting in the equation g(x) = (x+2)^2 + 5.

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

A: A horizontal translation involves shifting the function to the left or right, while a vertical translation involves shifting the function up or down. For example, if you have a function f(x) = x^2 and you want to shift it 2 units to the left, you would subtract 2 from x, resulting in the equation g(x) = (x-2)^2. If you want to shift it 5 units up, you would add 5 to the function, resulting in the equation g(x) = x^2 + 5.

Q: Can I have multiple translations?

A: Yes, you can have multiple translations. For example, if you have a function f(x) = x^2 and you want to shift it 2 units to the right and 5 units up, and then shift it 3 units to the left, you would add 2 to x, add 5 to the function, and then subtract 3 from x, resulting in the equation g(x) = (x-1)^2 + 5.

Q: How do I know which option is correct?

A: To determine which option is correct, you need to analyze the translation and the constant value used. For example, if you have a function f(x) = x^2 and you want to shift it 2 units to the right and 5 units up, you would look for an option that has (x+2)^2 + 5.

Q: What if I'm still unsure?

A: If you're still unsure, you can try plugging in some values to see which option works. For example, if you have a function f(x) = x^2 and you want to shift it 2 units to the right and 5 units up, you can plug in x = 0 and see which option gives you the correct result.

Common Mistakes

  • Not identifying the type of translation: Make sure you identify the type of translation (horizontal or vertical) and the constant value used.
  • Not adding or subtracting the correct value: Make sure you add or subtract the correct value to the input variable or the function.
  • Not checking the options: Make sure you check all the options to see which one works.

Conclusion

In conclusion, translated functions are an important concept in mathematics. By understanding how to find the equation of a translated function, you can solve a wide range of problems. Remember to identify the type of translation and the constant value used, and to check all the options to see which one works.