If F ( X ) = 5 ( X − 4 ) 2 − 1 F(x)=5(x-4)^2-1 F ( X ) = 5 ( X − 4 ) 2 − 1 Is Translated 2 Units To The Right, Which Of The Following Is The Transformed Function, G ( X G(x G ( X ]?A. G ( X ) = 5 ( X − 2 ) 2 − 1 G(x)=5(x-2)^2-1 G ( X ) = 5 ( X − 2 ) 2 − 1 B. G ( X ) = 5 ( X − 4 ) 2 − 3 G(x)=5(x-4)^2-3 G ( X ) = 5 ( X − 4 ) 2 − 3 C. G ( X ) = 5 ( X − 6 ) 2 − 1 G(x)=5(x-6)^2-1 G ( X ) = 5 ( X − 6 ) 2 − 1 D.

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Introduction

In mathematics, function translations are a crucial concept in understanding how functions change when they are shifted or transformed in various ways. In this article, we will focus on understanding how to translate a function to the right and how to identify the transformed function.

What is a Function Translation?

A function translation is a process of shifting a function to a new position on the coordinate plane. This can be done in various ways, including shifting the function to the left or right, up or down, or even reflecting it across the x-axis or y-axis.

Shifting a Function to the Right

When a function is shifted to the right, it means that the function is moved to the right by a certain number of units. This can be achieved by replacing the variable x with (x - h), where h is the number of units the function is shifted to the right.

Applying the Concept to the Given Function

The given function is f(x) = 5(x - 4)^2 - 1. To translate this function 2 units to the right, we need to replace x with (x - 2).

Calculating the Transformed Function

To calculate the transformed function, we need to substitute (x - 2) for x in the original function.

f(x) = 5(x - 4)^2 - 1

g(x) = 5((x - 2) - 4)^2 - 1

g(x) = 5(x - 6)^2 - 1

Conclusion

Based on the calculation above, the transformed function g(x) is 5(x - 6)^2 - 1.

Answer

The correct answer is C. g(x) = 5(x - 6)^2 - 1.

Discussion

When a function is translated to the right, the value of x is replaced with (x - h), where h is the number of units the function is shifted to the right. In this case, the function f(x) = 5(x - 4)^2 - 1 is translated 2 units to the right, so we need to replace x with (x - 2).

Key Takeaways

  • Function translations are a crucial concept in understanding how functions change when they are shifted or transformed in various ways.
  • Shifting a function to the right can be achieved by replacing the variable x with (x - h), where h is the number of units the function is shifted to the right.
  • To calculate the transformed function, we need to substitute (x - h) for x in the original function.

Real-World Applications

Function translations have numerous real-world applications in various fields, including physics, engineering, and economics. For example, in physics, function translations can be used to model the motion of objects, while in engineering, they can be used to design and optimize systems.

Conclusion

In conclusion, function translations are a fundamental concept in mathematics that has numerous real-world applications. By understanding how to translate functions to the right, we can better analyze and solve problems in various fields.

Final Thoughts

Function translations are a powerful tool for analyzing and solving problems in mathematics and other fields. By mastering this concept, we can gain a deeper understanding of how functions change when they are shifted or transformed in various ways.

References

Additional Resources

Glossary

  • Function Translation: A process of shifting a function to a new position on the coordinate plane.
  • Shifting a Function to the Right: Replacing the variable x with (x - h), where h is the number of units the function is shifted to the right.
  • Transformed Function: The new function obtained after shifting the original function to a new position on the coordinate plane.

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Introduction

In our previous article, we discussed the concept of function translations and how to translate a function to the right. In this article, we will provide a Q&A guide to help you better understand function translations and how to apply them in various situations.

Q&A

Q: What is a function translation?

A: A function translation is a process of shifting a function to a new position on the coordinate plane. This can be done in various ways, including shifting the function to the left or right, up or down, or even reflecting it across the x-axis or y-axis.

Q: How do I translate a function to the right?

A: To translate a function to the right, you need to replace the variable x with (x - h), where h is the number of units the function is shifted to the right.

Q: What is the difference between shifting a function to the right and shifting it to the left?

A: Shifting a function to the right means moving it to the right by a certain number of units, while shifting it to the left means moving it to the left by a certain number of units. To shift a function to the left, you need to replace the variable x with (x + h), where h is the number of units the function is shifted to the left.

Q: How do I calculate the transformed function?

A: To calculate the transformed function, you need to substitute (x - h) or (x + h) for x in the original function, depending on whether you are shifting the function to the right or left.

Q: What are some real-world applications of function translations?

A: Function translations have numerous real-world applications in various fields, including physics, engineering, and economics. For example, in physics, function translations can be used to model the motion of objects, while in engineering, they can be used to design and optimize systems.

Q: How do I determine the correct answer when given multiple options?

A: To determine the correct answer when given multiple options, you need to carefully analyze the problem and apply the concept of function translations. Make sure to substitute the correct value for x and simplify the expression to obtain the transformed function.

Q: What are some common mistakes to avoid when working with function translations?

A: Some common mistakes to avoid when working with function translations include:

  • Not substituting the correct value for x
  • Not simplifying the expression correctly
  • Not considering the direction of the shift (right or left)
  • Not using the correct formula for the transformed function

Conclusion

In conclusion, function translations are a fundamental concept in mathematics that has numerous real-world applications. By understanding how to translate functions to the right and left, we can better analyze and solve problems in various fields.

Final Thoughts

Function translations are a powerful tool for analyzing and solving problems in mathematics and other fields. By mastering this concept, we can gain a deeper understanding of how functions change when they are shifted or transformed in various ways.

References

Additional Resources

Glossary

  • Function Translation: A process of shifting a function to a new position on the coordinate plane.
  • Shifting a Function to the Right: Replacing the variable x with (x - h), where h is the number of units the function is shifted to the right.
  • Transformed Function: The new function obtained after shifting the original function to a new position on the coordinate plane.