What Value Represents The Horizontal Translation From The Graph Of The Parent Function $f(x)=x^2$ To The Graph Of The Function $g(x)=(x-4)^2+2$?A. \[$-4\$\]B. \[$-2\$\]C. \[$2\$\]D. \[$4\$\]

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What Value Represents the Horizontal Translation from the Graph of the Parent Function to the Graph of the Function?

Understanding Horizontal Translation in Graphs

When dealing with functions, particularly quadratic functions, understanding the concept of horizontal translation is crucial. Horizontal translation refers to the movement of the graph of a function to the left or right along the x-axis. This concept is essential in mathematics, particularly in algebra and calculus.

The Parent Function

The parent function in this case is f(x)=x2f(x) = x^2. This is a basic quadratic function that represents a parabola opening upwards with its vertex at the origin (0, 0). The graph of this function is a simple U-shaped curve.

The Function with Horizontal Translation

The function g(x)=(x−4)2+2g(x) = (x - 4)^2 + 2 represents a quadratic function with a horizontal translation. The term (x−4)(x - 4) indicates that the graph of the function has been shifted 4 units to the right along the x-axis. The addition of 2 inside the parentheses shifts the graph 2 units upwards along the y-axis.

Determining the Horizontal Translation

To determine the horizontal translation from the graph of the parent function to the graph of the function g(x)g(x), we need to identify the value that represents this translation. In this case, the horizontal translation is represented by the value inside the parentheses, which is 4.

Why is the Horizontal Translation 4?

The horizontal translation of 4 is due to the term (x−4)(x - 4) in the function g(x)g(x). This term indicates that the graph of the function has been shifted 4 units to the right along the x-axis. Therefore, the value that represents the horizontal translation from the graph of the parent function to the graph of the function g(x)g(x) is 4.

Conclusion

In conclusion, the value that represents the horizontal translation from the graph of the parent function f(x)=x2f(x) = x^2 to the graph of the function g(x)=(x−4)2+2g(x) = (x - 4)^2 + 2 is 4. This is because the term (x−4)(x - 4) in the function g(x)g(x) indicates that the graph of the function has been shifted 4 units to the right along the x-axis.

Answer

The correct answer is D. ${4\$}.

Additional Information

It's worth noting that the horizontal translation can also be represented by the value that is subtracted from x in the function. In this case, the value that is subtracted from x is 4, which represents the horizontal translation.

Example

To illustrate this concept further, let's consider another example. Suppose we have the function h(x)=(x+2)2−3h(x) = (x + 2)^2 - 3. In this case, the horizontal translation is represented by the value inside the parentheses, which is -2. This means that the graph of the function h(x)h(x) has been shifted 2 units to the left along the x-axis.

Conclusion

In conclusion, understanding horizontal translation is crucial in mathematics, particularly in algebra and calculus. The value that represents the horizontal translation from the graph of the parent function to the graph of the function can be determined by identifying the term that indicates the shift along the x-axis. In this case, the value that represents the horizontal translation is 4.
Q&A: Understanding Horizontal Translation in Graphs

Frequently Asked Questions

In this article, we will address some of the most frequently asked questions about horizontal translation in graphs.

Q: What is horizontal translation in graphs?

A: Horizontal translation in graphs refers to the movement of the graph of a function to the left or right along the x-axis. This concept is essential in mathematics, particularly in algebra and calculus.

Q: How do I determine the horizontal translation of a function?

A: To determine the horizontal translation of a function, you need to identify the term that indicates the shift along the x-axis. In the case of a quadratic function, this term is usually in the form of (x - a), where 'a' is the value that represents the horizontal translation.

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

A: Horizontal translation refers to the movement of the graph of a function to the left or right along the x-axis, while vertical translation refers to the movement of the graph of a function up or down along the y-axis. Vertical translation is represented by a constant term added to or subtracted from the function.

Q: Can you give an example of horizontal translation?

A: Consider the function f(x) = (x - 2)^2 + 1. In this case, the horizontal translation is represented by the value 2, which means that the graph of the function has been shifted 2 units to the right along the x-axis.

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

A: To apply horizontal translation to a function, you need to replace the x-term in the function with (x - a), where 'a' is the value that represents the horizontal translation. For example, if you want to apply a horizontal translation of 3 to the function f(x) = x^2, you would replace x with (x - 3) to get the new function f(x) = (x - 3)^2.

Q: What are some common mistakes to avoid when working with horizontal translation?

A: Some common mistakes to avoid when working with horizontal translation include:

  • Not identifying the correct term that represents the horizontal translation
  • Not applying the horizontal translation correctly to the function
  • Confusing horizontal and vertical translation

Q: Can you provide some practice problems to help me understand horizontal translation?

A: Here are a few practice problems to help you understand horizontal translation:

  1. Find the horizontal translation of the function f(x) = (x + 4)^2 - 2.
  2. Apply a horizontal translation of 2 to the function f(x) = x^2.
  3. Determine the horizontal translation of the function f(x) = (x - 1)^2 + 3.

Q: How do I know if I have applied the horizontal translation correctly?

A: To check if you have applied the horizontal translation correctly, you can graph the original function and the new function with the applied horizontal translation. If the graphs match, then you have applied the horizontal translation correctly.

Q: Can you provide some real-world applications of horizontal translation?

A: Horizontal translation has many real-world applications, including:

  • Physics: Horizontal translation is used to describe the motion of objects along a straight line.
  • Engineering: Horizontal translation is used to design and optimize systems that involve movement along a straight line.
  • Computer Science: Horizontal translation is used in computer graphics to create animations and special effects.

Q: How do I use horizontal translation to solve problems in mathematics?

A: To use horizontal translation to solve problems in mathematics, you need to identify the term that represents the horizontal translation and apply it to the function. You can then use the new function to solve the problem.

Q: Can you provide some tips for mastering horizontal translation?

A: Here are some tips for mastering horizontal translation:

  • Practice, practice, practice: The more you practice, the more comfortable you will become with horizontal translation.
  • Start with simple problems: Begin with simple problems and gradually move on to more complex ones.
  • Use visual aids: Graphs and diagrams can help you visualize the horizontal translation and make it easier to understand.
  • Seek help when needed: Don't be afraid to ask for help if you are struggling with horizontal translation.