Find \[$ G(x) \$\], Where \[$ G(x) \$\] Is The Translation 3 Units Down Of \[$ F(x) = X^2 \$\].Write Your Answer In The Form \[$ A(x - H)^2 + K \$\], Where \[$ A \$\], \[$ H \$\], And \[$ K \$\]

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Understanding the Problem

In this problem, we are given a quadratic function { f(x) = x^2 $}$ and asked to find the translation 3 units down of this function, denoted as { g(x) $}$. The translation of a function means that we need to shift the graph of the function up or down by a certain number of units. In this case, we need to shift the graph of { f(x) = x^2 $}$ 3 units down.

Recall the Standard Form of a Quadratic Function

The standard form of a quadratic function is { a(x - h)^2 + k $}$, where { a $}$, { h $}$, and { k $}$ are constants. The graph of a quadratic function in standard form is a parabola that opens upwards or downwards, depending on the sign of { a $}$.

Translation of a Quadratic Function

When we translate a quadratic function, we need to adjust the values of { a $}$, { h $}$, and { k $}$ accordingly. In this case, we need to shift the graph of { f(x) = x^2 $}$ 3 units down, which means that we need to decrease the value of { k $}$ by 3.

Finding the Translation of { f(x) = x^2 $}$

To find the translation of { f(x) = x^2 $}$, we need to rewrite the function in standard form. Since { f(x) = x^2 $}$ is already in standard form, we can see that { a = 1 $}$, { h = 0 $}$, and { k = 0 $}$.

Now, we need to shift the graph of { f(x) = x^2 $}$ 3 units down. To do this, we need to decrease the value of { k $}$ by 3. Therefore, the translation of { f(x) = x^2 $}$ is { g(x) = (x - 0)^2 + (-3) $}$.

Simplifying the Translation

We can simplify the translation by combining the terms inside the parentheses. Therefore, { g(x) = (x - 0)^2 + (-3) $}$ can be simplified to { g(x) = x^2 - 3 $}$.

Conclusion

In this problem, we found the translation 3 units down of the quadratic function { f(x) = x^2 $}$. We used the standard form of a quadratic function to rewrite the function and then shifted the graph 3 units down by decreasing the value of { k $}$. The translation of { f(x) = x^2 $}$ is { g(x) = x^2 - 3 $}$.

Final Answer

The final answer is { g(x) = x^2 - 3 $}$.

Step-by-Step Solution

Step 1: Recall the Standard Form of a Quadratic Function

The standard form of a quadratic function is { a(x - h)^2 + k $}$, where { a $}$, { h $}$, and { k $}$ are constants.

Step 2: Rewrite the Function in Standard Form

Since { f(x) = x^2 $}$ is already in standard form, we can see that { a = 1 $}$, { h = 0 $}$, and { k = 0 $}$.

Step 3: Shift the Graph 3 Units Down

To shift the graph of { f(x) = x^2 $}$ 3 units down, we need to decrease the value of { k $}$ by 3. Therefore, the translation of { f(x) = x^2 $}$ is { g(x) = (x - 0)^2 + (-3) $}$.

Step 4: Simplify the Translation

We can simplify the translation by combining the terms inside the parentheses. Therefore, { g(x) = (x - 0)^2 + (-3) $}$ can be simplified to { g(x) = x^2 - 3 $}$.

Key Concepts

  • Translation of a Quadratic Function: When we translate a quadratic function, we need to adjust the values of { a $}$, { h $}$, and { k $}$ accordingly.
  • Standard Form of a Quadratic Function: The standard form of a quadratic function is { a(x - h)^2 + k $}$, where { a $}$, { h $}$, and { k $}$ are constants.
  • Shifting the Graph of a Quadratic Function: To shift the graph of a quadratic function, we need to adjust the value of { k $}$ accordingly.

Common Mistakes

  • Not Using the Standard Form of a Quadratic Function: When working with quadratic functions, it is essential to use the standard form to ensure that we are working with the correct values of { a $}$, { h $}$, and { k $}$.
  • Not Shifting the Graph Correctly: When shifting the graph of a quadratic function, we need to adjust the value of { k $}$ accordingly. Failure to do so can result in an incorrect translation.

Real-World Applications

  • Physics: Quadratic functions are used to model the motion of objects under the influence of gravity. Shifting the graph of a quadratic function can be used to model the effect of gravity on an object's motion.
  • Engineering: Quadratic functions are used to model the behavior of electrical circuits. Shifting the graph of a quadratic function can be used to model the effect of resistance on an electrical circuit's behavior.

Practice Problems

  • Find the translation 2 units up of the quadratic function { f(x) = x^2 - 4 $}$.
  • Find the translation 1 unit down of the quadratic function { f(x) = x^2 + 2 $}$.

Conclusion

Q: What is the translation of a quadratic function?

A: The translation of a quadratic function is the process of shifting the graph of the function up or down by a certain number of units. This is achieved by adjusting the value of { k $}$ in the standard form of a quadratic function.

Q: How do I find the translation of a quadratic function?

A: To find the translation of a quadratic function, you need to rewrite the function in standard form and then adjust the value of { k $}$ accordingly. For example, if you want to shift the graph of { f(x) = x^2 $}$ 3 units down, you would rewrite the function as { g(x) = (x - 0)^2 + (-3) $}$.

Q: What is the standard form of a quadratic function?

A: The standard form of a quadratic function is { a(x - h)^2 + k $}$, where { a $}$, { h $}$, and { k $}$ are constants.

Q: How do I shift the graph of a quadratic function?

A: To shift the graph of a quadratic function, you need to adjust the value of { k $}$ accordingly. For example, if you want to shift the graph of { f(x) = x^2 $}$ 3 units down, you would decrease the value of { k $}$ by 3.

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

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

  • Not using the standard form of a quadratic function
  • Not shifting the graph correctly
  • Not adjusting the value of { k $}$ accordingly

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

A: Some real-world applications of quadratic function translations include:

  • Physics: Quadratic functions are used to model the motion of objects under the influence of gravity. Shifting the graph of a quadratic function can be used to model the effect of gravity on an object's motion.
  • Engineering: Quadratic functions are used to model the behavior of electrical circuits. Shifting the graph of a quadratic function can be used to model the effect of resistance on an electrical circuit's behavior.

Q: How do I practice quadratic function translations?

A: You can practice quadratic function translations by working through practice problems, such as finding the translation of a given quadratic function or shifting the graph of a quadratic function by a certain number of units.

Practice Problems

  • Find the translation 2 units up of the quadratic function { f(x) = x^2 - 4 $}$.
  • Find the translation 1 unit down of the quadratic function { f(x) = x^2 + 2 $}$.

Conclusion

In this article, we have discussed the concept of quadratic function translations and provided answers to some common questions. We have also highlighted some common mistakes to avoid and provided some real-world applications of quadratic function translations. By practicing quadratic function translations, you can develop a deeper understanding of this important concept in mathematics.

Key Concepts

  • Translation of a Quadratic Function: The process of shifting the graph of a quadratic function up or down by a certain number of units.
  • Standard Form of a Quadratic Function: The standard form of a quadratic function is { a(x - h)^2 + k $}$, where { a $}$, { h $}$, and { k $}$ are constants.
  • Shifting the Graph of a Quadratic Function: The process of adjusting the value of { k $}$ to shift the graph of a quadratic function.

Common Mistakes

  • Not Using the Standard Form of a Quadratic Function: When working with quadratic functions, it is essential to use the standard form to ensure that we are working with the correct values of { a $}$, { h $}$, and { k $}$.
  • Not Shifting the Graph Correctly: When shifting the graph of a quadratic function, we need to adjust the value of { k $}$ accordingly. Failure to do so can result in an incorrect translation.

Real-World Applications

  • Physics: Quadratic functions are used to model the motion of objects under the influence of gravity. Shifting the graph of a quadratic function can be used to model the effect of gravity on an object's motion.
  • Engineering: Quadratic functions are used to model the behavior of electrical circuits. Shifting the graph of a quadratic function can be used to model the effect of resistance on an electrical circuit's behavior.

Practice Problems

  • Find the translation 2 units up of the quadratic function { f(x) = x^2 - 4 $}$.
  • Find the translation 1 unit down of the quadratic function { f(x) = x^2 + 2 $}$.