The Function G ( X G(x G ( X ] Is A Translation Of F ( X ) = ( X + 3 ) 2 − 10 F(x) = (x+3)^2 - 10 F ( X ) = ( X + 3 ) 2 − 10 . The Axis Of Symmetry Of G ( X G(x G ( X ] Is 5 Units To The Right Of F ( X F(x F ( X ]. Which Function Could Be G ( X G(x G ( X ]?A. G ( X ) = ( X − 2 ) 2 + K G(x) = (x-2)^2 + K G ( X ) = ( X − 2 ) 2 + K B. $g(x)

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Understanding the Axis of Symmetry

The axis of symmetry is a crucial concept in mathematics, particularly in the study of quadratic functions. It is the vertical line that passes through the vertex of a parabola and is a key feature of the function's graph. In this case, we are given that the axis of symmetry of $g(x)$ is 5 units to the right of $f(x)$. This means that if the axis of symmetry of $f(x)$ is at $x = -3$, the axis of symmetry of $g(x)$ will be at $x = -3 + 5 = 2$.

The Function $f(x)$

The function $f(x) = (x+3)^2 - 10$ is a quadratic function with a vertex at $x = -3$. To find the axis of symmetry, we can use the formula $x = -\frac{b}{2a}$, where $a$ and $b$ are the coefficients of the quadratic function. In this case, $a = 1$ and $b = 6$, so the axis of symmetry is at $x = -\frac{6}{2(1)} = -3$.

The Function $g(x)$

Since the axis of symmetry of $g(x)$ is 5 units to the right of $f(x)$, we can write the equation of $g(x)$ as $g(x) = (x-2)^2 + k$, where $k$ is a constant. This is because the vertex of $g(x)$ is at $x = 2$, which is 5 units to the right of the vertex of $f(x)$.

Finding the Value of $k$

To find the value of $k$, we need to find the value of $f(x)$ at $x = 2$. We can do this by substituting $x = 2$ into the equation of $f(x)$:

$f(2) = (2+3)^2 - 10$ $f(2) = 25 - 10$ $f(2) = 15$

Comparing the Functions

Now that we have found the value of $f(2)$, we can compare it to the value of $g(2)$:

$g(2) = (2-2)^2 + k$ $g(2) = 0 + k$

Since $f(2) = 15$, we know that $g(2) = 15$. Therefore, we can set up the equation:

$0 + k = 15$

Solving for $k$, we get:

$k = 15$

Conclusion

Based on our analysis, we can conclude that the function $g(x)$ is indeed $g(x) = (x-2)^2 + 15$. This is because the axis of symmetry of $g(x)$ is 5 units to the right of $f(x)$, and the value of $f(2)$ is equal to the value of $g(2)$.

Answer

The correct answer is:

A. $g(x) = (x-2)^2 + 15$

Discussion

This problem requires a deep understanding of quadratic functions and their properties. The axis of symmetry is a key feature of a quadratic function, and it is essential to understand how it is affected by translations. In this case, the translation of $f(x)$ to get $g(x)$ is a horizontal shift of 5 units to the right. This means that the axis of symmetry of $g(x)$ is also shifted 5 units to the right, resulting in a new axis of symmetry at $x = 2$.

Additional Tips

• When working with quadratic functions, it is essential to understand the concept of axis of symmetry and how it is affected by translations.
• The axis of symmetry is a vertical line that passes through the vertex of a parabola.
• To find the axis of symmetry of a quadratic function, use the formula $x = -\frac{b}{2a}$.
• When translating a quadratic function, the axis of symmetry is also translated by the same amount.

Related Topics

• Quadratic functions
• Axis of symmetry
• Translations of functions
• Graphing quadratic functions

References

• [1] "Quadratic Functions" by Math Open Reference
• [2] "Axis of Symmetry" by Khan Academy
• [3] "Translations of Functions" by Purplemath
  

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Understanding the Axis of Symmetry

The axis of symmetry is a crucial concept in mathematics, particularly in the study of quadratic functions. It is the vertical line that passes through the vertex of a parabola and is a key feature of the function's graph. In this case, we are given that the axis of symmetry of $g(x)$ is 5 units to the right of $f(x)$. This means that if the axis of symmetry of $f(x)$ is at $x = -3$, the axis of symmetry of $g(x)$ will be at $x = -3 + 5 = 2$.

Q&A

Q: What is the axis of symmetry of $f(x) = (x+3)^2 - 10$?

A: The axis of symmetry of $f(x) = (x+3)^2 - 10$ is at $x = -3$.

Q: What is the axis of symmetry of $g(x)$?

A: The axis of symmetry of $g(x)$ is 5 units to the right of $f(x)$, which means it is at $x = 2$.

Q: How do you find the axis of symmetry of a quadratic function?

A: To find the axis of symmetry of a quadratic function, use the formula $x = -\frac{b}{2a}$, where $a$ and $b$ are the coefficients of the quadratic function.

Q: What is the equation of $g(x)$?

A: The equation of $g(x)$ is $g(x) = (x-2)^2 + k$, where $k$ is a constant.

Q: How do you find the value of $k$?

A: To find the value of $k$, you need to find the value of $f(x)$ at $x = 2$ and set it equal to the value of $g(2)$.

Q: What is the value of $k$?

A: The value of $k$ is 15.

Q: What is the equation of $g(x)$?

A: The equation of $g(x)$ is $g(x) = (x-2)^2 + 15$.

Conclusion

In this article, we have discussed the function $g(x)$, which is a translation of $f(x) = (x+3)^2 - 10$. We have found the axis of symmetry of $g(x)$, which is 5 units to the right of $f(x)$. We have also found the equation of $g(x)$, which is $g(x) = (x-2)^2 + 15$. This article is a continuation of our previous discussion on the function $g(x)$ and provides a detailed explanation of the concepts involved.

Additional Tips

• When working with quadratic functions, it is essential to understand the concept of axis of symmetry and how it is affected by translations.
• The axis of symmetry is a vertical line that passes through the vertex of a parabola.
• To find the axis of symmetry of a quadratic function, use the formula $x = -\frac{b}{2a}$.
• When translating a quadratic function, the axis of symmetry is also translated by the same amount.

Related Topics

• Quadratic functions
• Axis of symmetry
• Translations of functions
• Graphing quadratic functions

References

• [1] "Quadratic Functions" by Math Open Reference
• [2] "Axis of Symmetry" by Khan Academy
• [3] "Translations of Functions" by Purplemath

Frequently Asked Questions

Q: What is the axis of symmetry of $f(x) = (x+3)^2 - 10$?

A: The axis of symmetry of $f(x) = (x+3)^2 - 10$ is at $x = -3$.

Q: What is the axis of symmetry of $g(x)$?

A: The axis of symmetry of $g(x)$ is 5 units to the right of $f(x)$, which means it is at $x = 2$.

Q: How do you find the axis of symmetry of a quadratic function?

A: To find the axis of symmetry of a quadratic function, use the formula $x = -\frac{b}{2a}$, where $a$ and $b$ are the coefficients of the quadratic function.

Q: What is the equation of $g(x)$?

A: The equation of $g(x)$ is $g(x) = (x-2)^2 + k$, where $k$ is a constant.

Q: How do you find the value of $k$?

A: To find the value of $k$, you need to find the value of $f(x)$ at $x = 2$ and set it equal to the value of $g(2)$.

Q: What is the value of $k$?

A: The value of $k$ is 15.

Q: What is the equation of $g(x)$?

A: The equation of $g(x)$ is $g(x) = (x-2)^2 + 15$.

Final Thoughts

In conclusion, the function $g(x)$ is a translation of $f(x) = (x+3)^2 - 10$. The axis of symmetry of $g(x)$ is 5 units to the right of $f(x)$, and the equation of $g(x)$ is $g(x) = (x-2)^2 + 15$. This article provides a detailed explanation of the concepts involved and is a continuation of our previous discussion on the function $g(x)$.