Which Translation Maps The Vertex Of The Graph Of The Function F ( X ) = X 2 F(x)=x^2 F ( X ) = X 2 Onto The Vertex Of The Function G ( X ) = − 8 X + X 2 + 7 G(x)=-8x+x^2+7 G ( X ) = − 8 X + X 2 + 7 ?A. Left 4, Down 9 B. Left 4, Up 23 C. Right 4, Down 9 D. Right 4, Up 23

Understanding Vertex Translation

In mathematics, vertex translation is a process of shifting the vertex of a quadratic function to a new location on the coordinate plane. This concept is crucial in algebra and calculus, as it helps in understanding the behavior of quadratic functions and their transformations. In this article, we will explore the process of vertex translation and apply it to the given problem.

The Function $f(x)=x^2$

The function $f(x)=x^2$ is a quadratic function with a vertex at the origin (0, 0). This function represents a parabola that opens upwards, with its vertex at the point (0, 0).

The Function $g(x)=-8x+x^2+7$

The function $g(x)=-8x+x^2+7$ is also a quadratic function, but it has a different vertex. To find the vertex of this function, we need to complete the square.

Completing the Square

To complete the square, we need to rewrite the function in the form $g(x)=(x-h)^2+k$, where (h, k) is the vertex of the function.

import sympy as sp
x = sp.symbols('x')

g = -8*x + x**2 + 7

g_completed_square = sp.expand((x - 4)**2 + 7)
print(g_completed_square)

The output of the code above is:

x**2 - 8*x + 15

Comparing this with the original function, we can see that the vertex of the function $g(x)$ is at the point (4, 15).

Vertex Translation

Now that we have found the vertex of the function $g(x)$, we can apply the concept of vertex translation to map the vertex of the function $f(x)$ onto the vertex of the function $g(x)$.

Translation Vectors

To map the vertex of the function $f(x)$ onto the vertex of the function $g(x)$, we need to find the translation vector. The translation vector is a vector that points from the vertex of the function $f(x)$ to the vertex of the function $g(x)$.

In this case, the vertex of the function $f(x)$ is at the point (0, 0), and the vertex of the function $g(x)$ is at the point (4, 15). Therefore, the translation vector is:

translation_vector = (4, 15)

Applying the Translation Vector

To apply the translation vector, we need to add the x-coordinate and y-coordinate of the translation vector to the x-coordinate and y-coordinate of the vertex of the function $f(x)$.

# Define the vertex of the function f(x)
vertex_f = (0, 0)

translation_vector = (4, 15)

vertex_g = (vertex_f[0] + translation_vector[0], vertex_f[1] + translation_vector[1])
print(vertex_g)

The output of the code above is:

(4, 15)

Therefore, the vertex of the function $g(x)$ is at the point (4, 15).

Conclusion

In this article, we have explored the concept of vertex translation and applied it to the given problem. We have found the vertex of the function $g(x)$ and applied the translation vector to map the vertex of the function $f(x)$ onto the vertex of the function $g(x)$. The correct answer is:

• A. left 4, down 9: This option is incorrect, as the vertex of the function $g(x)$ is at the point (4, 15), not (0, 6).
• B. left 4, up 23: This option is incorrect, as the vertex of the function $g(x)$ is at the point (4, 15), not (0, 38).
• C. right 4, down 9: This option is incorrect, as the vertex of the function $g(x)$ is at the point (4, 15), not (8, 6).
• D. right 4, up 23: This option is correct, as the vertex of the function $g(x)$ is at the point (4, 15), and the translation vector is (4, 15).

Therefore, the correct answer is:

• D. right 4, up 23
  Vertex Translation: A Comprehensive Guide =============================================

Q&A: Vertex Translation

In this article, we will provide a comprehensive guide to vertex translation, including a Q&A section to help you understand the concept better.

Q: What is vertex translation?

A: Vertex translation is a process of shifting the vertex of a quadratic function to a new location on the coordinate plane. This concept is crucial in algebra and calculus, as it helps in understanding the behavior of quadratic functions and their transformations.

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

A: To find the vertex of a quadratic function, you need to complete the square. This involves rewriting the function in the form $(x-h)^2+k$, where $(h, k)$ is the vertex of the function.

Q: What is the translation vector?

A: The translation vector is a vector that points from the vertex of the original function to the vertex of the translated function. It is used to apply the translation to the original function.

Q: How do I apply the translation vector?

A: To apply the translation vector, you need to add the x-coordinate and y-coordinate of the translation vector to the x-coordinate and y-coordinate of the vertex of the original function.

Q: What are the different types of vertex translation?

A: There are two main types of vertex translation:

• Horizontal translation: This involves shifting the vertex of the function to the left or right.
• Vertical translation: This involves shifting the vertex of the function up or down.

Q: How do I determine the direction of the translation?

A: To determine the direction of the translation, you need to look at the sign of the x-coordinate and y-coordinate of the translation vector. If the sign is positive, the translation is in the positive direction. If the sign is negative, the translation is in the negative direction.

Q: What are some common mistakes to avoid when applying vertex translation?

A: Some common mistakes to avoid when applying vertex translation include:

• Not completing the square: Failing to complete the square can lead to incorrect results.
• Not applying the translation vector correctly: Failing to apply the translation vector correctly can lead to incorrect results.
• Not considering the direction of the translation: Failing to consider the direction of the translation can lead to incorrect results.

Q: How do I practice vertex translation?

A: To practice vertex translation, you can try the following:

• Work through examples: Try working through examples of vertex translation to get a feel for the concept.
• Use online resources: There are many online resources available that can help you practice vertex translation.
• Take practice quizzes: Taking practice quizzes can help you assess your understanding of vertex translation.

Conclusion

In this article, we have provided a comprehensive guide to vertex translation, including a Q&A section to help you understand the concept better. We have also discussed some common mistakes to avoid when applying vertex translation and provided tips on how to practice the concept.

Vertex Translation: Key Takeaways

• Vertex translation is a process of shifting the vertex of a quadratic function to a new location on the coordinate plane.
• To find the vertex of a quadratic function, you need to complete the square.
• The translation vector is a vector that points from the vertex of the original function to the vertex of the translated function.
• To apply the translation vector, you need to add the x-coordinate and y-coordinate of the translation vector to the x-coordinate and y-coordinate of the vertex of the original function.
• There are two main types of vertex translation: horizontal translation and vertical translation.
• To determine the direction of the translation, you need to look at the sign of the x-coordinate and y-coordinate of the translation vector.

Vertex Translation: Resources

• Online resources: There are many online resources available that can help you practice vertex translation, including Khan Academy, Mathway, and Wolfram Alpha.
• Textbooks: There are many textbooks available that cover vertex translation, including "Algebra and Trigonometry" by Michael Sullivan and "Calculus" by Michael Spivak.
• Practice quizzes: Taking practice quizzes can help you assess your understanding of vertex translation and identify areas where you need to improve.

Vertex Translation: Final Thoughts

Vertex translation is a powerful tool for understanding the behavior of quadratic functions and their transformations. By mastering the concept of vertex translation, you can gain a deeper understanding of algebra and calculus and improve your problem-solving skills.