Which Translation Maps The Vertex Of The Graph Of The Function $f(x)=x^2$ Onto The Vertex Of The Function $g(x)=-8x+x^2+7$?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 fundamental concept in graphing functions. It involves shifting the vertex of a function to a new location on the coordinate plane. This concept is crucial in understanding various mathematical operations and transformations. In this article, we will explore the process of vertex translation and how it applies to the given functions $f(x)=x^2$ and $g(x)=-8x+x^2+7$.

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

The function $f(x)=x^2$ is a quadratic function that represents a parabola opening upwards. The vertex of this parabola is located at the origin (0, 0). To understand the concept of vertex translation, let's first analyze the graph of this function.

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 than the function $f(x)=x^2$. 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 $g(x)=-8x+x^2+7$ in the form $(x-h)^2+k$, where $(h, k)$ is the vertex of the parabola.

g(x) = -8x + x^2 + 7
g(x) = (x^2 - 8x) + 7
g(x) = (x^2 - 8x + 16) - 16 + 7
g(x) = (x - 4)^2 - 9

From the completed square form, we can see that the vertex of the function $g(x)=-8x+x^2+7$ is located at the point (4, -9).

Vertex Translation

Now that we have found the vertex of the function $g(x)=-8x+x^2+7$, we need to determine the translation that maps the vertex of the function $f(x)=x^2$ onto the vertex of the function $g(x)=-8x+x^2+7$. This translation involves shifting the vertex of the function $f(x)=x^2$ to the right by 4 units and down by 9 units.

Conclusion

In conclusion, the vertex translation that maps the vertex of the function $f(x)=x^2$ onto the vertex of the function $g(x)=-8x+x^2+7$ is a translation to the right by 4 units and down by 9 units. This translation is represented by the option C. right 4, down 9.

Final Answer

Q&A: Vertex Translation

In the previous article, we explored the concept of vertex translation and how it applies to the given functions $f(x)=x^2$ and $g(x)=-8x+x^2+7$. In this article, we will provide a comprehensive Q&A guide to help you understand vertex translation better.

Q: What is vertex translation?

A: Vertex translation is a process of shifting the vertex of a function to a new location on the coordinate plane. This concept is crucial in understanding various mathematical operations and transformations.

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

A: To find the vertex of a 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 parabola.

Q: What is the vertex of the function $f(x)=x^2$?

A: The vertex of the function $f(x)=x^2$ is located at the origin (0, 0).

Q: What is the vertex of the function $g(x)=-8x+x^2+7$?

A: The vertex of the function $g(x)=-8x+x^2+7$ is located at the point (4, -9).

Q: How do I determine the translation that maps the vertex of one function onto the vertex of another function?

A: To determine the translation, you need to compare the vertices of the two functions. The translation involves shifting the vertex of the first function to the right by the difference in the x-coordinates and down by the difference in the y-coordinates.

Q: What is the translation that maps the vertex of the function $f(x)=x^2$ onto the vertex of the function $g(x)=-8x+x^2+7$?

A: The translation that maps the vertex of the function $f(x)=x^2$ onto the vertex of the function $g(x)=-8x+x^2+7$ is a translation to the right by 4 units and down by 9 units.

Q: What are the options for the translation that maps the vertex of the function $f(x)=x^2$ onto the vertex of the function $g(x)=-8x+x^2+7$?

A: The options for the translation are:

A. left 4, down 9 B. left 4, up 23 C. right 4, down 9 D. right 4, up 23

Q: Which option is the correct translation?

A: The correct translation is C. right 4, down 9.

Conclusion

In conclusion, vertex translation is a fundamental concept in mathematics that involves shifting the vertex of a function to a new location on the coordinate plane. By understanding vertex translation, you can better comprehend various mathematical operations and transformations. We hope this Q&A guide has helped you understand vertex translation better.

Final Answer

The final answer is C.