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 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 the behavior of functions and their graphs. 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 find the vertex of a quadratic function in the form $f(x)=ax^2+bx+c$, we can use the formula:

$$\text{Vertex} = \left(-\frac{b}{2a}, f\left(-\frac{b}{2a}\right)\right)$$

In this case, $a=1$ and $b=0$, so the vertex of the function $f(x)=x^2$ is indeed located at the origin (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 form. To find the vertex of this function, we can use the same formula:

$$\text{Vertex} = \left(-\frac{b}{2a}, f\left(-\frac{b}{2a}\right)\right)$$

In this case, $a=1$ and $b=-8$, so the vertex of the function $g(x)=-8x+x^2+7$ is located at:

$$\text{Vertex} = \left(-\frac{-8}{2(1)}, g\left(-\frac{-8}{2(1)}\right)\right)$$

$$\text{Vertex} = \left(4, g(4)\right)$$

To find the value of $g(4)$, we can substitute $x=4$ into the function:

$$g(4) = -8(4) + (4)^2 + 7$$

$$g(4) = -32 + 16 + 7$$

$$g(4) = -9$$

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

Vertex Translation

Now that we have found the vertices of both functions, we can apply vertex translation to map the vertex of the function $f(x)=x^2$ onto the vertex of the function $g(x)=-8x+x^2+7$. Vertex translation involves shifting the vertex of a function by a certain amount in the horizontal and vertical directions.

To map the vertex of the function $f(x)=x^2$ onto the vertex of the function $g(x)=-8x+x^2+7$, we need to find the translation that will move the vertex of $f(x)=x^2$ from (0, 0) to (4, -9).

The translation that will move the vertex of $f(x)=x^2$ from (0, 0) (4, -9) is:

• Left 4 units (since the x-coordinate of the new vertex is 4, which is 4 units to the left of the original vertex)
• Down 9 units (since the y-coordinate of the new vertex is -9, which is 9 units below the original vertex)

Therefore, the correct translation is:

Left 4, down 9

This is option A.

Conclusion

In conclusion, vertex translation is a fundamental concept in graphing functions. By understanding how to apply vertex translation, we can map the vertex of one function onto the vertex of another function. In this article, we applied vertex translation to map the vertex of the function $f(x)=x^2$ onto the vertex of the function $g(x)=-8x+x^2+7$. We found that the correct translation is Left 4, down 9.

Final Answer

The final answer is:

Understanding Vertex Translation

In our 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 answer some frequently asked questions about vertex translation to help you better understand this concept.

Q: What is vertex translation?

A: Vertex translation is a process of shifting the vertex of a function by a certain amount in the horizontal and vertical directions. This process involves changing the coordinates of the vertex of a function to a new location on the coordinate plane.

Q: Why is vertex translation important?

A: Vertex translation is important because it helps us understand the behavior of functions and their graphs. By applying vertex translation, we can map the vertex of one function onto the vertex of another function, which is useful in various mathematical applications.

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

A: To find the vertex of a function, you can use the formula:

$$\text{Vertex} = \left(-\frac{b}{2a}, f\left(-\frac{b}{2a}\right)\right)$$

This formula works for quadratic functions in the form $f(x)=ax^2+bx+c$.

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

A: Vertex translation involves shifting the vertex of a function by a certain amount in both the horizontal and vertical directions. Horizontal and vertical shifts, on the other hand, involve shifting the entire graph of a function by a certain amount in either the horizontal or vertical direction.

Q: Can I apply vertex translation to any function?

A: Yes, you can apply vertex translation to any function, but it's most useful for quadratic functions. Vertex translation is a powerful tool for understanding the behavior of quadratic functions and their graphs.

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

A: To apply vertex translation to a function, you need to find the translation that will move the vertex of the function from its original location to a new location. This involves finding the horizontal and vertical shifts required to move the vertex to the new location.

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

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

• Not using the correct formula to find the vertex of a function
• Not considering the horizontal and vertical shifts required to move the vertex to the new location
• Not checking the work to ensure that the vertex has been translated correctly

Q: Can I use vertex translation to solve real-world problems?

A: Yes, vertex translation can be used to solve real-world problems. For example, you can use vertex translation to model the motion of an object or to understand the behavior of a system.

Conclusion

In conclusion, vertex translation is a powerful tool for understanding the behavior of functions and their graphs. By answering some frequently asked questions about vertex translation, we hope to have provided you with a better understanding of this concept and its applications.

Final Tips

• Make sure to use the correct formula to find the vertex of a function
• Consider the horizontal and vertical shifts required to move the vertex to the new location
• Check your work to ensure that the vertex has been translated correctly

We hope this article has been helpful in your understanding of vertex translation. If you have any further questions or need additional clarification, please don't hesitate to ask.