Question

Which translation maps the vertex of the graph of the function f(x) = x2 onto the vertex of the function g(x) = –8x + x2 + 7 ?

left 4, down 9
left 4, up 23
right 4, down 9
right 4, up 23

Answer 1

To find the translation that maps the vertex of the graph of the function \( f(x) = x^2 \) onto the vertex of the function \( g(x) = -8x + x^2 + 7 \), we first need to determine the vertices of both functions.

1. **Vertex of \( f(x) = x^2 \):**
The function \( f(x) = x^2 \) is a standard quadratic function. Its vertex is at the point \( (0, 0) \).

2. **Vertex of \( g(x) = -8x + x^2 + 7 \):**
The function can be rewritten in standard quadratic form as:
\[
g(x) = x^2 - 8x + 7
\]
To find the vertex, we can use the vertex formula \( x = -\frac{b}{2a} \) where \( a = 1 \) and \( b = -8 \):
\[
x = -\frac{-8}{2 \cdot 1} = \frac{8}{2} = 4
\]
Now we substitute \( x = 4 \) into \( g(x) \) to find the corresponding \( y \)-coordinate:
\[
g(4) = (4)^2 - 8(4) + 7 = 16 - 32 + 7 = -16 + 7 = -9
\]
Thus, the vertex of \( g(x) \) is at the point \( (4, -9) \).

3. **Translation from \( (0, 0) \) to \( (4, -9) \):**
To find the translation vector from \( (0, 0) \) to \( (4, -9) \), we subtract the coordinates:
\[
\text{Translation vector} = (4 - 0, -9 - 0) = (4, -9)
\]
Thus, the translation is **right 4** and **down 9**.

Therefore, the translation that maps the vertex of \( f(x) = x^2 \) onto the vertex of \( g(x) = -8x + x^2 + 7 \) is:

**right 4, down 9**.