To determine the translation that maps the vertex of the graph of the function [tex]\( f(x) = x^2 \)[/tex] onto the vertex of the function [tex]\( g(x) = -8x + x^2 + 7 \)[/tex], we need to find the vertices of both functions and then calculate the translation.
### Step 1: Finding the vertex of [tex]\( f(x) = x^2 \)[/tex]
The function [tex]\( f(x) = x^2 \)[/tex] is a simple quadratic function in standard form [tex]\( y = ax^2 + bx + c \)[/tex] with [tex]\( a = 1 \)[/tex], [tex]\( b = 0 \)[/tex], and [tex]\( c = 0 \)[/tex]. The vertex form of a quadratic function is [tex]\( y = a(x-h)^2 + k \)[/tex], where [tex]\((h, k)\)[/tex] is the vertex. Since [tex]\( f(x) = x^2 \)[/tex] is already in the form [tex]\( y = (x-0)^2 + 0 \)[/tex], the vertex is at
[tex]\[ (0, 0). \][/tex]
### Step 2: Finding the vertex of [tex]\( g(x) = -8x + x^2 + 7 \)[/tex]
The function [tex]\( g(x) = -8x + x^2 + 7 \)[/tex] can be rewritten as [tex]\( g(x) = x^2 - 8x + 7 \)[/tex]. Again, this is a standard quadratic function in the form [tex]\( y = ax^2 + bx + c \)[/tex] where [tex]\( a = 1 \)[/tex], [tex]\( b = -8 \)[/tex], and [tex]\( c = 7 \)[/tex]. To find the vertex, we use the vertex formula for a parabola [tex]\( x = -\frac{b}{2a} \)[/tex]:
[tex]\[
x = -\frac{-8}{2 \cdot 1} = \frac{8}{2} = 4
\][/tex]
Next, we substitute [tex]\( x = 4 \)[/tex] back into the function to find the [tex]\( y \)[/tex]-coordinate of the vertex:
[tex]\[
g(4) = (4)^2 - 8(4) + 7 = 16 - 32 + 7 = -16 + 7 = -9
\][/tex]
So, the vertex of [tex]\( g(x) \)[/tex] is at
[tex]\[ (4, -9). \][/tex]
### Step 3: Calculating the translation
To map the vertex of [tex]\( f(x) \)[/tex] which is at [tex]\( (0, 0) \)[/tex] to the vertex of [tex]\( g(x) \)[/tex] which is at [tex]\( (4, -9) \)[/tex], we calculate the horizontal and vertical shifts.
- The horizontal translation is from [tex]\( x = 0 \)[/tex] to [tex]\( x = 4 \)[/tex], which is a shift to the right by [tex]\( 4 \)[/tex] units.
- The vertical translation is from [tex]\( y = 0 \)[/tex] to [tex]\( y = -9 \)[/tex], which is a shift down by [tex]\( 9 \)[/tex] units.
Thus, the translation that maps the vertex of [tex]\( f(x) = x^2 \)[/tex] onto the vertex of [tex]\( g(x) = -8x + x^2 + 7 \)[/tex] is:
[tex]\[ \text{right 4, down 9} \][/tex]
So the correct answer is:
[tex]\[ \boxed{\text{right 4, down 9}} \][/tex]
