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.
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Vertex of \( f(x) = x^2 \): The function \( f(x) = x^2 \) is a standard quadratic function. Its vertex is at the point \( (0, 0) \).
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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 \( (4, -9) \).
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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.