To find which translation maps the vertex of the graph of the function \( f(x) = x^2 \) onto the vertex of the function \( g(x) = x^2 + 2x + 1 \), we first need to identify the vertices of both functions.
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The function \( f(x) = x^2 \) is in standard form, and its vertex is at the point \( (0, 0) \).
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The function \( g(x) = x^2 + 2x + 1 \) can be rewritten in vertex form. To do this, we can complete the square: \[ g(x) = x^2 + 2x + 1 = (x + 1)^2 \] The vertex of \( g(x) \) is at \( (-1, 0) \).
Now we compare the vertices:
- Vertex of \( f(x) \) is \( (0, 0) \)
- Vertex of \( g(x) \) is \( (-1, 0) \)
To move from the vertex of \( f(x) \) at \( (0, 0) \) to the vertex of \( g(x) \) at \( (-1, 0) \), we need to translate left by 1 unit.
Therefore, the correct translation is left 1 unit.