To find the function \( g(x) \) after applying the specified translations and transformations to the function \( f(x) = x^2 \), we will use the vertex form of a quadratic function, which is given by \( a(x-h)^2 + k \), where \( (h, k) \) is the vertex of the parabola.
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Translation 4 units left of \( f(x) = x^2 \):
- Translating 4 units left means we replace \( x \) with \( x + 4 \). \[ g(x) = (x + 4)^2 = 1(x + 4)^2 + 0 \] So, \( g(x) = (x + 4)^2 \).
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Translation 5 units up of \( f(x) = x^2 \):
- Translating 5 units up means we add 5 to the entire function. \[ g(x) = x^2 + 5 = 1(x - 0)^2 + 5 \] So, \( g(x) = x^2 + 5 \).
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Reflection across the x-axis of \( f(x) = x^2 \):
- Reflecting across the x-axis means we take the negative of the function. \[ g(x) = -x^2 = -1(x - 0)^2 + 0 \] So, \( g(x) = -x^2 \).
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Translation 2 units right and 1 unit down of \( f(x) = x^2 \):
- Translating 2 units right means we replace \( x \) with \( x - 2 \) and translating 1 unit down means we subtract 1 from the entire function. \[ g(x) = (x - 2)^2 - 1 = 1(x - 2)^2 - 1 \] So, \( g(x) = (x - 2)^2 - 1 \).
To summarize:
- \( g(x) = (x + 4)^2 \)
- \( g(x) = x^2 + 5 \)
- \( g(x) = -x^2 \)
- \( g(x) = (x - 2)^2 - 1 \)