To find the function \( g(x) \) that represents the translation of the function \( f(x) = x^2 \) 2 units to the left, we need to use the formula for translating a function horizontally.
When translating a function \( f(x) \) to the left by \( h \) units, we replace \( x \) with \( x + h \). In this case, since we are translating 2 units to the left, we will replace \( x \) with \( x + 2 \).
Therefore, we have:
\[ g(x) = f(x + 2) = (x + 2)^2 \]
Now we need to express \( g(x) \) in the form \( a(x - h)^2 + k \):
- First, expand \( (x + 2)^2 \):
\[ g(x) = (x + 2)(x + 2) = x^2 + 4x + 4 \]
- Next, we can rewrite the quadratic in vertex form. The standard form we want is \( a(x - h)^2 + k \). The vertex form of a parabola \( ax^2 + bx + c \) can be found by completing the square:
Starting from \( x^2 + 4x + 4 \):
\[ g(x) = (x^2 + 4x + 4) \] \[ = (x + 2)^2 + 0 \]
So, we have:
- \( a = 1 \)
- \( h = -2 \)
- \( k = 0 \)
Thus, we can write:
\[ g(x) = 1(x - (-2))^2 + 0 \]
Finally, the answer is:
\[ g(x) = 1(x + 2)^2 + 0 \]
In summary:
\[ g(x) = (x + 2)^2 \] or in the specified vertex form:
\[ g(x) = 1(x - (-2))^2 + 0 \]
Putting it all together, we can state:
\[ g(x) = 1(x + 2)^2 + 0 \]
So the final answer in the requested form is:
\[ g(x) = 1(x - (-2))^2 + 0 \]
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