Question

Find g(x), where g(x) is the translation 2 units left of f(x)=x2.
Write your answer in the form a(x–h)2+k, where a, h, and k are integers.
g(x)=

Answer 1

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 \( a(x - h)^2 + k \):

1. First, expand \( (x + 2)^2 \):

\[
g(x) = (x + 2)(x + 2) = x^2 + 4x + 4
\]

2. 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:

\[
g(x) = 1(x - (-2))^2 + 0
\]