Question

The three points on the graph are (0,2) (-2,6) (2,6)

The graph shows g(x), which is a translation of f(x)=x^2. Write the function rule for g(x).
Write your answer in the form a(x–h)^2+k, where a, h, and k are integers or simplified fractions.
g(x)=

Answer 1

To find the function rule for \( g(x) \) as a translation of \( f(x) = x^2 \), we need to determine the values of \( a \), \( h \), and \( k \) in the form \( g(x) = a(x - h)^2 + k \).

First, we identify the vertex of the parabola formed by the points given: (0, 2), (-2, 6), and (2, 6). The point (0, 2) will be the vertex because it is a minimum point on the graph.

From the vertex, we can determine \( h \) and \( k \):
- \( h = 0 \)
- \( k = 2 \)

The equation now looks like this:
\[
g(x) = a(x - 0)^2 + 2 = a x^2 + 2
\]

Next, we can use one of the other points to find \( a \). We can use the point (-2, 6):
\[
g(-2) = a(-2)^2 + 2 = 6
\]
This simplifies to:
\[
4a + 2 = 6
\]
Subtracting 2 from both sides gives:
\[
4a = 4
\]
Dividing by 4, we find:
\[
a = 1
\]

Now, we substitute back into our equation for \( g(x) \):
\[
g(x) = 1(x - 0)^2 + 2 = x^2 + 2
\]

Thus, the final form of the function \( g(x) \) is:
\[
g(x) = (x - 0)^2 + 2
\]

In the required form, \( g(x) \) can be expressed as:
\[
g(x) = 1(x - 0)^2 + 2
\]

So, the final answer is:
\[
g(x) = 1(x - 0)^2 + 2
\]