Question

The graph shows g(x), which is a translation of f(x)=x^2. Write the function rule for g(x).
(-2,8)(0,4)(2,8)
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 \( g(x) \), which is a translation of \( f(x) = x^2 \), based on the points provided: (-2, 8), (0, 4), and (2, 8), we need to determine the values of \( a \), \( h \), and \( k \) in the vertex form of a quadratic function, \( g(x) = a(x - h)^2 + k \).

1. **Identify the vertex (h, k)**:
- The points (-2, 8) and (2, 8) imply that the maximum or minimum is at x=0, where the value is lower (4).
- Therefore, the vertex of the parabola is at (0, 4). This gives us \( h = 0 \) and \( k = 4 \).

2. **Determine the value of \( a \)**:
- We can use one of the points to find \( a \). Let's use the point (2, 8).
- Substitute \( x = 2 \) and \( g(x) = 8 \) into the function:
\[
8 = a(2 - 0)^2 + 4.
\]
\[
8 = 4a + 4.
\]
\[
8 - 4 = 4a.
\]
\[
4 = 4a.
\]
\[
a = 1.
\]

Now we have identified the values: \( a = 1 \), \( h = 0 \), and \( k = 4 \).

Thus, the function rule for \( g(x) \) is:

\[
g(x) = 1(x - 0)^2 + 4, \text{ or simplified, } g(x) = x^2 + 4.
\]

In the requested form, it can be expressed as:

\[
g(x) = (x - 0)^2 + 4.
\]

Final answer:
\[
g(x) = (x - 0)^2 + 4.
\]