Question

The graph shows g(x), which is a translation of f(x)=x^2. Write the function rule for g(x).
(-1,8)(0,7)(1,8)
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 \( g(x) \) that is a translation of \( f(x) = x^2 \), we start with the general form of a quadratic function:

\[
g(x) = a(x - h)^2 + k
\]

where \( (h, k) \) represents the vertex of the parabola, and \( a \) determines the direction and width of the parabola.

Given that \( g(x) \) passes through the points \((-1, 8)\), \( (0, 7)\), and \( (1, 8)\), we can start by determining the vertex from these points.

### Step 1: Find the vertex
Based on the points provided, notice that \((-1, 8)\) and \( (1, 8)\) have the same \( y \)-value (8). This indicates that the vertex is located directly between these two points, which is at \( (0, 7) \).

Thus, we have:
- \( h = 0 \)
- \( k = 7 \)

### Step 2: Determine the value of \( a \)
Now we need to determine the value of \( a \). We can use one of the points, such as \((1, 8)\):

Substituting \( x = 1 \) and \( g(x) = 8 \) into the equation:

\[
8 = a(1 - 0)^2 + 7
\]

Simplifying this, we have:

\[
8 = a(1)^2 + 7
\]
\[
8 = a + 7
\]
\[
a = 8 - 7 = 1
\]

### Step 3: Write the function rule
Now we have:
- \( a = 1 \)
- \( h = 0 \)
- \( k = 7 \)

Substituting these values into the function rule:

\[
g(x) = 1(x - 0)^2 + 7
\]

This simplifies to:

\[
g(x) = (x - 0)^2 + 7
\]

So the final function rule for \( g(x) \) is:

\[
g(x) = x^2 + 7
\]

In the form requested:

\[
g(x) = 1(x - 0)^2 + 7
\]

Thus, the answer is:

\[
g(x) = 1(x - 0)^2 + 7
\]