Question

The graph shows g(x), which is a translation of f(x)=x^2. Write the function rule for g(x).
(1,10)(0,9)(1,10)
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 write the function rule for \( g(x) \) as a translation of \( f(x) = x^2 \), we can analyze the points given and find how the function has been transformed.

From your points:

1. The vertex of the parabola \( f(x) = x^2 \) is at \( (0, 0) \).
2. The point \( (0, 9) \) indicates that when \( x = 0 \), \( g(x) = 9 \). This means the parabola has been shifted up vertically by 9 units.

Thus, \( k = 9 \).

We also know that the standard form for a parabola includes a horizontal shift represented by \( h \) in \( (x - h) \).

From the point \( (1, 10) \):
- If \( x = 1 \),
- We know that \( g(1) = 10 \) but the base function \( f(1) = 1^2 = 1 \).

Since \( g(1) = 10 \) while \( f(1) = 1 \), we see that:

\[
g(1) = a(1 - h)^2 + k = 10
\]
\[
f(1) = a(1 - h)^2 + 9
\]
**Using the values of k we calculated earlier:**
\[
10 - 9 = a(1 - h)^2
\]
Which simplifies to:
\[
1 = a(1 - h)^2
\]
**Now checking the condition for "a" when h=0, Inequality**
- \( a \) could be any number as long as it translates vertically.

To find if there is a horizontal shift, with \( h = 0 \):
If no horizontal shift, \( a \) must equal 1 for both shifts. Therefore:
- It's quite possibly:
**g(x) = 1(x - 0)^2 + 9 (Not displacing h only k goes with the form for multiples)**
Thus collectively if we rerun all cases, we can derive the full:

\[
g(x) = 1(x - 0)^2 + 9
\]

So the explicit conclusion is:
The function rule for \( g(x) \) is
\[
\boxed{g(x) = (x - 0)^2 + 9} \text{ or simplified as } g(x) = x^2 + 9.
\]