Question

The graph of g(x), shown below in pink, has the same shape as the graph of f(x) = x2, shown in gray.
The equation for g(x) is
f(x)
5
g(x)/
十一-
-5
(o. 0)
5
(3,-1)
-5|
choose your answer...
へ
choose your answer...
g(x) = (x+3)2-1
8(x) = (x-1)3-3
g(x) = (x+1)? -3
g(x) = (x-3) - 1

Answer 1

To determine the correct equation for \( g(x) \), we need to analyze the transformations applied to the original function \( f(x) = x^2 \) to obtain \( g(x) \).

From the description, it seems that:

- The graph of \( g(x) \) maintains the same shape as \( f(x) = x^2 \) but has has undergone some transformations (a horizontal shift, vertical shift, or vertical stretch/compression).

1. **Identify the vertex of the function**: From the point given (3, -1), we see that \( g(x) \) has its vertex at this point.

2. **Working with the vertex**: The standard form for a quadratic function is:
\[
g(x) = a(x-h)^2 + k
\]
where \( (h,k) \) is the vertex.

Here, \( (h, k) = (3, -1) \).

3. **Finding the value of \( a \)**: If the function \( g(x) \) is the same shape as \( f(x) = x^2 \), then \( a \) must equal 1 (indicating no vertical stretching or compressing).

So, substituting \( h \) and \( k \) into the vertex form:
\[
g(x) = 1(x-3)^2 - 1
\]
or more simply,
\[
g(x) = (x-3)^2 - 1
\]

4. **Select from the options**:
The last choice you presented is:
\[
g(x) = (x-3)^2 - 1
\]
This matches our derived equation.

Thus, the correct equation for \( g(x) \) is:
\[
\boxed{g(x) = (x-3)^2 - 1}
\]