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The graphs of f and g are given. Find a formula for the function g.
g(x) = 


The graph shows two parabolas: one in blue representing \( f(x) = x^2 \) and one in red representing the function \( g(x) \).

### Observations:
1. **Blue Parabola (\( f(x) = x^2 \))**: This parabola is the basic quadratic function with its vertex at the origin \((0,0)\) and opens upwards.

2. **Red Parabola (\( g(x) \))**: This parabola also opens upwards, suggesting a positive leading coefficient, similar to \( f(x) \). The vertex of this red parabola appears to be shifted to the right compared to the blue parabola.

### Steps to Find the Formula for \( g(x) \):
To find the formula for \( g(x) \), we will use the standard form of a quadratic function:
\[
g(x) = a(x - h)^2 + k
\]
where \((h, k)\) is the vertex of the parabola, and \( a \) is the leading coefficient.

From the graph:
- The vertex of the blue parabola \( f(x) = x^2 \) is at \( (0, 0) \).
- The vertex of the red parabola \( g(x) \) appears to be at \( (3, 0) \). This indicates a horizontal shift of 3 units to the right from the vertex of \( f(x) \).

### Determining the Formula:
Since the vertex of \( g(x) \) is at \((3, 0)\) and the parabola has the same shape as \( f(x) = x^2 \) (meaning the coefficient \( a = 1 \) is the same), we can write:
\[
g(x) = (x - 3)^2
\]

### Conclusion:
The formula for the function \( g \) is:
\[
g(x) = (x - 3)^2
\]

Would you like to go into more detail or have any questions about this? Here are five related questions that might further expand your understanding:

1. What is the effect of changing the value of \( a \) in the quadratic function \( g(x) = a(x - h)^2 + k \)?
2. How would the graph of \( g(x) = (x - 3)^2 + 4 \) differ from \( g(x) = (x - 3)^2 \)?
3. How can we determine the vertex of a parabola given its equation?
4. What does the value of \( k \) represent in the vertex form of a quadratic function?
5. How would the graph of \( g(x) = (x + 3)^2 \) compare to that of \( g(x) = (x - 3)^2 \)?

**Tip:** When identifying transformations of functions, pay close attention to horizontal and vertical shifts, as these directly impact the vertex and position of the graph on the coordinate plane.

Explore how to find the formula for the function g(x) using graphical analysis. Detailed steps include identifying the vertex form of a quadratic function and analyzing shifts in the graph. Suitable for high school students.

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