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


To find the formula for the function \(g(x)\) based on the graph provided, let's analyze the characteristics of the given graphs.

### Step-by-Step Analysis

1. **Identify the Given Functions:**
   - The blue graph is labeled \(f(x) = x^2\). This is a standard parabola that opens upwards with its vertex at the origin \((0, 0)\).
   - The red graph is the function \(g(x)\), which is another parabola opening upwards. We need to find its equation.

2. **Observe the Properties of \(g(x)\):**
   - The vertex of \(g(x)\) appears to be at the point \((3, 0)\).
   - Since \(g(x)\) is a parabola and shares the same shape as \(f(x) = x^2\), it is likely a transformation of \(f(x)\).

3. **Determine the Transformation:**
   - The red parabola is shifted 3 units to the right compared to \(f(x)\).
   - This indicates a horizontal shift.

4. **Formulate the Equation:**
   - A horizontal shift of a function \(f(x)\) by \(h\) units to the right is represented by \(f(x - h)\).
   - Since \(f(x) = x^2\) and \(g(x)\) is \(f(x)\) shifted 3 units to the right, the equation for \(g(x)\) can be written as:
   \[
   g(x) = (x - 3)^2
   \]

### Conclusion

The formula for the function \(g(x)\) is:

\[
g(x) = (x - 3)^2
\]

Would you like more details or have any questions?

### Related Questions:

1. What would be the formula for \(g(x)\) if the graph was shifted 3 units to the left instead of to the right?
2. How do vertical shifts affect the equation of a parabola?
3. How can you determine if a function represents a parabola from its equation?
4. What is the impact of reflecting a parabola across the x-axis on its equation?
5. How can we determine the axis of symmetry for a parabola given its equation?

### Tip:

When analyzing transformations of functions, always note changes in position (horizontal or vertical shifts) and shape (stretching, compressing, reflecting). These changes help in formulating the new equation from the parent function.

Learn how to find the formula for function g(x) given the graphs of f(x) = x^2 and g(x). Understand the transformation principles of functions and apply them to determine the equation of g(x). Suitable for high school students studying functions and transformations.

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