https://file.aiquickdraw.com/mathbot/file/1727269654386pdjn1as2.jpg

The given function is \( f(x) = x^2 \), and the problem asks for the equation of the graphed function shown in the image. The options provided are:

A. \( y = f(x - 3) - 2 \)  
B. \( y = f(x + 3) - 2 \)  
C. \( y = f(x + 2) - 3 \)  
D. \( y = f(x - 2) + 3 \)  

Let's analyze the transformations step-by-step:

1. **Horizontal Shifts:**  
   A shift left or right is represented by \( f(x \pm h) \). If the graph shifts left by \( h \) units, it becomes \( f(x + h) \), and if it shifts right by \( h \) units, it becomes \( f(x - h) \).

2. **Vertical Shifts:**  
   A vertical shift is represented by adding or subtracting a constant \( k \) outside the function: \( f(x) \pm k \).

### Analysis of the Graph:
1. **Horizontal Shift:**  
   The graph of \( f(x) = x^2 \) is typically centered at the origin \((0, 0)\). The graph in the image appears to have been shifted **right** by 2 units. This corresponds to the transformation \( f(x - 2) \).

2. **Vertical Shift:**  
   The vertex of the parabola \( f(x) = x^2 \) normally starts at \((0, 0)\). In the graph, it has moved up by 3 units. This corresponds to adding 3 to the function.

### Combining Transformations:
Combining the horizontal and vertical transformations, the equation of the transformed function is:  
\[ y = f(x - 2) + 3 = (x - 2)^2 + 3 \]

This corresponds to option **D**: \( y = f(x - 2) + 3 \).

Would you like more details or have any questions?

### Related Questions:
1. How do vertical transformations affect the graph of a quadratic function?
2. How can you identify horizontal shifts in polynomial functions?
3. What are the effects of reflecting a function over the x-axis or y-axis?
4. How do you find the vertex of a transformed quadratic function?
5. What is the general form of a transformed quadratic function?

**Tip:** Always check both the horizontal and vertical shifts independently to accurately determine the function's transformation.

If f(x) = x^2, what is the equation of the graphed function?

Analyze the graph of a transformed quadratic function. Learn how horizontal and vertical shifts affect the equation f(x) = x^2. This example demonstrates how the graph shifts right by 2 units and up by 3 units to yield the equation f(x-2) + 3.

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