Use transformations of the graph of $$ f(x)=x^{2} $$ to determine the graph of the given function. $$ g(x)=(x-4)^{2} $$

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To determine the graph of $$ g(x) = (x - 4)^2 $$ using transformations of the parent function $$ f(x) = x^2 $$, follow these steps:

### 1. Start with the Parent Function
The parent function $$ f(x) = x^2 $$ is a parabola with:
- **Vertex:** $$(0, 0)$$
- **Axis of Symmetry:** $$ x = 0 $$
- **Direction:** Opens upward

### 2. Identify the Transformation
The function $$ g(x) = (x - 4)^2 $$ involves a **horizontal shift** of the parent function. Here's how:

- **Inside the Function (Horizontal Shift):**
  - The expression $$ (x - 4) $$ indicates a shift to the **right** by 4 units.
  
  Recall that for a function $$ f(x - h) $$:
  - If $$ h > 0 $$, the graph shifts **right** by $$ h $$ units.
  - If $$ h < 0 $$, the graph shifts **left** by $$ |h| $$ units.

### 3. Apply the Transformation
- **Shift Right by 4 Units:**
  - Move every point of $$ f(x) = x^2 $$ 4 units to the right.
  
- **New Vertex:**
  - The original vertex $$(0, 0)$$ moves to $$(4, 0)$$.

### 4. Sketch the Transformed Graph
- **Vertex:** $$(4, 0)$$
- **Axis of Symmetry:** $$ x = 4 $$
- **Shape:** Same as the parent parabola, opening upward.

### 5. Summary of Transformation
- **Type of Transformation:** Horizontal shift
- **Direction and Distance:** 4 units to the right
- **Effect on Vertex:** Moves from $$(0, 0)$$ to $$(4, 0)$$

### 6. Visual Representation

Here's a comparison of the two graphs:

**Parent Function $$ f(x) = x^2 $$:**
```
    |
  2 |     *
    |    * *
  1 |   *   *
    |  *     *
  0 |*-------*-------
    |        
```
**Transformed Function $$ g(x) = (x - 4)^2 $$:**
```
        |
      2 |         *
        |        * *
      1 |       *   *
        |      *     *
      0 |-----*-------*-----
        |             
```
*Note:* In the transformed graph, the vertex has moved to the point $$(4, 0)$$.

### 7. Conclusion
By applying a horizontal shift of 4 units to the right to the parent function $$ f(x) = x^2 $$, you obtain the graph of $$ g(x) = (x - 4)^2 $$. The parabola retains its shape and orientation but is repositioned so that its vertex is at $$(4, 0)$$.
To graph $$ g(x) = (x - 4)^2 $$, shift the parent function $$ f(x) = x^2 $$ 4 units to the right. The vertex moves from $$(0, 0)$$ to $$(4, 0)$$, and the parabola opens upward with the same shape.

Use transformations of the graph of \( f(x)=x^{2} \) to determine the graph of the given function. \[ g(x)=(x-4)^{2} \]

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