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 graph of $$ f(x) = x^2 $$, follow these steps:

### 1. Start with the Basic Function
- **Basic Function:** $$ f(x) = x^2 $$
- **Graph:** A parabola opening upwards with its vertex at the origin $$(0, 0)$$.

### 2. Identify the Transformation
- **Given Function:** $$ g(x) = (x - 4)^2 $$
- **Transformation Applied:** The expression $$ (x - 4) $$ inside the squared term indicates a **horizontal shift**.

### 3. Determine the Type and Direction of Shift
- **Expression Analysis:** $$ (x - h)^2 $$ shifts the basic parabola horizontally.
  - If $$ h > 0 $$, the shift is **to the right**.
  - If $$ h < 0 $$, the shift is **to the left**.
- **In $$ g(x) = (x - 4)^2 $$:**
  - $$ h = 4 $$ (since $$ x $$ is replaced by $$ x - 4 $$)
  - **Direction:** 4 units **to the right**.

### 4. Apply the Transformation
- **Shift the Vertex:** Move the vertex of the parabola from $$(0, 0)$$ to $$(4, 0)$$.
  
### 5. Sketch the Transformed Graph
- **New Vertex:** $$(4, 0)$$
- **Shape:** Same as $$ f(x) = x^2 $$ (a parabola opening upwards)
- **Axis of Symmetry:** $$ x = 4 $$

### 6. Summary of Transformation
- **Original Function:** $$ f(x) = x^2 $$ with vertex at $$(0, 0)$$.
- **Transformed Function:** $$ g(x) = (x - 4)^2 $$ with vertex shifted to $$(4, 0)$$.

### 7. Visual Representation

Here's a comparison of both graphs:

**Graph of $$ f(x) = x^2 $$:**
```
    |
  4 |       *
    |     *   *
  2 |   *       *
    | *           *
--- +--------------->
  -2 | *           *
    |   *       *
  -4 |     *   *
    |       *
```

**Graph of $$ g(x) = (x - 4)^2 $$:**
```
    |
  4 |             *
    |           *   *
  2 |         *       *
    |       *           *
--- +----------------------->
  -2 |       *           *
    |         *       *
  -4 |           *   *
    |             *
```

In the graph of $$ g(x) $$, the entire parabola has shifted **4 units to the right** compared to the graph of $$ f(x) = x^2 $$.

### 8. Conclusion
By applying a horizontal shift of **4 units to the right**, the graph of $$ f(x) = x^2 $$ becomes the graph of $$ g(x) = (x - 4)^2 $$. The vertex moves from $$(0, 0)$$ to $$(4, 0)$$, and the overall shape of the parabola remains unchanged.
The graph of $$ g(x) = (x - 4)^2 $$ is the graph of $$ f(x) = x^2 $$ shifted 4 units to the right.

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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