Use transformations of the graph of $$ f(x)=x^{2} $$ to determine the graph of the given function. $$ g(x)=(x+6)^{2} $$

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To determine the graph of $$ g(x) = (x + 6)^2 $$ using transformations of the basic quadratic function $$ f(x) = x^2 $$, follow these steps:

### Step 1: Understand the Basic Function
The basic function $$ f(x) = x^2 $$ is a parabola with its vertex at the origin $$(0, 0)$$, opening upwards.

### Step 2: Identify the Transformation
The function $$ g(x) = (x + 6)^2 $$ can be seen as a transformation of $$ f(x) = x^2 $$. Specifically, it's in the form:
$$ g(x) = f(x + 6) $$

This represents a **horizontal shift** of the graph of $$ f(x) $$.

### Step 3: Determine the Direction and Distance of the Shift
- **Inside the Function (Horizontal Shift):** For $$ f(x + c) $$, the graph shifts **horizontally**.
  - If $$ c $$ is positive, the graph shifts **left** by $$ c $$ units.
  - If $$ c $$ is negative, the graph shifts **right** by $$ |c| $$ units.

In $$ g(x) = f(x + 6) $$:
- $$ c = 6 $$ (positive), so the graph shifts **6 units to the left**.

### Step 4: Plot the Transformed Graph
- **Vertex Shift:** The original vertex at $$(0, 0)$$ shifts to $$(-6, 0)$$.
- **Shape:** The parabola still opens upwards and retains its original shape (width and direction).

### Summary of the Transformation
- **Original Function:** $$ f(x) = x^2 $$ with vertex at $$(0, 0)$$.
- **Transformed Function:** $$ g(x) = (x + 6)^2 $$ with vertex shifted to $$(-6, 0)$$.
- **Transformation Applied:** Shift the graph of $$ f(x) $$ **6 units to the left**.

### Visual Representation
While I can't provide a visual graph here, imagine the standard parabola $$ f(x) = x^2 $$ and then move every point on the graph 6 units to the left along the x-axis. This results in the vertex moving from $$(0, 0)$$ to $$(-6, 0)$$, and the overall shape of the parabola remains unchanged.

### Conclusion
The graph of $$ g(x) = (x + 6)^2 $$ is obtained by shifting the graph of $$ f(x) = x^2 $$ **6 units to the left**. The vertex of the parabola moves from the origin $$(0, 0)$$ to $$(-6, 0)$$, and the parabola continues to open upwards with the same shape.
The graph of $$ g(x) = (x + 6)^2 $$ is obtained by shifting the graph of $$ f(x) = x^2 $$ **6 units to the left**.

Use transformations of the graph of \( f(x)=x^{2} \) to determine the graph of the given function. \( g(x)=(x+6)^{2} \)

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