Describe how the graph of the quadratic function changes if the equation is altered from $$ y = x^2 $$ to $$ y = (x + 2)^2 $$.

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When altering the quadratic function from $$ y = x^2 $$ to $$ y = (x + 2)^2 $$, the graph undergoes a horizontal shift. Here's a detailed breakdown of the changes:

1. **Horizontal Shift:**
   - **Direction:** The graph shifts **2 units to the left**.
   - **Reason:** The equation $$ y = (x + 2)^2 $$ can be rewritten as $$ y = (x - (-2))^2 $$, indicating a horizontal shift of $$ -2 $$ units (left) from the original graph.

2. **Vertex Movement:**
   - **Original Vertex:** For $$ y = x^2 $$, the vertex is at the origin $$ (0, 0) $$.
   - **New Vertex:** After the shift, the vertex moves to $$ (-2, 0) $$.

3. **Axis of Symmetry:**
   - **Original Axis:** The axis of symmetry for $$ y = x^2 $$ is the vertical line $$ x = 0 $$.
   - **New Axis:** For $$ y = (x + 2)^2 $$, the axis of symmetry shifts to $$ x = -2 $$.

4. **Shape and Orientation:**
   - **Same Shape:** The parabola retains its **U-shape** and **width**; there is no vertical stretching, compression, or reflection.
   - **Opening Direction:** It continues to **open upwards**, just like the original graph.

5. **Key Points:**
   - **No Vertical Shift:** There's no change in the $$ y $$-values; only the $$ x $$-values are affected.
   - **Example Point Transformation:** A point $$ (a, b) $$ on $$ y = x^2 $$ moves to $$ (a - 2, b) $$ on $$ y = (x + 2)^2 $$.

**Visual Illustration:**

- **Original Graph ($$ y = x^2 $$):**
  - Vertex at $$ (0, 0) $$
  - Passes through points like $$ (1, 1) $$ and $$ (-1, 1) $$

- **Shifted Graph ($$ y = (x + 2)^2 $$):**
  - Vertex at $$ (-2, 0) $$
  - Passes through points like $$ (-1, 1) $$ and $$ (-3, 1) $$

**Conclusion:**
Changing the equation from $$ y = x^2 $$ to $$ y = (x + 2)^2 $$ results in the entire parabola shifting **2 units to the left** on the Cartesian plane, with the vertex moving from the origin to the point $$ (-2, 0) $$. The overall shape and orientation of the graph remain unchanged.
The graph shifts 2 units to the left, moving the vertex from (0,0) to (-2,0).

Describe how the graph of the quadratic function changes if the equation is altered from \( y = x^2 \) to \( y = (x + 2)^2 \).

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