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

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.