(b) Describe the transformations that have occurred to $$ y=x^{2} $$ to get the equation $$ y=-2 x^{2}+7 $$

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To transform the parent function $$ y = x^{2} $$ into the equation $$ y = -2x^{2} + 7 $$, a series of transformations are applied. Here's a step-by-step description:

1. **Vertical Stretch by a Factor of 2:**
   - **Transformation:** Multiply the original function by 2.
   - **Effect:** This stretches the graph of $$ y = x^{2} $$ vertically, making it "narrower."
   - **Equation After Transformation:** $$ y = 2x^{2} $$

2. **Reflection Across the X-Axis:**
   - **Transformation:** Multiply the function by -1.
   - **Effect:** This reflects the graph of $$ y = 2x^{2} $$ across the x-axis, turning it upside down.
   - **Equation After Transformation:** $$ y = -2x^{2} $$

3. **Vertical Shift (Translation) Upwards by 7 Units:**
   - **Transformation:** Add 7 to the function.
   - **Effect:** This moves the entire graph upward by 7 units.
   - **Final Equation:** $$ y = -2x^{2} + 7 $$

**Summary of Transformations:**
- **Vertical Stretch:** The graph is stretched vertically by a factor of 2.
- **Reflection:** The graph is reflected over the x-axis, changing its direction.
- **Vertical Shift:** The graph is shifted upward by 7 units.

**Visual Representation:**

1. **Start with $$ y = x^{2} $$:**
   - A standard parabola opening upwards with its vertex at the origin (0,0).

2. **Apply Vertical Stretch and Reflection to get $$ y = -2x^{2} $$:**
   - The parabola becomes narrower due to the stretch and opens downward because of the reflection.

3. **Shift Up to Get $$ y = -2x^{2} + 7 $$:**
   - The vertex of the parabola moves from (0,0) to (0,7).

This sequence of transformations results in the final equation $$ y = -2x^{2} + 7 $$.
To transform $$ y = x^{2} $$ into $$ y = -2x^{2} + 7 $$, the following transformations are applied: 1. **Vertical Stretch by a Factor of 2:** The graph becomes narrower. 2. **Reflection Across the X-Axis:** The parabola opens downward. 3. **Vertical Shift Upwards by 7 Units:** The vertex moves up to (0,7). So, the final equation is $$ y = -2x^{2} + 7 $$.

(b) Describe the transformations that have occurred to to get the equation

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