1. **Problem Statement:** We start with the graph of $y = x^2$ and want to find the transformations for: (a) $y = \left(\frac{1}{2}x\right)^2$ (b) $y = -\frac{1}{2}x^2$ 2. **Recall the base function:** The base function is $y = x^2$, a parabola opening upwards centered at the origin. 3. **Transformation rules:** - Horizontal scaling by a factor $k$ changes $x$ to $kx$ inside the function. - Vertical scaling by a factor $a$ multiplies the whole function by $a$. - Reflection about the x-axis multiplies the function by $-1$. 4. **Part (a):** Given $y = \left(\frac{1}{2}x\right)^2 = \frac{1}{4}x^2$. - This is a horizontal stretch by a factor of 2 (since $x$ is multiplied by $\frac{1}{2}$ inside the function). - Equivalently, it can be seen as a vertical compression by a factor of $\frac{1}{4}$. So the graph is the original parabola stretched horizontally by 2 or compressed vertically by 4. 5. **Part (b):** Given $y = -\frac{1}{2}x^2$. - The negative sign reflects the parabola about the x-axis (it opens downward). - The factor $\frac{1}{2}$ compresses the parabola vertically by $\frac{1}{2}$. 6. **Summary:** - (a) Horizontal stretch by 2 or vertical compression by 4. - (b) Vertical compression by $\frac{1}{2}$ and reflection about the x-axis. **Final answers:** (a) $y = \left(\frac{1}{2}x\right)^2 = \frac{1}{4}x^2$ (b) $y = -\frac{1}{2}x^2$