1. **Problem Statement:** We start with the graph of the function $y = x^2$ and apply transformations to get new graphs: (a) Find the graph of $y = \left(\frac{1}{2}x\right)^2$. (b) Find the graph of $y = -\frac{1}{2}x^2$. 2. **Recall the base function:** The base function is $y = x^2$, a parabola opening upwards with vertex 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): $y = \left(\frac{1}{2}x\right)^2$** - Rewrite as $y = \left(\frac{1}{2}x\right)^2 = \frac{1}{4}x^2$. - This means the graph is vertically compressed by a factor of $\frac{1}{4}$ compared to $y = x^2$. - Alternatively, since the input $x$ is multiplied by $\frac{1}{2}$, the graph is horizontally stretched by a factor of 2. 5. **Part (b): $y = -\frac{1}{2}x^2$** - The negative sign reflects the graph about the x-axis, flipping it upside down. - The factor $\frac{1}{2}$ vertically compresses the graph by half. **Final answers:** - (a) The graph of $y = \left(\frac{1}{2}x\right)^2$ is a parabola opening upwards, horizontally stretched by 2, or equivalently vertically compressed by $\frac{1}{4}$. - (b) The graph of $y = -\frac{1}{2}x^2$ is a parabola opening downwards, vertically compressed by $\frac{1}{2}$.