1. **Stating the problem:** We start with the base graph $y = x^2$ and need to find the equations of graphs A, B, C, and D by considering transformations of this base graph. 2. **Recall transformations of $y = x^2$:** - Horizontal shifts: $y = (x - h)^2$ shifts the graph $h$ units to the right. - Vertical shifts: $y = x^2 + k$ shifts the graph $k$ units up. - Reflections and stretches/compressions are not explicitly mentioned here. 3. **Graph A (pink):** It is a waveform above the x-axis around $x=3$ to $5$, suggesting a local modification rather than a simple quadratic transformation. Since the problem states transformations of $y=x^2$, and the waveform is above the x-axis near $x=3$ to $5$, this likely represents a vertical shift or addition of a positive bump. Without explicit function details, we approximate it as a vertical shift upwards near $x=4$. So, approximate equation: $$y = x^2 + 3$$ 4. **Graph B (blue):** Shifted slightly upwards and rightwards compared to $y=x^2$. This means a horizontal shift right by $h$ and vertical shift up by $k$ with small values. Approximate shifts: $h=1$, $k=1$. Equation: $$y = (x - 1)^2 + 1$$ 5. **Graph C (orange):** Downward-shifted curve, so vertical shift down by $k$. Approximate $k = -3$. Equation: $$y = x^2 - 3$$ 6. **Graph D (green):** Shifted downward and rightward significantly. Approximate horizontal shift $h=2$, vertical shift $k=-4$. Equation: $$y = (x - 2)^2 - 4$$ **Final answers:** - Graph A: $y = x^2 + 3$ - Graph B: $y = (x - 1)^2 + 1$ - Graph C: $y = x^2 - 3$ - Graph D: $y = (x - 2)^2 - 4$