1. **Problem:** Determine the equation of the parabola after applying the transformation $(x,y) \to (x+4, \frac{1}{4}y)$ to the graph of $y = x^2$. 2. **Original equation:** $y = x^2$ 3. **Transformation explanation:** The transformation changes the $x$-coordinate to $x+4$ and the $y$-coordinate to $\frac{1}{4}y$. This means the new parabola's points $(X,Y)$ relate to the original points $(x,y)$ by: $$X = x + 4, \quad Y = \frac{1}{4} y$$ 4. **Express original $x$ and $y$ in terms of $X$ and $Y$:** $$x = X - 4, \quad y = 4Y$$ 5. **Substitute into original equation:** $$y = x^2 \implies 4Y = (X - 4)^2$$ 6. **Solve for $Y$:** $$Y = \frac{(X - 4)^2}{4}$$ 7. **Rename variables for clarity:** Let $x = X$ and $y = Y$ for the transformed graph, so the equation is: $$y = \frac{(x - 4)^2}{4}$$ **Final answer:** $$\boxed{y = \frac{(x - 4)^2}{4}}$$ This is the equation of the parabola after the given transformation.