1. **Problem statement:** (a)(i) Describe the single transformation that maps the graph of $y=\sqrt{x}$ onto $C_1$ with equation $y=\sqrt{2x}$. (a)(ii) Describe the single transformation that maps the graph of $y=-\sqrt{x}$ onto $C_2$ with equation $y=12 - \sqrt{x}$. (b)(i) Show that the $x$-coordinate of the intersection point $P$ of $C_1$ and $C_2$ satisfies $\sqrt{x} = 12(\sqrt{2} - 1)$. (b)(ii) Find the exact coordinates of $P$ in simplest form. 2. **Step (a)(i): Transformation from $y=\sqrt{x}$ to $y=\sqrt{2x}$** - The original function is $y=\sqrt{x}$. - The new function is $y=\sqrt{2x} = \sqrt{2} \sqrt{x}$. - This means the $y$-values are multiplied by $\sqrt{2}$ for the same $x$. - **Transformation:** Vertical stretch by a factor of $\sqrt{2}$. 3. **Step (a)(ii): Transformation from $y=-\sqrt{x}$ to $y=12 - \sqrt{x}$** - Original function: $y = -\sqrt{x}$. - New function: $y = 12 - \sqrt{x} = 12 + (-\sqrt{x})$. - This is the original graph shifted vertically upwards by 12 units. - **Transformation:** Vertical translation (shift) upwards by 12 units. 4. **Step (b)(i): Find $x$ where $C_1$ and $C_2$ meet** - At intersection point $P$, $y$ values are equal: $$\sqrt{2x} = 12 - \sqrt{x}$$ - Rearrange: $$\sqrt{2x} + \sqrt{x} = 12$$ - Factor out $\sqrt{x}$: $$\sqrt{x}(\sqrt{2} + 1) = 12$$ - Divide both sides by $(\sqrt{2} + 1)$: $$\sqrt{x} = \frac{12}{\sqrt{2} + 1}$$ - Rationalize denominator: $$\sqrt{x} = \frac{12}{\sqrt{2} + 1} \times \frac{\sqrt{2} - 1}{\sqrt{2} - 1} = \frac{12(\sqrt{2} - 1)}{(\sqrt{2} + 1)(\sqrt{2} - 1)}$$ - Since $(\sqrt{2} + 1)(\sqrt{2} - 1) = 2 - 1 = 1$: $$\sqrt{x} = 12(\sqrt{2} - 1)$$ - This proves the required equation. 5. **Step (b)(ii): Find exact coordinates of $P$** - From above: $$\sqrt{x} = 12(\sqrt{2} - 1)$$ - Square both sides: $$x = \left(12(\sqrt{2} - 1)\right)^2 = 144(\sqrt{2} - 1)^2$$ - Expand $(\sqrt{2} - 1)^2$: $$ (\sqrt{2})^2 - 2 \times \sqrt{2} \times 1 + 1^2 = 2 - 2\sqrt{2} + 1 = 3 - 2\sqrt{2}$$ - So: $$x = 144(3 - 2\sqrt{2})$$ - Find $y$ coordinate using $C_1$: $y = \sqrt{2x}$ - Substitute $x$: $$y = \sqrt{2 \times 144(3 - 2\sqrt{2})} = \sqrt{288(3 - 2\sqrt{2})}$$ - Simplify inside the root: $$288 = 144 \times 2$$ - So: $$y = \sqrt{144 \times 2 (3 - 2\sqrt{2})} = 12 \sqrt{2(3 - 2\sqrt{2})}$$ - Simplify inside the square root: $$2(3 - 2\sqrt{2}) = 6 - 4\sqrt{2}$$ - Thus: $$y = 12 \sqrt{6 - 4\sqrt{2}}$$ **Final answer:** $$P = \left(144(3 - 2\sqrt{2}), 12 \sqrt{6 - 4\sqrt{2}}\right)$$