| Exam Board | Edexcel |
|---|---|
| Module | C3 (Core Mathematics 3) |
| Marks | 13 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Function Transformations |
| Type | Completing square for transformations |
| Difficulty | Moderate -0.3 This is a standard C3 completing-the-square question with routine transformations and inverse function work. Part (a) is straightforward algebra, (b) requires standard transformation knowledge, (c) is a typical inverse function calculation with domain restriction, and (d) is basic sketching. All parts follow textbook procedures with no novel problem-solving required, making it slightly easier than average. |
| Spec | 1.02e Complete the square: quadratic polynomials and turning points1.02v Inverse and composite functions: graphs and conditions for existence1.02w Graph transformations: simple transformations of f(x) |
| Answer | Marks | Guidance |
|---|---|---|
| (a) | \(= 2[x^2 + 2x] + 2 = 2[(x+1)^2 - 1] + 2\) | M1 |
| \(= 2(x+1)^2\) | A1 | |
| (b) | Translation by 1 unit in negative \(x\) direction | |
| Stretch by scale factor of 2 in \(y\) direction (either first) | B3 | |
| (c) | \(y = 2(x+1)^2, \quad \frac{y}{2} = (x+1)^2\) | M1 |
| \(x + 1 = \pm\sqrt{\frac{y}{2}}\) | M1 | |
| \(x = -1 \pm \sqrt{\frac{y}{2}}\) (domain \(\Rightarrow +\)), \(\quad f^{-1}(x) = -1 + \sqrt{\frac{x}{2}}, \quad x \in \mathbb{R}, x \geq 0\) | A2 | |
| (d) | [Graph showing \(y = f(x)\) and \(y = f^{-1}(x)\) as reflections in line \(y = x\)] | B3 |
| \(y = f^{-1}(x)\) is reflection of \(y = f(x)\) in line \(y = x\) | B1 | |
| (13) |
(a) | $= 2[x^2 + 2x] + 2 = 2[(x+1)^2 - 1] + 2$ | M1 |
| $= 2(x+1)^2$ | A1 |
(b) | Translation by 1 unit in negative $x$ direction | |
| Stretch by scale factor of 2 in $y$ direction (either first) | B3 |
(c) | $y = 2(x+1)^2, \quad \frac{y}{2} = (x+1)^2$ | M1 |
| $x + 1 = \pm\sqrt{\frac{y}{2}}$ | M1 |
| $x = -1 \pm \sqrt{\frac{y}{2}}$ (domain $\Rightarrow +$), $\quad f^{-1}(x) = -1 + \sqrt{\frac{x}{2}}, \quad x \in \mathbb{R}, x \geq 0$ | A2 |
(d) | [Graph showing $y = f(x)$ and $y = f^{-1}(x)$ as reflections in line $y = x$] | B3 |
| $y = f^{-1}(x)$ is reflection of $y = f(x)$ in line $y = x$ | B1 |
| | (13) |5.
$$\mathrm { f } ( x ) \equiv 2 x ^ { 2 } + 4 x + 2 , \quad x \in \mathbb { R } , \quad x \geq - 1 .$$
\begin{enumerate}[label=(\alph*)]
\item Express $\mathrm { f } ( x )$ in the form $a ( x + b ) ^ { 2 } + c$.
\item Describe fully two transformations that would map the graph of $y = x ^ { 2 } , x \geq 0$ onto the graph of $y = \mathrm { f } ( x )$.
\item Find an expression for $\mathrm { f } ^ { - 1 } ( x )$ and state its domain.
\item Sketch the graphs of $y = \mathrm { f } ( x )$ and $y = \mathrm { f } ^ { - 1 } ( x )$ on the same diagram and state the relationship between them.
\end{enumerate}
\hfill \mbox{\textit{Edexcel C3 Q5 [13]}}