| Exam Board | SPS |
|---|---|
| Module | SPS FM (SPS FM) |
| Year | 2026 |
| Session | November |
| Marks | 10 |
| Topic | Completing the square and sketching |
| Type | Sketch quadratic curve |
| Difficulty | Moderate -0.8 This is a routine completing-the-square question with standard follow-ups (sketching, transformations, and finding range using the completed square form). All parts use well-practiced techniques with no novel problem-solving required. Part (c)(i) requires simplifying g(x) first, adding a minor computational step, but overall this is easier than average A-level content. |
| Spec | 1.02e Complete the square: quadratic polynomials and turning points1.02n Sketch curves: simple equations including polynomials1.02w Graph transformations: simple transformations of f(x) |
$f(x) = 2x^2 + 4x + 9 \quad x \in \mathbb{R}$
\begin{enumerate}[label=(\alph*)]
\item Write $f(x)$ in the form $a(x + b)^2 + c$, where $a$, $b$ and $c$ are integers to be found.
[3]
\item Sketch the curve with equation $y = f(x)$ showing any points of intersection with the coordinate axes and the coordinates of any turning point.
[3]
\item \begin{enumerate}[label=(\roman*)]
\item Describe fully the transformation that maps the curve with equation $y = f(x)$ onto the curve with equation $y = g(x)$ where
$$g(x) = 2(x - 2)^2 + 4x - 3 \quad x \in \mathbb{R}$$
\item Find the range of the function
$$h(x) = \frac{21}{2x^2 + 4x + 9} \quad x \in \mathbb{R}$$
[4]
\end{enumerate}
\end{enumerate}
\hfill \mbox{\textit{SPS SPS FM 2026 Q7 [10]}}