| Exam Board | SPS |
|---|---|
| Module | SPS FM Pure (SPS FM Pure) |
| Year | 2023 |
| Session | November |
| Marks | 10 |
| Topic | Completing the square and sketching |
| Type | Complete the square |
| Difficulty | Moderate -0.8 This is a routine multi-part question on completing the square, sketching quadratics, and transformations. Part (a) is standard AS-level completing the square, part (b) is basic sketching, part (c)(i) requires simplifying g(x) to identify the transformation (translation), and part (c)(ii) uses the completed square form to find the range of a reciprocal function. All techniques are straightforward applications of standard methods with no novel problem-solving required, making it easier than average. |
| Spec | 1.02e Complete the square: quadratic polynomials and turning points1.02n Sketch curves: simple equations including polynomials1.02u Functions: definition and vocabulary (domain, range, mapping)1.02w Graph transformations: simple transformations of f(x) |
7.
$$\mathrm { f } ( x ) = 2 x ^ { 2 } + 4 x + 9 \quad x \in \mathbb { R }$$
\begin{enumerate}[label=(\alph*)]
\item Write $\mathrm { f } ( x )$ in the form $a ( x + b ) ^ { 2 } + c$, where $a , b$ and $c$ are integers to be found.
\item Sketch the curve with equation $y = \mathrm { f } ( x )$ showing any points of intersection with the coordinate axes and the coordinates of any turning point.
\item \begin{enumerate}[label=(\roman*)]
\item Describe fully the transformation that maps the curve with equation $y = \mathrm { f } ( x )$ onto the curve with equation $y = \mathrm { g } ( x )$ where
$$\mathrm { g } ( x ) = 2 ( x - 2 ) ^ { 2 } + 4 x - 3 \quad x \in \mathbb { R }$$
\item Find the range of the function
$$\mathrm { h } ( x ) = \frac { 21 } { 2 x ^ { 2 } + 4 x + 9 } \quad x \in \mathbb { R }$$
\end{enumerate}\end{enumerate}
\hfill \mbox{\textit{SPS SPS FM Pure 2023 Q7 [10]}}