| Exam Board | SPS |
|---|---|
| Module | SPS SM Pure (SPS SM Pure) |
| Year | 2022 |
| Session | June |
| Marks | 8 |
| Topic | Function Transformations |
| Type | Algebraic to algebraic transformation description |
| Difficulty | Moderate -0.3 This is a slightly below-average A-level question. Part (a) requires simple substitution of x=2 into the polynomial (1 mark routine work). Part (b) involves factorizing a cubic after finding k, which is standard. Parts (c)(i-ii) ask students to identify transformations by comparing coefficients, requiring recognition that g(x) = -f(x) and h(x) = -f(2x), which are straightforward pattern-matching exercises rather than problem-solving. The 8-mark total and multi-part structure are typical, but each component is procedural with no novel insight required. |
| Spec | 1.02j Manipulate polynomials: expanding, factorising, division, factor theorem1.02w Graph transformations: simple transformations of f(x) |
The function $f(x)$ is such that $f(x) = -x^3 + 2x^2 + kx - 10$
The graph of $y = f(x)$ crosses the $x$-axis at the points with coordinates $(a, 0)$, $(2, 0)$ and $(b, 0)$ where $a < b$
\begin{enumerate}[label=(\alph*)]
\item Show that $k = 5$ [1 mark]
\item Find the exact value of $a$ and the exact value of $b$ [3 marks]
\item The functions $g(x)$ and $h(x)$ are such that
$$g(x) = x^3 + 2x^2 - 5x - 10$$
$$h(x) = -8x^3 + 8x^2 + 10x - 10$$
\begin{enumerate}[label=(\roman*)]
\item Explain how the graph of $y = f(x)$ can be transformed into the graph of $y = g(x)$
Fully justify your answer. [2 marks]
\item Explain how the graph of $y = f(x)$ can be transformed into the graph of $y = h(x)$
Fully justify your answer. [2 marks]
\end{enumerate}
\end{enumerate}
\hfill \mbox{\textit{SPS SPS SM Pure 2022 Q8 [8]}}