| Exam Board | SPS |
|---|---|
| Module | SPS FM Pure (SPS FM Pure) |
| Year | 2022 |
| Session | June |
| Marks | 9 |
| Topic | Modulus function |
| Type | Transformations of modulus graphs from given f(x) sketch |
| Difficulty | Standard +0.3 This is a standard Further Maths question on graph transformations and modulus functions, but it's relatively straightforward. Parts (a) and (b) test routine horizontal translation and reflection transformations. Part (c) requires simple substitution into the given function. Part (d) involves solving a modulus equation which splits into two cases, but this is a well-practiced technique. The question requires multiple steps but no novel insight, making it slightly easier than average even for Further Maths. |
| Spec | 1.02l Modulus function: notation, relations, equations and inequalities1.02q Use intersection points: of graphs to solve equations1.02t Solve modulus equations: graphically with modulus function1.02w Graph transformations: simple transformations of f(x) |
\includegraphics{figure_1}
Figure 1 shows part of the graph of $y = f(x)$, $x \in \mathbb{R}$. The graph consists of two line segments that meet at the point $(1, a)$, $a < 0$. One line meets the $x$-axis at $(3, 0)$. The other line meets the $x$-axis at $(-1, 0)$ and the $y$-axis at $(0, b)$, $b < 0$.
In separate diagrams, sketch the graph with equation
\begin{enumerate}[label=(\alph*)]
\item $y = f(x + 1)$, [2]
\item $y = f(|x|)$. [2]
\end{enumerate}
Indicate clearly on each sketch the coordinates of any points of intersection with the axes.
Given that $f(x) = |x - 1| - 2$, find
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{2}
\item the value of $a$ and the value of $b$, [2]
\item the value of $x$ for which $f(x) = 5x$. [3]
\end{enumerate}
\hfill \mbox{\textit{SPS SPS FM Pure 2022 Q2 [9]}}