| Exam Board | SPS |
|---|---|
| Module | SPS SM Mechanics (SPS SM Mechanics) |
| Year | 2022 |
| Session | February |
| Marks | 4 |
| Topic | Completing the square and sketching |
| Type | Complete square then find vertex/turning point |
| Difficulty | Easy -1.8 This is a routine completing-the-square exercise followed by reading off basic features of a quadratic curve. Part (a) requires standard algebraic manipulation, while part (b) involves simple substitution and identifying the vertex from completed square form—both are fundamental AS-level skills with no problem-solving or novel insight required. |
| Spec | 1.02d Quadratic functions: graphs and discriminant conditions1.02e Complete the square: quadratic polynomials and turning points |
Given that
$$f(x) = x^2 - 4x + 5 \quad x \in \mathbb{R}$$
\begin{enumerate}[label=(\alph*)]
\item express $f(x)$ in the form $(x + a)^2 + b$ where $a$ and $b$ are integers to be found.
[2]
\end{enumerate}
The curve with equation $y = f(x)$
• meets the $y$-axis at the point $P$
• has a minimum turning point at the point $Q$
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{1}
\item Write down
\begin{enumerate}[label=(\roman*)]
\item the coordinates of $P$
\item the coordinates of $Q$
\end{enumerate}
[2]
\end{enumerate}
\hfill \mbox{\textit{SPS SPS SM Mechanics 2022 Q2 [4]}}