| Exam Board | AQA |
|---|---|
| Module | C1 (Core Mathematics 1) |
| Year | 2008 |
| Session | January |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Completing the square and sketching |
| Type | Complete square then find vertex/turning point |
| Difficulty | Moderate -0.8 This is a straightforward C1 question testing basic quadratic manipulation and sketching. Part (a) is routine factorisation, part (b) is simple algebraic verification by expansion, and part (c) involves standard techniques (completing the square to find vertex/symmetry, finding intercepts). All steps are textbook exercises with no problem-solving or novel insight required, making it easier than average. |
| Spec | 1.02d Quadratic functions: graphs and discriminant conditions1.02e Complete the square: quadratic polynomials and turning points1.02j Manipulate polynomials: expanding, factorising, division, factor theorem1.02n Sketch curves: simple equations including polynomials |
| Answer | Marks | Guidance |
|---|---|---|
| \((9+x)(1-x)\) | M1, A1 | \(\pm(9 \pm x)(1 \pm x)\); Correct factors |
| 2 |
| Answer | Marks | Guidance |
|---|---|---|
| \(25 - (x^2 + 8x + 16) = 9 - 8x - x^2\) | B1 | AG |
| 1 |
| Answer | Marks |
|---|---|
| \(x = -4\) is line of symmetry | B1 |
| 1 |
| Answer | Marks |
|---|---|
| Vertex is \((-4, 25)\) | B1, B1 |
| 2 |
| Answer | Marks |
|---|---|
| [Graph showing: General \(\cap\) shape, \(-9\) and \(1\) marked on x-axis or stated, 9 marked on y-axis and maximum to the left of y-axis, Must continue below x-axis at both ends] | M1, B1, A1 |
| 3 | |
| 9 |
## Part (a)
$(9+x)(1-x)$ | M1, A1 | $\pm(9 \pm x)(1 \pm x)$; Correct factors
| | 2 |
## Part (b)
$25 - (x^2 + 8x + 16) = 9 - 8x - x^2$ | B1 | AG
| | 1 |
## Part (c)(i)
$x = -4$ is line of symmetry | B1 |
| | 1 |
## Part (c)(ii)
Vertex is $(-4, 25)$ | B1, B1 |
| | 2 |
## Part (c)(iii)
[Graph showing: General $\cap$ shape, $-9$ and $1$ marked on x-axis or stated, 9 marked on y-axis and maximum to the left of y-axis, Must continue below x-axis at both ends] | M1, B1, A1 |
| | 3 |
| | 9 |
---5
\begin{enumerate}[label=(\alph*)]
\item Factorise $9 - 8 x - x ^ { 2 }$.
\item Show that $25 - ( x + 4 ) ^ { 2 }$ can be written as $9 - 8 x - x ^ { 2 }$.
\item A curve has equation $y = 9 - 8 x - x ^ { 2 }$.
\begin{enumerate}[label=(\roman*)]
\item Write down the equation of its line of symmetry.
\item Find the coordinates of its vertex.
\item Sketch the curve, indicating the values of the intercepts on the $x$-axis and the $y$-axis.
\end{enumerate}\end{enumerate}
\hfill \mbox{\textit{AQA C1 2008 Q5 [9]}}