| Exam Board | AQA |
|---|---|
| Module | C1 (Core Mathematics 1) |
| Year | 2012 |
| Session | January |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Completing the square and sketching |
| Type | Sketch quadratic curve |
| Difficulty | Moderate -0.8 This is a routine multi-part question covering standard C1 quadratic techniques: factorising, sketching, completing the square, and transformations. All parts are textbook exercises requiring straightforward application of learned methods with no problem-solving or novel insight needed. Slightly easier than average due to the simple coefficients and step-by-step scaffolding. |
| 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 polynomials1.02w Graph transformations: simple transformations of f(x) |
| Answer | Marks | Guidance |
|---|---|---|
| Working | Mark | Guidance |
| \((x-6)(x+2)\) | B1 | ISW for \(x=6, x=-2\) etc |
| Answer | Marks | Guidance |
|---|---|---|
| Working | Mark | Guidance |
| \(x = -2\), \(x = 6\) | B1\(\checkmark\) | Correct \(x\) values or FT 'their' factors (\(x\)-intercepts stated or marked on sketch); may be seen in (a) |
| \(y = -12\) | B1 | Stated or \(-12\) marked on sketch |
| \(\cup\)-shaped curve | M1 | Approximately |
| "Correct" shape in all 4 quadrants with minimum to right of \(y\)-axis | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Working | Mark | Guidance |
| \((x-2)^2\) | M1 | \(p = 2\) |
| \((x-2)^2 - 16\) | A1 | \(p = 2\) and \(q = 16\) |
| Answer | Marks | Guidance |
|---|---|---|
| Working | Mark | Guidance |
| (Minimum value is) \(-16\) | B1\(\checkmark\) | FT 'their \(-q\)' |
| Answer | Marks | Guidance |
|---|---|---|
| Working | Mark | Guidance |
| Replacing each \(x\) by \(x+3\) OR adding 2 to their quadratic | M1 | In original equation or 'their' completed square or factorised form, or replacing \(y\) by \(y-2\) |
| \(y = [(x+3)^2 - 4(x+3) - 12] + 2\) or \(y = (x+1)^2 - 14\) or \(y = x^2 + 2x - 13\) or \(y - 2 = (x-3)(x+5)\) | A1 | OE any correct equation in \(x\) and \(y\) unsimplified |
# Question 2:
## Part (a)
| Working | Mark | Guidance |
|---------|------|----------|
| $(x-6)(x+2)$ | B1 | ISW for $x=6, x=-2$ etc |
## Part (b)
| Working | Mark | Guidance |
|---------|------|----------|
| $x = -2$, $x = 6$ | B1$\checkmark$ | Correct $x$ values or FT 'their' factors ($x$-intercepts stated or marked on sketch); may be seen in (a) |
| $y = -12$ | B1 | Stated or $-12$ marked on sketch |
| $\cup$-shaped curve | M1 | Approximately |
| "Correct" shape in all 4 quadrants with minimum to right of $y$-axis | A1 | |
## Part (c)(i)
| Working | Mark | Guidance |
|---------|------|----------|
| $(x-2)^2$ | M1 | $p = 2$ |
| $(x-2)^2 - 16$ | A1 | $p = 2$ and $q = 16$ |
## Part (c)(ii)
| Working | Mark | Guidance |
|---------|------|----------|
| (Minimum value is) $-16$ | B1$\checkmark$ | FT 'their $-q$' |
## Part (d)
| Working | Mark | Guidance |
|---------|------|----------|
| Replacing each $x$ by $x+3$ **OR** adding 2 to their quadratic | M1 | In original equation or 'their' completed square or factorised form, or replacing $y$ by $y-2$ |
| $y = [(x+3)^2 - 4(x+3) - 12] + 2$ or $y = (x+1)^2 - 14$ or $y = x^2 + 2x - 13$ or $y - 2 = (x-3)(x+5)$ | A1 | OE any correct equation in $x$ and $y$ **unsimplified** |
---2
\begin{enumerate}[label=(\alph*)]
\item Factorise $x ^ { 2 } - 4 x - 12$.
\item Sketch the graph with equation $y = x ^ { 2 } - 4 x - 12$, stating the values where the curve crosses the coordinate axes.
\item \begin{enumerate}[label=(\roman*)]
\item Express $x ^ { 2 } - 4 x - 12$ in the form $( x - p ) ^ { 2 } - q$, where $p$ and $q$ are positive integers.
\item Hence find the minimum value of $x ^ { 2 } - 4 x - 12$.
\end{enumerate}\item The curve with equation $y = x ^ { 2 } - 4 x - 12$ is translated by the vector $\left[ \begin{array} { r } - 3 \\ 2 \end{array} \right]$. Find an equation of the new curve. You need not simplify your answer.
\end{enumerate}
\hfill \mbox{\textit{AQA C1 2012 Q2 [10]}}