| Exam Board | AQA |
|---|---|
| Module | C1 (Core Mathematics 1) |
| Year | 2005 |
| Session | June |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Completing the square and sketching |
| Type | Complete the square |
| Difficulty | Easy -1.2 This is a straightforward completing-the-square question with standard follow-up parts requiring minimal problem-solving. Part (a) is routine algebraic manipulation, parts (b)(i-iii) are direct applications of the completed square form, and part (c) requires only recognition of a translation. All techniques are basic C1 content with no novel insight required. |
| Spec | 1.02e Complete the square: quadratic polynomials and turning points1.02n Sketch curves: simple equations including polynomials1.02w Graph transformations: simple transformations of f(x) |
| Answer | Marks | Guidance |
|---|---|---|
| Working | Marks | Guidance |
| \((x-3)^2\) | B1 | \(p = 3\) |
| \(+ 7\) | B1 | \(q = 7\) |
| Answer | Marks | Guidance |
|---|---|---|
| Working | Marks | Guidance |
| Vertex \((3, 7)\) | B1\(\checkmark\) | ft their \(p\) |
| B1\(\checkmark\) | ft their \(q\) |
| Answer | Marks | Guidance |
|---|---|---|
| Working | Marks | Guidance |
| Graph (parabola with \(y\)-intercept 16, vertex shown) | M1 | Parabola (ft on vertex approx position) |
| A1 | Correct with \(y = 16\) marked or stated |
| Answer | Marks | Guidance |
|---|---|---|
| Working | Marks | Guidance |
| Line of symmetry \(x = 3\) | B1 | Must have correct equation |
| Answer | Marks | Guidance |
|---|---|---|
| Working | Marks | Guidance |
| Translation (and no additional transformation) | E1 | Not shift, move, transformation, etc |
| through \(\begin{bmatrix} 3 \\ 7 \end{bmatrix}\) | M1 | One part correct e.g. 7 units up |
| A1 | All correct — if not vector must say 3 units in positive \(x\)-direction etc |
## Question 2:
**Part (a)**
| Working | Marks | Guidance |
|---------|-------|----------|
| $(x-3)^2$ | B1 | $p = 3$ |
| $+ 7$ | B1 | $q = 7$ |
**Part (b)(i)**
| Working | Marks | Guidance |
|---------|-------|----------|
| Vertex $(3, 7)$ | B1$\checkmark$ | ft their $p$ |
| | B1$\checkmark$ | ft their $q$ |
**Part (b)(ii)**
| Working | Marks | Guidance |
|---------|-------|----------|
| Graph (parabola with $y$-intercept 16, vertex shown) | M1 | Parabola (ft on vertex approx position) |
| | A1 | Correct with $y = 16$ marked or stated |
**Part (b)(iii)**
| Working | Marks | Guidance |
|---------|-------|----------|
| Line of symmetry $x = 3$ | B1 | Must have correct equation |
**Part (c)**
| Working | Marks | Guidance |
|---------|-------|----------|
| Translation (and no additional transformation) | E1 | Not shift, move, transformation, etc |
| through $\begin{bmatrix} 3 \\ 7 \end{bmatrix}$ | M1 | One part correct e.g. 7 units up |
| | A1 | All correct — if not vector must say 3 units in positive $x$-direction etc |
**Total: 10 marks**
---2
\begin{enumerate}[label=(\alph*)]
\item Express $x ^ { 2 } - 6 x + 16$ in the form $( x - p ) ^ { 2 } + q$.
\item A curve has equation $y = x ^ { 2 } - 6 x + 16$.
Using your answer from part (a), or otherwise:
\begin{enumerate}[label=(\roman*)]
\item find the coordinates of the vertex (minimum point) of the curve;
\item sketch the curve, indicating the value where the curve crosses the $y$-axis;
\item state the equation of the line of symmetry of the curve.
\end{enumerate}\item Describe geometrically the transformation that maps the graph of $y = x ^ { 2 }$ onto the graph of $y = x ^ { 2 } - 6 x + 16$.
\end{enumerate}
\hfill \mbox{\textit{AQA C1 2005 Q2 [10]}}