| Exam Board | AQA |
|---|---|
| Module | C1 (Core Mathematics 1) |
| Year | 2006 |
| Session | June |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Completing the square and sketching |
| Type | Transformations of quadratic graphs |
| Difficulty | Moderate -0.8 This is a straightforward completing-the-square question with standard follow-up parts. Part (a) is routine algebraic manipulation, (b) follows directly from (a), (c) requires basic sketching skills using the completed square form, and (d) tests understanding of transformations. All parts are textbook exercises requiring recall and application of standard techniques with no problem-solving or novel insight needed. |
| Spec | 1.02d Quadratic functions: graphs and discriminant conditions1.02e Complete the square: quadratic polynomials and turning points1.02n Sketch curves: simple equations including polynomials1.02w Graph transformations: simple transformations of f(x) |
2
\begin{enumerate}[label=(\alph*)]
\item Express $x ^ { 2 } + 8 x + 19$ in the form $( x + p ) ^ { 2 } + q$, where $p$ and $q$ are integers.
\item Hence, or otherwise, show that the equation $x ^ { 2 } + 8 x + 19 = 0$ has no real solutions.
\item Sketch the graph of $y = x ^ { 2 } + 8 x + 19$, stating the coordinates of the minimum point and the point where the graph crosses the $y$-axis.
\item Describe geometrically the transformation that maps the graph of $y = x ^ { 2 }$ onto the graph of $y = x ^ { 2 } + 8 x + 19$.
\end{enumerate}
\hfill \mbox{\textit{AQA C1 2006 Q2 [10]}}