Question 5 - A-Level Maths

Write \(x ^ { 2 } - 7 x + 6\) in the form \(( x - a ) ^ { 2 } + b\).
State the coordinates of the minimum point on the graph of \(y = x ^ { 2 } - 7 x + 6\).
Find the coordinates of the points where the graph of \(y = x ^ { 2 } - 7 x + 6\) crosses the axes and sketch the graph.
Show that the graphs of \(y = x ^ { 2 } - 7 x + 6\) and \(y = x ^ { 2 } - 3 x + 4\) intersect only once. Find the \(x\)-coordinate of the point of intersection. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{6978a576-e86a-48d4-93a4-268845ea6699-3_643_864_390_622} \captionsetup{labelformat=empty} \caption{Fig. 11}
\end{figure} Fig. 11 shows a sketch of the curve with equation \(y = ( x - 4 ) ^ { 2 } - 3\).
Write down the equation of the line of symmetry of the curve and the coordinates of the minimum point.
Find the coordinates of the points of intersection of the curve with the \(x\)-axis and the \(y\)-axis, using surds where necessary.
The curve is translated by \(\binom { 2 } { 0 }\). Show that the equation of the translated curve may be written as \(y = x ^ { 2 } - 12 x + 33\).
Show that the line \(y = 8 - 2 x\) meets the curve \(y = x ^ { 2 } - 12 x + 33\) at just one point, and find the coordinates of this point.