2
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{3e867e1a-db4e-4fc1-ad93-b5c49a9506e6-2_1263_1219_322_463}
\captionsetup{labelformat=empty}
\caption{Fig. 12}
\end{figure}
Fig. 12 shows the graph of \(y = \frac { 1 } { x - 2 }\).
- Draw accurately the graph of \(y = 2 x + 3\) on the copy of Fig. 12 and use it to estimate the coordinates of the points of intersection of \(y = \frac { 1 } { x - 2 }\) and \(y = 2 x + 3\).
- Show algebraically that the \(x\)-coordinates of the points of intersection of \(y = \frac { 1 } { x - 2 }\) and \(y = 2 x + 3\) satisfy the equation \(2 x ^ { 2 } - x - 7 = 0\). Hence find the exact values of the \(x\)-coordinates of the points of intersection.
- Find the quadratic equation satisfied by the \(x\)-coordinates of the points of intersection of \(y = \frac { 1 } { x - 2 }\) and \(y = - x + k\). Hence find the exact values of \(k\) for which \(y = - x + k\) is a tangent to \(y = \frac { 1 } { x - 2 }\). [4]
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{3e867e1a-db4e-4fc1-ad93-b5c49a9506e6-3_1292_1404_364_353}
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\caption{Fig. 12}
\end{figure}
Fig. 12 shows the graph of \(y = \frac { 1 } { x - 3 }\). - Draw accurately, on the copy of Fig. 12, the graph of \(y = x ^ { 2 } - 4 x + 1\) for \(- 1 \leqslant x \leqslant 5\). Use your graph to estimate the coordinates of the intersections of \(y = \frac { 1 } { x - 3 }\) and \(y = x ^ { 2 } - 4 x + 1\).
- Show algebraically that, where the curves intersect, \(x ^ { 3 } - 7 x ^ { 2 } + 13 x - 4 = 0\).
- Use the fact that \(x = 4\) is a root of \(x ^ { 3 } - 7 x ^ { 2 } + 13 x - 4 = 0\) to find a quadratic factor of \(x ^ { 3 } - 7 x ^ { 2 } + 13 x - 4\). Hence find the exact values of the other two roots of this equation.
- Find algebraically the coordinates of the points of intersection of the curve \(y = 4 x ^ { 2 } + 24 x + 31\) and the line \(x + y = 10\).
- Express \(4 x ^ { 2 } + 24 x + 31\) in the form \(a ( x + b ) ^ { 2 } + c\).
- For the curve \(y = 4 x ^ { 2 } + 24 x + 31\),
(A) write down the equation of the line of symmetry,
(B) write down the minimum \(y\)-value on the curve.