Reflections

6. \begin{figure}[h] \end{figure} Figure 1 shows a sketch of the curve with equation \(y = \mathrm { f } ( x )\). The curve crosses the \(x\)-axis at the points \(( 2,0 )\) and \(( 4,0 )\). The minimum point on the curve is \(P ( 3 , - 2 )\). In separate diagrams sketch the curve with equation

1. \(y = - \mathrm { f } ( x )\),
2. \(y = \mathrm { f } ( 2 x )\). On each diagram, give the coordinates of the points at which the curve crosses the \(x\)-axis, and the coordinates of the image of \(P\) under the given transformation.

3

1. Sketch the curve \(y = x ^ { 3 }\).
2. Describe a transformation that transforms the curve \(y = x ^ { 3 }\) to the curve \(y = - x ^ { 3 }\).
3. The curve \(y = x ^ { 3 }\) is translated by \(p\) units, parallel to the \(x\)-axis. State the equation of the curve after it has been transformed.

5 The diagram shows the curve \(y = \mathrm { f } ( x )\) where \(a\) is a positive constant.
\includegraphics[max width=\textwidth, alt={}, center]{c55a5f04-3573-4f36-a12c-3755bdd4a45b-3_551_962_255_476} Sketch the following curves on separate diagrams, in each case stating the coordinates of points where they meet the \(x\) - and \(y\)-axes.

1. \(\quad y = - \mathrm { f } ( x )\)
2. \(\quad y = \mathrm { f } ( - x )\)

2
\includegraphics[max width=\textwidth, alt={}, center]{559e9f1a-340e-4478-adaa-a7361dd70fe8-2_325_479_468_794} The graph of \(y = \mathrm { f } ( x )\) for \(- 2 \leqslant x \leqslant 2\) is shown above.

1. Sketch the graph of \(y = \mathrm { f } ( - x )\) for \(- 2 \leqslant x \leqslant 2\).
2. Sketch the graph of \(y = \mathrm { f } ( x ) + 2\) for \(- 2 \leqslant x \leqslant 2\).

3 The diagram shows the curve \(y = \mathrm { f } ( x )\), where \(\mathrm { f } ( x )\) is a cubic polynomial in \(x\). This diagram is repeated in the Printed Answer Booklet.
\includegraphics[max width=\textwidth, alt={}, center]{d44919ed-806d-48c0-9726-c5fd67764504-03_896_1467_1382_244}

1. State the values of \(x\) for which \(\mathrm { f } ( x ) < \frac { 1 } { 2 }\), giving your answer in set notation.
2. On the diagram in the Printed Answer Booklet, draw the graph of \(y = \mathrm { f } ( - x )\).
3. Explain how you can tell that \(\mathrm { f } ( x )\) cannot be expressed as the product of three real linear factors.

5 The diagram shows the graph of \(y = \sin x ^ { \circ }\) for \(0 \leqslant x \leqslant 360\).
\includegraphics[max width=\textwidth, alt={}, center]{85de9a39-f8be-40ee-b0c8-e2e632be93d8-4_597_965_1909_539} Find an equation for the transformed curve when the curve \(y = \sin x ^ { \circ }\) is reflected in

1. the \(x\)-axis,
2. the line \(y = 0.5\).