Multiple transformation descriptions

1. \begin{figure}[h] \end{figure} Figure 1 shows the graph of \(y = \mathrm { f } ( x ) , - 5 \leqslant x \leqslant 5\).
The point \(M ( 2,4 )\) is the maximum turning point of the graph.
Sketch, on separate diagrams, the graphs of

1. \(y = \mathrm { f } ( x ) + 3\),
2. \(y = | \mathrm { f } ( x ) |\),
3. \(y = \mathrm { f } ( | x | )\). Show on each graph the coordinates of any maximum turning points.

3. \begin{figure}[h] \end{figure} Figure 1 shows the graph of \(y = \mathrm { f } ( x ) , \quad 1 < x < 9\).
The points \(T ( 3,5 )\) and \(S ( 7,2 )\) are turning points on the graph.
Sketch, on separate diagrams, the graphs of

1. \(y = 2 \mathrm { f } ( x ) - 4\),
2. \(y = | \mathrm { f } ( x ) |\). Indicate on each diagram the coordinates of any turning points on your sketch.

3. \begin{figure}[h] \end{figure} Figure 1 shows part of the graph of \(y = \mathrm { f } ( x ) , x \in \mathbb { R }\). The graph consists of two line segments that meet at the point \(R ( 4 , - 3 )\), as shown in Figure 1. Sketch, on separate diagrams, the graphs of

1. \(y = 2 \mathrm { f } ( x + 4 )\),
2. \(y = | \mathrm { f } ( - x ) |\). On each diagram, show the coordinates of the point corresponding to \(R\).

2. \begin{figure}[h] \end{figure} Figure 1 shows a sketch of the curve with equation \(y = \mathrm { f } ( x ) , x > 0\), where f is an increasing function of \(x\). The curve crosses the \(x\)-axis at the point \(( 1,0 )\) and the line \(x = 0\) is an asymptote to the curve. On separate diagrams, sketch the curve with equation

1. \(y = \mathrm { f } ( 2 x ) , x > 0\)
2. \(y = | \mathrm { f } ( x ) | , x > 0\) Indicate clearly on each sketch the coordinates of the point at which the curve crosses or meets the \(x\)-axis.

6. \begin{figure}[h] \end{figure} Figure 1 shows a sketch of part of the curve with equation \(y = \mathrm { f } ( x ) , x \in \mathbb { R }\). The curve has a minimum point at \(( - 0.5 , - 2 )\) and a maximum point at \(( 0.4 , - 4 )\). The lines \(x = 1\), the \(x\)-axis and the \(y\)-axis are asymptotes of the curve, as shown in Fig. 1. On a separate diagram sketch the graphs of

1. \(y = | \mathrm { f } ( x ) |\),
2. \(y = \mathrm { f } ( x - 3 )\),
3. \(y = \mathrm { f } ( | x | )\). In each case show clearly
   1. the coordinates of any points at which the curve has a maximum or minimum point,
   2. how the curve approaches the asymptotes of the curve.
      6. continued

\begin{figure}[h] \end{figure} Figure 1 shows a sketch of part of the curve with equation \(y = \mathrm { f } ( x )\).
The curve crosses the coordinate axes at the points \(( - 6,0 )\) and \(( 0,3 )\), has a stationary point at \(( - 3,9 )\) and has an asymptote with equation \(y = 1\) On separate diagrams, sketch the curve with equation

1. \(y = - \mathrm { f } ( x )\)
2. \(y = \mathrm { f } \left( \frac { 3 } { 2 } x \right)\) On each diagram, show clearly the coordinates of the points of intersection of the curve with the two coordinate axes, the coordinates of the stationary point, and the equation of the asymptote. \includegraphics[max width=\textwidth, alt={}, center]{de511cb3-35c7-4225-b459-a136b6304b78-07_2255_45_316_36}

5

1. Sketch the curve \(y = x ^ { 3 } + 2\).
2. Sketch the curve \(y = 2 \sqrt { x }\).
3. Describe a transformation that transforms the curve \(y = 2 \sqrt { x }\) to the curve \(y = 3 \sqrt { x }\).

5

1. On the same axes, sketch the graphs of \(y = \cos x\) and \(y = \cos 2 x\) for values of \(x\) from 0 to \(2 \pi\).
2. Describe the transformation which maps the graph of \(y = \cos x\) onto the graph of \(y = 3 \cos x\).

14

1. On the same axes, sketch the graphs of \(y = \cos x\) and \(y = \cos 2 x\) for values of \(x\) from 0 to \(2 \pi\).
2. Describe the transformation which maps the graph of \(y = \cos x\) onto the graph of \(y = 3 \cos x\).

7

1. Sketch the curve \(y = x ^ { 2 } ( 3 - x )\) stating the coordinates of points of intersection with the axes.
2. The curve \(y = x ^ { 2 } ( 3 - x )\) is translated by 2 units in the positive direction parallel to the \(x\)-axis. State the equation of the curve after it has been translated.
3. Describe fully a transformation that transforms the curve \(y = x ^ { 2 } ( 3 - x )\) to \(y = \frac { 1 } { 2 } x ^ { 2 } ( 3 - x )\).