1.02x Combinations of transformations: multiple transformations

2

1. The graph of \(y = \mathrm { f } ( x )\) is transformed to the graph of \(y = 2 \mathrm { f } ( x - 1 )\).
   Describe fully the two single transformations which have been combined to give the resulting transformation.
2. The curve \(y = \sin 2 x - 5 x\) is reflected in the \(y\)-axis and then stretched by scale factor \(\frac { 1 } { 3 }\) in the \(x\)-direction. Write down the equation of the transformed curve.

5 \includegraphics[max width=\textwidth, alt={}, center]{54f3f051-e124-470d-87b5-8e25c35248a9-07_775_768_260_685} In the diagram, the graph of \(y = \mathrm { f } ( x )\) is shown with solid lines. The graph shown with broken lines is a transformation of \(y = \mathrm { f } ( x )\).

1. Describe fully the two single transformations of \(y = \mathrm { f } ( x )\) that have been combined to give the resulting transformation.
2. State in terms of \(y\), f and \(x\), the equation of the graph shown with broken lines.

2 A function f is defined by \(\mathrm { f } ( x ) = x ^ { 2 } - 2 x + 5\) for \(x \in \mathbb { R }\). A sequence of transformations is applied in the following order to the graph of \(y = \mathrm { f } ( x )\) to give the graph of \(y = \mathrm { g } ( x )\). Stretch parallel to the \(x\)-axis with scale factor \(\frac { 1 } { 2 }\) Reflection in the \(y\)-axis
Stretch parallel to the \(y\)-axis with scale factor 3
Find \(\mathrm { g } ( x )\), giving your answer in the form \(a x ^ { 2 } + b x + c\), where \(a , b\) and \(c\) are constants.

2 \includegraphics[max width=\textwidth, alt={}, center]{b5eb378d-a9cb-40e0-9203-374b58f1dcf9-03_451_597_255_735} The diagram shows part of the curve with equation \(\mathrm { y } = \mathrm { ksin } \frac { 1 } { 2 } \mathrm { x }\), where \(k\) is a positive constant and \(x\) is measured in radians. The curve has a minimum point \(A\).

1. State the coordinates of \(A\).
2. A sequence of transformations is applied to the curve in the following order. Translation of 2 units in the negative \(y\)-direction
   Reflection in the \(x\)-axis
   Find the equation of the new curve and determine the coordinates of the point on the new curve corresponding to \(A\).

5 The graph with equation \(y = \mathrm { f } ( x )\) is transformed to the graph with equation \(y = \mathrm { g } ( x )\) by a stretch in the \(x\)-direction with factor 0.5 , followed by a translation of \(\binom { 0 } { 1 }\).

1. The diagram below shows the graph of \(y = \mathrm { f } ( x )\). On the diagram sketch the graph of \(y = \mathrm { g } ( x )\). \includegraphics[max width=\textwidth, alt={}, center]{5d26c357-ea9f-47d9-8eca-2152901cf2f1-07_613_1527_623_342}
2. Find an expression for \(\mathrm { g } ( x )\) in terms of \(\mathrm { f } ( x )\).

5 \includegraphics[max width=\textwidth, alt={}, center]{8eb3d21b-dc45-493c-9e5c-3c0535c505e8-06_743_750_269_687} The diagram shows a curve which has a maximum point at \(( 8,12 )\) and a minimum point at \(( 8,0 )\). The curve is the result of applying a combination of two transformations to a circle. The first transformation applied is a translation of \(\binom { 7 } { - 3 }\). The second transformation applied is a stretch in the \(y\)-direction.

1. State the scale factor of the stretch.
2. State the radius of the original circle.
3. State the coordinates of the centre of the circle after the translation has been completed but before the stretch is applied.
4. State the coordinates of the centre of the original circle.

4 The transformation R denotes a reflection in the \(x\)-axis and the transformation T denotes a translation of \(\binom { 3 } { - 1 }\).

1. Find the equation, \(y = \mathrm { g } ( x )\), of the curve with equation \(y = x ^ { 2 }\) after it has been transformed by the sequence of transformations R followed by T .
2. Find the equation, \(y = \mathrm { h } ( x )\), of the curve with equation \(y = x ^ { 2 }\) after it has been transformed by the sequence of transformations T followed by R .
3. State fully the transformation that maps the curve \(y = \mathrm { g } ( x )\) onto the curve \(y = \mathrm { h } ( x )\).

8. \begin{figure}[h] \end{figure} The curve \(C\) with equation \(y = \mathrm { f } ( x )\) is shown in Figure 4. The curve \(C\)

• has a single turning point, a maximum at ( 4,9 )
• crosses the coordinate axes at only two places, \(( - 3,0 )\) and \(( 0,6 )\)
• has a single asymptote with equation \(y = 4\) as shown in Figure 4.
  1. state the possible values for \(k\). The curve \(C\) is transformed to a new curve that passes through the origin.
  2. 1. Given that the new curve has equation \(y = \mathrm { f } ( x ) - a\), state the value of the constant \(a\).
     2. Write down an equation for another single transformation of \(C\) that also passes through the origin.

8. \begin{figure}[h] \end{figure} Figure 1 shows a sketch of the curve \(C\) with equation \(y = \mathrm { f } ( x )\).
The curve \(C\) passes through the origin and through \(( 6,0 )\).
The curve \(C\) has a minimum at the point \(( 3 , - 1 )\). On separate diagrams, sketch the curve with equation

1. \(y = \mathrm { f } ( 2 x )\),
2. \(y = - \mathrm { f } ( x )\),
3. \(y = \mathrm { f } ( x + p )\), where \(p\) is a constant and \(0 < p < 3\). On each diagram show the coordinates of any points where the curve intersects the \(x\)-axis and of any minimum or maximum points.

10. \begin{figure}[h] \end{figure} Figure 1 shows a sketch of the curve \(C\) with equation \(y = \mathrm { f } ( x )\) where $$f ( x ) = x ^ { 2 } ( 9 - 2 x )$$ There is a minimum at the origin, a maximum at the point \(( 3,27 )\) and \(C\) cuts the \(x\)-axis at the point \(A\).

1. Write down the coordinates of the point \(A\).
2. On separate diagrams sketch the curve with equation
   1. \(y = \mathrm { f } ( x + 3 )\)
   2. \(y = \mathrm { f } ( 3 x )\) On each sketch you should indicate clearly the coordinates of the maximum point and any points where the curves cross or meet the coordinate axes. The curve with equation \(y = \mathrm { f } ( x ) + k\), where \(k\) is a constant, has a maximum point at \(( 3,10 )\).
3. Write down the value of \(k\).

2. \begin{figure}[h] \end{figure} Figure 1 shows a sketch of the curve with equation \(y = \mathrm { f } ( x )\), where \(x \in \mathbb { R }\) and \(\mathrm { f } ( x )\) is a polynomial. The curve passes through the origin and touches the \(x\)-axis at the point \(( 3,0 )\) There is a maximum turning point at \(( 1,2 )\) and a minimum turning point at \(( 3,0 )\) On separate diagrams, sketch the curve with equation

1. \(y = 3 f ( 2 x )\)
2. \(y = \mathrm { f } ( - x ) - 1\) On each sketch, show clearly the coordinates of
   • the point where the curve crosses the \(y\)-axis
   • any maximum or minimum turning points

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.

6. \begin{figure}[h] \end{figure} Figure 1 shows a sketch of the graph of \(y = \mathrm { f } ( x )\).
The graph intersects the \(y\)-axis at the point \(( 0,1 )\) and the point \(A ( 2,3 )\) is the maximum turning point. Sketch, on separate axes, the graphs of

1. \(y = \mathrm { f } ( - x ) + 1\),
2. \(y = \mathrm { f } ( x + 2 ) + 3\),
3. \(y = 2 \mathrm { f } ( 2 x )\). On each sketch, show the coordinates of the point at which your graph intersects the \(y\)-axis and the coordinates of the point to which \(A\) is transformed.

6. \begin{figure}[h] \end{figure} Figure 2 shows a sketch of the curve with the equation \(y = \mathrm { f } ( x ) , x \in \mathbb { R }\). The curve has a turning point at \(A ( 3 , - 4 )\) and also passes through the point \(( 0,5 )\).

1. Write down the coordinates of the point to which \(A\) is transformed on the curve with equation
   1. \(y = | \mathrm { f } ( x ) |\),
   2. \(y = 2 f \left( \frac { 1 } { 2 } x \right)\).
2. Sketch the curve with equation $$y = \mathrm { f } ( | x | )$$ On your sketch show the coordinates of all turning points and the coordinates of the point at which the curve cuts the \(y\)-axis. The curve with equation \(y = \mathrm { f } ( x )\) is a translation of the curve with equation \(y = x ^ { 2 }\).
3. Find \(\mathrm { f } ( x )\).
4. Explain why the function f does not have an inverse.

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 \(R\).

4. \begin{figure}[h] \end{figure} Figure 2 shows part of the curve with equation \(y = \mathrm { f } ( x )\) The curve passes through the points \(P ( - 1.5,0 )\) and \(Q ( 0,5 )\) as shown.
On separate diagrams, sketch the curve with equation

1. \(y = | f ( x ) |\)
2. \(y = \mathrm { f } ( | x | )\)
3. \(y = 2 f ( 3 x )\) Indicate clearly on each sketch the coordinates of the points at which the curve crosses or meets the axes.

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.

4. \begin{figure}[h] \end{figure} Figure 1 shows part of the graph with equation \(y = \mathrm { f } ( x ) , x \in \mathbb { R }\). The graph consists of two line segments that meet at the point \(Q ( 6 , - 1 )\). The graph crosses the \(y\)-axis at the point \(P ( 0,11 )\). Sketch, on separate diagrams, the graphs of

1. \(y = | f ( x ) |\)
2. \(y = 2 f ( - x ) + 3\) On each diagram, show the coordinates of the points corresponding to \(P\) and \(Q\).
   Given that \(\mathrm { f } ( x ) = a | x - b | - 1\), where \(a\) and \(b\) are constants,
3. state the value of \(a\) and the value of \(b\).

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

Figure 1 shows the graph of equation \(y = \mathrm { f } ( x )\).
The points \(P ( - 3,0 )\) and \(Q ( 2 , - 4 )\) are stationary points on the graph.
Sketch, on separate diagrams, the graphs of

1. \(y = 3 \mathrm { f } ( x + 2 )\)
2. \(y = | \mathrm { f } ( x ) |\) On each diagram, show the coordinates of any stationary points.

\begin{figure}[h] \end{figure} Figure 1 shows part of the curve with equation \(y = \mathrm { f } ( x ) , x \in \mathbb { R }\), where f is an increasing function of \(x\). The curve passes through the points \(P ( 0 , - 2 )\) and \(Q ( 3,0 )\) as shown. In separate diagrams, sketch the curve with equation

1. \(y = | f ( x ) |\),
2. \(y = \mathrm { f } ^ { - 1 } ( x )\),
3. \(y = \frac { 1 } { 2 } \mathrm { f } ( 3 x )\). Indicate clearly on each sketch the coordinates of the points at which the curve crosses or meets the axes.

3. \includegraphics[max width=\textwidth, alt={}, center]{14ec6709-e1cb-42d7-af99-91365e50e4fc-1_535_810_877_406} The diagram shows the curve \(y = \mathrm { f } ( x )\) which has a maximum point at \(( - 3,2 )\) and a minimum point at \(( 2 , - 4 )\).

1. Showing the coordinates of any stationary points, sketch on separate diagrams the graphs of
   1. \(y = | \mathrm { f } ( x ) |\),
   2. \(y = 3 \mathrm { f } ( 2 x )\).
2. Write down the values of the constants \(a\) and \(b\) such that the curve with equation \(y = a + \mathrm { f } ( x + b )\) has a minimum point at the origin \(O\).

7 The curve \(y = \ln x\) is transformed to the curve \(y = \ln \left( \frac { 1 } { 2 } x - a \right)\) by means of a translation followed by a stretch. It is given that \(a\) is a positive constant.

1. Give full details of the translation and stretch involved.
2. Sketch the graph of \(y = \ln \left( \frac { 1 } { 2 } x - a \right)\).
3. Sketch, on another diagram, the graph of \(y = \left| \ln \left( \frac { 1 } { 2 } x - a \right) \right|\).
4. State, in terms of \(a\), the set of values of \(x\) for which \(\left| \ln \left( \frac { 1 } { 2 } x - a \right) \right| = - \ln \left( \frac { 1 } { 2 } x - a \right)\).

2 \includegraphics[max width=\textwidth, alt={}, center]{774bb427-5392-45d3-8e4e-47d08fb8a792-02_538_1061_388_541} The diagram shows the curve with equation \(y = \mathrm { f } ( x )\). It is given that \(\mathrm { f } ( - 7 ) = 0\) and that there are stationary points at \(( - 2 , - 6 )\) and \(( 0,0 )\). Sketch the curve with equation \(y = - 4 \mathrm { f } ( x + 3 )\), indicating the coordinates of the stationary points.

7 The sketch shows part of the curve with equation \(y = \mathrm { f } ( x )\). \includegraphics[max width=\textwidth, alt={}, center]{d3c66c34-b09c-4223-8383-cf0a68419bf9-5_632_1029_712_541}

1. On Figure 2 on page 6, sketch the curve with equation \(y = | \mathrm { f } ( x ) |\).
2. On Figure 3 on page 6, sketch the curve with equation \(y = \mathrm { f } ( | x | )\).
3. Describe a sequence of two geometrical transformations that maps the graph of \(y = \mathrm { f } ( x )\) onto the graph of \(y = \frac { 1 } { 2 } \mathrm { f } ( x + 1 )\).
4. The maximum point of the curve with equation \(y = \mathrm { f } ( x )\) has coordinates \(( - 1,10 )\). Find the coordinates of the maximum point of the curve with equation \(y = \frac { 1 } { 2 } \mathrm { f } ( x + 1 )\).
   (2 marks)
   1. \begin{figure}[h]
      \captionsetup{labelformat=empty} \caption{Figure 2} \includegraphics[alt={},max width=\textwidth]{d3c66c34-b09c-4223-8383-cf0a68419bf9-6_785_1022_358_548}
      \end{figure}
   2. \begin{figure}[h]
      \captionsetup{labelformat=empty} \caption{Figure 3} \includegraphics[alt={},max width=\textwidth]{d3c66c34-b09c-4223-8383-cf0a68419bf9-6_776_1022_1395_548}
      \end{figure}