Multiple separate transformations

3. \begin{figure}[h] \end{figure} Figure 1 shows a sketch of the curve with equation \(y = \mathrm { f } ( x )\).
The curve passes through the points \(( - 1,0 )\) and \(( 0,2 )\) and touches the \(x\)-axis at the point \(( 3,0 )\). On separate diagrams, sketch the curve with equation

1. \(y = \mathrm { f } ( \mathrm { x } + 3 )\)
2. \(y = \mathrm { f } ( - 3 x )\) On each diagram, show clearly the coordinates of all the points where the curve cuts or touches the coordinate axes.

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.

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

1. Find the value of ff(-3). On separate diagrams, sketch the curve with equation
2. \(y = \mathrm { f } ^ { - 1 } ( x )\),
3. \(y = \mathrm { f } ( | x | ) - 2\),
4. \(y = 2 \mathrm { f } \left( \frac { 1 } { 2 } x \right)\). Indicate clearly on each sketch the coordinates of the points at which the curve crosses or meets the axes.

\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.

6
\includegraphics[max width=\textwidth, alt={}, center]{32f90420-e1eb-47ab-b588-e3806b64813f-3_641_837_1306_657} The diagram shows the graph of \(y = - \sin ^ { - 1 } ( x - 1 )\).

1. Give details of the pair of geometrical transformations which transforms the graph of \(y = - \sin ^ { - 1 } ( x - 1 )\) to the graph of \(y = \sin ^ { - 1 } x\).
2. Sketch the graph of \(y = \left| - \sin ^ { - 1 } ( x - 1 ) \right|\).
3. Find the exact solutions of the equation \(\left| - \sin ^ { - 1 } ( x - 1 ) \right| = \frac { 1 } { 3 } \pi\).

2
\includegraphics[max width=\textwidth, alt={}, center]{5c501214-b41c-43a8-b9c6-986758e83e7d-2_529_855_397_646} The diagram shows the graph of \(y = \mathrm { f } ( x )\). It is given that \(\mathrm { f } ( - 3 ) = 0\) and \(\mathrm { f } ( 0 ) = 2\). Sketch, on separate diagrams, the following graphs, indicating in each case the coordinates of the points where the graph crosses the axes:

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

3. \begin{figure}[h] \end{figure} Figure 1 shows a sketch of the curve with equation \(y = \mathrm { f } ( x )\). The curve has a maximum at \(( - 3,4 )\) and a minimum at \(( 1 , - 2 )\). Showing the coordinates of any turning points, sketch on separate diagrams the curves with equations

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

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)
5. \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}
6. \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}

7 The diagram shows a sketch of the curve with equation \(y = \mathrm { f } ( x )\).
\includegraphics[max width=\textwidth, alt={}, center]{063bbfa5-df49-44a1-8143-5e076397f63f-06_620_1216_356_422}

1. On Figure 1, below, sketch the curve with equation \(y = - \mathrm { f } ( 3 x )\), indicating the values where the curve cuts the coordinate axes.
2. On Figure 2, on page 7, sketch the curve with equation \(y = \mathrm { f } ( | x | )\), indicating the values where the curve cuts the coordinate axes.
3. Describe a sequence of two geometrical transformations that maps the graph of \(y = \mathrm { f } ( x )\) onto the graph of \(y = \mathrm { f } \left( - \frac { 1 } { 2 } x \right)\). \begin{figure}[h]
   \captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{063bbfa5-df49-44a1-8143-5e076397f63f-06_732_1237_1649_443}
   \end{figure} \begin{figure}[h]
   \captionsetup{labelformat=empty} \caption{Figure 2} \includegraphics[alt={},max width=\textwidth]{063bbfa5-df49-44a1-8143-5e076397f63f-07_727_1211_340_466}
   \end{figure}

4 The sketch shows part of the curve with equation \(y = \mathrm { f } ( x )\).
\includegraphics[max width=\textwidth, alt={}, center]{57412ec0-ad97-4418-8ba8-93f1f7d8aac1-08_536_1054_367_539}

1. On Figure 2 below, sketch the curve with equation \(y = - | \mathrm { f } ( x ) |\).
2. On Figure 3 on the page opposite, sketch the curve with equation \(y = \mathrm { f } ( | 2 x | )\).
3. 1. Describe a sequence of two geometrical transformations that maps the graph of \(y = \mathrm { f } ( x )\) onto the graph of \(y = \mathrm { f } ( 2 x + 2 )\).
   2. Find the coordinates of the image of the point \(P ( 4 , - 3 )\) under the sequence of transformations given in part (c)(i). \begin{figure}[h]
      \captionsetup{labelformat=empty} \caption{Figure 3} \includegraphics[alt={},max width=\textwidth]{57412ec0-ad97-4418-8ba8-93f1f7d8aac1-09_778_1032_424_529}
      \end{figure}

4. \begin{figure}[h] \end{figure} Figure 1 shows a sketch of the curve with equation \(y = \mathrm { f } ( x ) , - 1 \leq x \leq 3\). The curve touches the \(x\)-axis at the origin \(O\), crosses the \(x\)-axis at the point \(A ( 2,0 )\) and has a maximum at the point \(B \left( \frac { 4 } { 3 } , 1 \right)\). In separate diagrams, show a sketch of the curve with equation

1. \(y = \mathrm { f } ( x + 1 )\),
2. \(y = | \mathrm { f } ( x ) |\),
3. \(y = \mathrm { f } ( | x | )\),
   marking on each sketch the coordinates of points at which the curve
   1. has a turning point,
   2. meets the \(x\)-axis.

7. \begin{figure}[h] \end{figure} Figure 1 shows the graph of \(y = \mathrm { f } ( x )\) which meets the coordinate axes at the points \(( a , 0 )\) and \(( 0 , b )\), where \(a\) and \(b\) are constants.

1. Showing, in terms of \(a\) and \(b\), the coordinates of any points of intersection with the axes, sketch on separate diagrams the graphs of
   1. \(\quad y = \mathrm { f } ^ { - 1 } ( x )\),
   2. \(y = 2 \mathrm { f } ( 3 x )\). Given that $$\mathrm { f } ( x ) = 2 - \sqrt { x + 9 } , \quad x \in \mathbb { R } , \quad x \geq - 9 ,$$
2. find the values of \(a\) and \(b\),
3. find an expression for \(\mathrm { f } ^ { - 1 } ( x )\) and state its domain.

5 Answer this question on the insert provided. Fig. 5 shows the graph of \(y = \mathrm { f } ( x )\). \begin{figure}[h] \end{figure} On the insert, draw the graph of

1. \(y = \mathrm { f } ( x - 2 )\),
2. \(y = 3 \mathrm { f } ( x )\).

9 Answer parts (i) and (iii) on the insert provided. Fig. 9 shows a sketch graph of \(y = \mathrm { f } ( x )\). \begin{figure}[h] \end{figure}

1. On the Insert sketch graphs of
   (A) \(y = 2 \mathrm { f } ( x )\),
   (B) \(y = \mathrm { f } ( - x )\),
   (C) \(y = \mathrm { f } ( x - 2 )\) In each case describe the transformations.
2. Explain why the function \(y = \mathrm { f } ( x )\) does not have an inverse function.
3. The function \(\mathrm { g } ( x )\) is defined as follows: $$\mathrm { g } ( x ) = \mathrm { f } ( x ) \text { for } x \geq 0$$ On the Insert sketch the graph of \(y = \mathrm { g } ^ { - 1 } ( x )\).
4. You are given that \(\mathrm { f } ( x ) = x ^ { 2 } ( x + 2 )\). Calculate the gradient of the curve \(y = \mathrm { f } ( x )\) at the point \(( 1,3 )\).
   Deduce the gradient of the function \(\mathrm { g } ^ { - 1 } ( x )\) at the point where \(x = 3\).
5. Show that \(\mathrm { g } ( x )\) and \(\mathrm { g } ^ { - 1 } ( x )\) cross where \(x = - 1 + \sqrt { 2 }\). \section*{Insert for question 9.}
6. (A) On the axes below sketch the graph of \(y = 2 \mathrm { f } ( x )\). Describe the transformation.
   \includegraphics[max width=\textwidth, alt={}, center]{3f8be5ab-d241-4027-af26-c49da9394adc-5_563_1102_484_395} Description:
7. (B) On the axes below sketch the graph of \(y = \mathrm { f } ( - x )\). Describe the transformation.
   \includegraphics[max width=\textwidth, alt={}, center]{3f8be5ab-d241-4027-af26-c49da9394adc-5_588_1154_1576_404} Description:
8. (C) On the axes below sketch the graph of \(y = \mathrm { f } ( x - 2 )\). Describe the transformation.
   \includegraphics[max width=\textwidth, alt={}, center]{3f8be5ab-d241-4027-af26-c49da9394adc-6_615_1230_402_406} Description:
9. The function \(\mathrm { g } ( x )\) is defined as follows: $$\mathrm { g } ( x ) = \mathrm { f } ( x ) \text { for } x \geq 0$$ On the axes below sketch the graph of \(y = g ^ { - 1 } ( x )\).
   \includegraphics[max width=\textwidth, alt={}, center]{3f8be5ab-d241-4027-af26-c49da9394adc-6_677_1356_1567_312}

2. The figure above shows the graph of a curve with equation \(y = f ( x )\). The curve meets the \(x\) axis at \(( - 3,0 )\) and the \(y\) axis at \(( 0,2 )\). The curve has a maximum at \(( 3,4 )\) and a minimum at \(( - 3,0 )\). The line with equation \(y = 2\) is a horizontal asymptote to the curve. Sketch on separate diagrams the graph of ...
a) \(\ldots \quad y = f ( x + 3 )\).
b) \(. . \quad y = f ( x ) - 2\).
c) \(\ldots \quad y = \frac { 1 } { 2 } f ( x )\). Each of the sketches must include

• the coordinates of any points where the graph meets the coordinate axes.
• the coordinates of any minimum or maximum points of the curve.
• any asymptotes to the curve, clearly labelled.

3. \begin{figure}[h] \end{figure} Figure 1 shows the graph of \(y = \mathrm { f } ( x )\).

1. Write down the number of solutions that exist for the equation
   1. \(\mathrm { f } ( x ) = 1\),
   2. \(\mathrm { f } ( x ) = - x\).
2. Labelling the axes in a similar way, sketch on separate diagrams in the space provided the graphs of
   1. \(\quad y = \mathrm { f } ( x - 2 )\),
   2. \(y = \mathrm { f } ( 2 x )\).
      [0pt] [BLANK PAGE]

6. \begin{figure}[h] \end{figure} Figure 1 shows the graph of \(y = \mathrm { f } ( x )\).

1. Write down the number of solutions that exist for the equation
   1. \(\mathrm { f } ( x ) = 1\),
   2. \(\mathrm { f } ( x ) = - x\).
2. Labelling the axes in a similar way, sketch on separate diagrams in the space provided the graphs of
   1. \(\quad y = \mathrm { f } ( x - 2 )\),
   2. \(y = \mathrm { f } ( 2 x )\).
      [0pt] [BLANK PAGE]