Sequence of transformations order

8

1. The curve \(y = \sin x\) is transformed to the curve \(y = 4 \sin \left( \frac { 1 } { 2 } x - 30 ^ { \circ } \right)\).
   Describe fully a sequence of transformations that have been combined, making clear the order in which the transformations are applied.
2. Find the exact solutions of the equation \(4 \sin \left( \frac { 1 } { 2 } x - 30 ^ { \circ } \right) = 2 \sqrt { 2 }\) for \(0 ^ { \circ } \leqslant x \leqslant 360 ^ { \circ }\).

5

1. Express \(2 x ^ { 2 } - 8 x + 14\) in the form \(2 \left[ ( x - a ) ^ { 2 } + b \right]\).
   The functions \(f\) and \(g\) are defined by $$\begin{aligned} & \mathrm { f } ( x ) = x ^ { 2 } \quad \text { for } x \in \mathbb { R }
   & \mathrm {~g} ( x ) = 2 x ^ { 2 } - 8 x + 14 \quad \text { for } x \in \mathbb { R } \end{aligned}$$
2. Describe fully a sequence of transformations that maps the graph of \(y = \mathrm { f } ( x )\) onto the graph of \(y = \mathrm { g } ( x )\), making clear the order in which the transformations are applied.
   \includegraphics[max width=\textwidth, alt={}, center]{05e75fa2-81ae-44b1-b073-4100f5d911e0-08_679_1043_260_552} The circle with equation \(( x + 1 ) ^ { 2 } + ( y - 2 ) ^ { 2 } = 85\) and the straight line with equation \(y = 3 x - 20\) are shown in the diagram. The line intersects the circle at \(A\) and \(B\), and the centre of the circle is at \(C\).

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 \(y = \mathrm { h } ( x )\).

11 The function f is defined by \(\mathrm { f } ( x ) = 3 + 6 x - 2 x ^ { 2 }\) for \(x \in \mathbb { R }\).

1. Express \(\mathrm { f } ( x )\) in the form \(a - b ( x - c ) ^ { 2 }\), where \(a , b\) and \(c\) are constants, and state the range of f .
2. The graph of \(y = \mathrm { f } ( x )\) is transformed to the graph of \(y = \mathrm { h } ( x )\) by a reflection in one of the axes followed by a translation. It is given that the graph of \(y = \mathrm { h } ( x )\) has a minimum point at the origin. Give details of the reflection and translation involved.
   \includegraphics[max width=\textwidth, alt={}, center]{e32902b8-a259-4572-982e-2a35413b81b2-16_2715_38_109_2009} The function g is defined by \(\mathrm { g } ( x ) = 3 + 6 x - 2 x ^ { 2 }\) for \(x \leqslant 0\).
3. Sketch the graph of \(y = \mathrm { g } ( x )\) and explain why g is a one-one function. You are not required to find the coordinates of any intersections with the axes.
4. Sketch the graph of \(y = \mathrm { g } ^ { - 1 } ( x )\) on your diagram in (c), and find an expression for \(\mathrm { g } ^ { - 1 } ( x )\). You should label the two graphs in your diagram appropriately and show any relevant mirror line.
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.
   \includegraphics[max width=\textwidth, alt={}, center]{e32902b8-a259-4572-982e-2a35413b81b2-18_2714_38_109_2010}

8 Fig. 8 shows the curve \(y = \mathrm { f } ( x )\), where \(\mathrm { f } ( x ) = 1 + \sin 2 x\) for \(- \frac { 1 } { 4 } \pi \leqslant x \leqslant \frac { 1 } { 4 } \pi\). \begin{figure}[h] \end{figure}

1. State a sequence of two transformations that would map part of the curve \(y = \sin x\) onto the curve \(y = \mathrm { f } ( x )\).
2. Find the area of the region enclosed by the curve \(y = \mathrm { f } ( x )\), the \(x\)-axis and the line \(x = \frac { 1 } { 4 } \pi\).
3. Find the gradient of the curve \(y = \mathrm { f } ( x )\) at the point \(( 0,1 )\). Hence write down the gradient of the curve \(y = \mathrm { f } ^ { - 1 } ( x )\) at the point \(( 1,0 )\).
4. State the domain of \(\mathrm { f } ^ { - 1 } ( x )\). Add a sketch of \(y = \mathrm { f } ^ { - 1 } ( x )\) to a copy of Fig. 8.
5. Find an expression for \(\mathrm { f } ^ { - 1 } ( x )\).

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 \(\left| \ln \left( \frac { 1 } { 2 } x - a \right) \right| = - \ln \left( \frac { 1 } { 2 } x - a \right)\).

3 Fig. 8 shows the curve \(y = \mathrm { f } ( x )\), where \(\mathrm { f } ( x ) = 1 + \sin 2 x\) for \(- \frac { 1 } { 4 } \pi \leqslant x \leqslant \frac { 1 } { 4 } \pi\). \begin{figure}[h] \end{figure}

1. State a sequence of two transformations that would map part of the curve \(y = \sin x\) onto the curve \(y = \mathrm { f } ( x )\).
2. Find the area of the region enclosed by the curve \(y = \mathrm { f } ( x )\), the \(x\)-axis and the line \(x = \frac { 1 } { 4 } \pi\).
3. Find the gradient of the curve \(y = \mathrm { f } ( x )\) at the point \(( 0,1 )\). Hence write down the gradient of the curve \(y = \mathrm { f } ^ { - 1 } ( x )\) at the point \(( 1,0 )\).
4. State the domain of \(\mathrm { f } ^ { - 1 } ( x )\). Add a sketch of \(y = \mathrm { f } ^ { - 1 } ( x )\) to a copy of Fig. 8.
5. Find an expression for \(\mathrm { f } ^ { - 1 } ( x )\).

3 Fig. 8 shows the curve \(y = \mathrm { f } ( x )\), where \(\mathrm { f } ( x ) = 1 + \sin 2 x\) for \(- \frac { 1 } { 4 } \pi \leqslant x \leqslant \frac { 1 } { 4 } \pi\). \begin{figure}[h] \end{figure}

1. State a sequence of two transformations that would map part of the curve \(y = \sin x\) onto the curve \(y = \mathrm { f } ( x )\).
2. Find the area of the region enclosed by the curve \(y = \mathrm { f } ( x )\), the \(x\)-axis and the line \(x = \frac { 1 } { 4 } \pi\).
3. Find the gradient of the curve \(y = \mathrm { f } ( x )\) at the point ( 0,1 ). Hence write down the gradient of the curve \(y = \mathrm { f } ^ { - 1 } ( x )\) at the point \(( 1,0 )\).
4. State the domain of \(\mathrm { f } ^ { - 1 } ( x )\). Add a sketch of \(y = \mathrm { f } ^ { - 1 } ( x )\) to a copy of Fig. 8.
5. Find an expression for \(\mathrm { f } ^ { - 1 } ( x )\).

8

1. The curve \(y = 3 ^ { x }\) can be transformed to the curve \(y = 3 ^ { x - 2 }\) by a translation. Give details of the translation.
2. Alternatively, the curve \(y = 3 ^ { x }\) can be transformed to the curve \(y = 3 ^ { x - 2 }\) by a stretch. Give details of the stretch.
3. Sketch the curve \(y = 3 ^ { x - 2 }\), stating the coordinates of any points of intersection with the axes.
4. The point \(P\) on the curve \(y = 3 ^ { x - 2 }\) has \(y\)-coordinate equal to 180 . Use logarithms to find the \(x\)-coordinate of \(P\), correct to 3 significant figures.
5. Use the trapezium rule, with 2 strips each of width 1.5, to find an estimate for \(\int _ { 1 } ^ { 4 } 3 ^ { x - 2 } \mathrm {~d} x\). Give your answer correct to 3 significant figures.

8 Fig. 8 shows parts of the curves \(y = \mathrm { f } ( x )\) and \(y = \mathrm { g } ( x )\), where \(\mathrm { f } ( x ) = \tan x\) and \(\mathrm { g } ( x ) = 1 + \mathrm { f } \left( x - \frac { 1 } { 4 } \pi \right)\). \begin{figure}[h] \end{figure}

1. Describe a sequence of two transformations which maps the curve \(y = \mathrm { f } ( x )\) to the curve \(y = \mathrm { g } ( x )\). It can be shown that \(\mathrm { g } ( x ) = \frac { 2 \sin x } { \sin x + \cos x }\).
2. Show that \(\mathrm { g } ^ { \prime } ( x ) = \frac { 2 } { ( \sin x + \cos x ) ^ { 2 } }\). Hence verify that the gradient of \(y = \mathrm { g } ( x )\) at the point \(\left( \frac { 1 } { 4 } \pi , 1 \right)\) is the same as that of \(y = \mathrm { f } ( x )\) at the origin.
3. By writing \(\tan x = \frac { \sin x } { \cos x }\) and using the substitution \(u = \cos x\), show that \(\int _ { 0 } ^ { \frac { 1 } { 4 } \pi } \mathrm { f } ( x ) \mathrm { d } x = \int _ { \frac { 1 } { \sqrt { 2 } } } ^ { 1 } \frac { 1 } { u } \mathrm {~d} u\). Evaluate this integral exactly.
4. Hence find the exact area of the region enclosed by the curve \(y = \mathrm { g } ( x )\), the \(x\)-axis and the lines \(x = \frac { 1 } { 4 } \pi\) and \(x = \frac { 1 } { 2 } \pi\).

2 Given that \(\mathrm { f } ( x ) = x ^ { 3 }\) and \(\mathrm { g } ( x ) = 2 x ^ { 3 } - 1\), describe a sequence of two transformations which maps the curve \(y = \mathrm { f } ( x )\) onto the curve \(y = \mathrm { g } ( x )\).

2 Describe a sequence of two geometrical transformations that maps the graph of \(y = \sec x\) onto the graph of \(y = 1 + \sec 3 x\).

5

1. Describe a sequence of two geometrical transformations that maps the graph of \(y = \ln x\) onto the graph of \(y = 4 \ln ( x - \mathrm { e } )\).
2. Sketch, on the axes given below, the graph of \(y = | 4 \ln ( x - \mathrm { e } ) |\), indicating the exact value of the \(x\)-coordinate where the curve meets the \(x\)-axis.
3. 1. Solve the equation \(| 4 \ln ( x - e ) | = 4\).
   2. Hence, or otherwise, solve the inequality \(| 4 \ln ( x - e ) | \geqslant 4\).
      \includegraphics[max width=\textwidth, alt={}, center]{7aa76d26-e3c4-4374-ae4f-8bb61e61b135-3_655_1428_2023_315}

4 The diagram shows a sketch of the curve with equation \(y = \mathrm { f } ( x )\).
\includegraphics[max width=\textwidth, alt={}, center]{b8614dd6-2197-40c3-a673-5bef3e3653a5-5_629_1113_370_461}

1. On the axes below, sketch the curve with equation \(y = | \mathrm { f } ( x ) |\).
2. Describe a sequence of two geometrical transformations that maps the graph of \(y = \mathrm { f } ( x )\) onto the graph of \(y = \mathrm { f } ( 2 x - 1 )\).

8 The diagram shows part of the graph of \(y = \mathrm { e } ^ { 2 x } + 3\).
\includegraphics[max width=\textwidth, alt={}, center]{d5b78fa6-ea3c-497b-94d8-1d5f61288aa5-4_833_1034_1027_513}

1. Describe a sequence of two geometrical transformations that maps the graph of \(y = \mathrm { e } ^ { x }\) onto the graph of \(y = \mathrm { e } ^ { 2 x } + 3\).
2. Use the mid-ordinate rule with four strips of equal width to find an estimate for the area of the shaded region \(A\), giving your answer to three significant figures.
3. Find the exact value of the area of the shaded region \(A\).
4. The region \(B\) is indicated on the diagram. Find the area of the region \(B\), giving your answer in the form \(p \mathrm { e } ^ { 8 } + q \mathrm { e } ^ { 4 }\), where \(p\) and \(q\) are numbers to be determined.

2. The transformations \(\mathrm { R } , \mathrm { S }\) and T are defined as follows.
R : reflection in the \(x\)-axis
S : stretch in the \(x\)-direction with scale factor 3
T: translation in the positive \(x\)-direction by 4 units

1. The curve \(y = \ln x\) is transformed by R followed by T . Find the equation of the resulting curve.
2. Find, in terms of S and T, a sequence of transformations that transforms the curve \(y = x ^ { 3 }\) to the curve \(y = \left( \frac { 1 } { 9 } x - 4 \right) ^ { 3 }\). You should make clear the order of the transformations.
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10. The graph of \(y = \mathrm { e } ^ { x }\) can be transformed to the graph of \(y = \mathrm { e } ^ { 2 x - 1 }\) by a stretch parallel to the \(x\)-axis followed by a translation.

1. 1. State the scale factor of the stretch.
   2. Give full details of the translation. Alternatively the graph of \(y = \mathrm { e } ^ { x }\) can be transformed to the graph of \(y = \mathrm { e } ^ { 2 x - 1 }\) by a stretch parallel to the \(x\)-axis and a stretch parallel to the \(y\)-axis.
2. State the scale factor of the stretch parallel to the \(y\)-axis.
   [0pt] [BLANK PAGE]

10
\includegraphics[max width=\textwidth, alt={}, center]{a16ab26f-21fb-4a73-8b94-c16bef611bcb-7_524_714_274_246} The diagram shows the graph of \(y = - \tan ^ { - 1 } \left( \frac { 1 } { 2 } x - \frac { 1 } { 3 } \pi \right)\), which crosses the \(x\)-axis at the point \(A\) and the \(y\)-axis at the point \(B\).

1. Determine the coordinates of the points \(A\) and \(B\).
2. Give full details of a sequence of three geometrical transformations which transform the graph of \(y = \tan ^ { - 1 } x\) to the graph of \(y = - \tan ^ { - 1 } \left( \frac { 1 } { 2 } x - \frac { 1 } { 3 } \pi \right)\). The equation \(x = - \tan ^ { - 1 } \left( \frac { 1 } { 2 } x - \frac { 1 } { 3 } \pi \right)\) has only one root.
3. Show by calculation that this root lies between \(x = 0\) and \(x = 1\).
4. Use the iterative formula \(x _ { n + 1 } = - \tan ^ { - 1 } \left( \frac { 1 } { 2 } x _ { n } - \frac { 1 } { 3 } \pi \right)\), with a suitable starting value, to find the root correct to 3 significant figures. Show the result of each iteration.
5. Using the diagram in the Printed Answer Booklet, show how the iterative process converges to the root.

8 The sketch shows the graph of \(y = \cos ^ { - 1 } x\).
\includegraphics[max width=\textwidth, alt={}, center]{59b896ae-60ce-49ea-9c70-0f76fc5fffae-5_593_686_383_683}

1. Write down the coordinates of \(P\) and \(Q\), the end points of the graph.
2. Describe a sequence of two geometrical transformations that maps the graph of \(y = \cos ^ { - 1 } x\) onto the graph of \(y = 2 \cos ^ { - 1 } ( x - 1 )\).
3. Sketch the graph of \(y = 2 \cos ^ { - 1 } ( x - 1 )\).
4. 1. Write the equation \(y = 2 \cos ^ { - 1 } ( x - 1 )\) in the form \(x = \mathrm { f } ( y )\).
   2. Hence find the value of \(\frac { \mathrm { d } x } { \mathrm {~d} y }\) when \(y = 2\).