1.02w Graph transformations: simple transformations of f(x)

3 In each of parts (a), (b) and (c), the graph shown with solid lines has equation \(y = \mathrm { f } ( x )\). The graph shown with broken lines is a transformation of \(y = \mathrm { f } ( x )\).

1. \includegraphics[max width=\textwidth, alt={}, center]{aa4c496d-ce5f-4f46-ad37-d901644a9e7c-04_412_645_367_788} State, in terms of f , the equation of the graph shown with broken lines.
2. \includegraphics[max width=\textwidth, alt={}, center]{aa4c496d-ce5f-4f46-ad37-d901644a9e7c-04_650_423_1046_900} State, in terms of f , the equation of the graph shown with broken lines.
3. \includegraphics[max width=\textwidth, alt={}, center]{aa4c496d-ce5f-4f46-ad37-d901644a9e7c-04_550_631_1975_804} State, in terms of f , the equation of the graph shown with broken lines.

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.

6 Functions f and g are both defined for \(x \in \mathbb { R }\) and are given by $$\begin{aligned} & \mathrm { f } ( x ) = x ^ { 2 } - 2 x + 5 \\ & \mathrm {~g} ( x ) = x ^ { 2 } + 4 x + 13 \end{aligned}$$

1. By first expressing each of \(\mathrm { f } ( x )\) and \(\mathrm { g } ( x )\) in completed square form, express \(\mathrm { g } ( x )\) in the form \(\mathrm { f } ( x + p ) + q\), where \(p\) and \(q\) are constants.
2. Describe fully the transformation which transforms the graph of \(y = \mathrm { f } ( x )\) to the graph of \(y = \mathrm { g } ( x )\).

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 \(0 ^ { \circ } \leqslant x \leqslant 360 ^ { \circ }\).

4

1. The curve with equation \(y = x ^ { 2 } + 2 x - 5\) is translated by \(\binom { - 1 } { 3 }\).
   Find the equation of the translated curve, giving your answer in the form \(y = a x ^ { 2 } + b x + c\).
2. The curve with equation \(y = x ^ { 2 } + 2 x - 5\) is transformed to a curve with equation \(y = 4 x ^ { 2 } + 4 x - 5\). Describe fully the single transformation that has been applied.

3 \includegraphics[max width=\textwidth, alt={}, center]{77f27b11-b931-481f-b4ef-5e549eff8086-04_1150_1164_269_484} The diagram shows graphs with equations \(y = \mathrm { f } ( x )\) and \(y = \mathrm { g } ( x )\).
Describe fully a sequence of two transformations which transforms the graph of \(y = \mathrm { f } ( x )\) to \(y = \mathrm { g } ( x )\).

1 \includegraphics[max width=\textwidth, alt={}, center]{51bd3ba6-e1d1-4c07-81cd-d99dd77f9306-02_778_1061_269_532} The diagram shows the graph of \(y = \mathrm { f } ( x )\), which consists of the two straight lines \(A B\) and \(B C\). The lines \(A ^ { \prime } B ^ { \prime }\) and \(B ^ { \prime } C ^ { \prime }\) form the graph of \(y = \mathrm { g } ( x )\), which is the result of applying a sequence of two transformations, in either order, to \(y = \mathrm { f } ( x )\). State fully the two transformations.

6 A curve passes through the point \(\left( \frac { 4 } { 5 } , - 3 \right)\) and is such that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { - 20 } { ( 5 x - 3 ) ^ { 2 } }\).

1. Find the equation of the curve.
2. The curve is transformed by a stretch in the \(x\)-direction with scale factor \(\frac { 1 } { 2 }\) followed by a translation of \(\binom { 2 } { 10 }\). Find the equation of the new curve. \includegraphics[max width=\textwidth, alt={}, center]{d6976a4b-aecf-43f1-a3f2-bcad37d03585-08_2716_38_143_2009}

2 The graph of \(y = \mathrm { f } ( x )\) is transformed to the graph of \(y = 1 + \mathrm { f } \left( \frac { 1 } { 2 } x \right)\).
Describe fully the two single transformations which have been combined to give the resulting transformation.

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.

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 \(C\).

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.

9 The function f is defined by \(\mathrm { f } ( x ) = - 3 x ^ { 2 } + 2\) for \(x \leqslant - 1\).

1. State the range of f.
2. Find an expression for \(\mathrm { f } ^ { - 1 } ( x )\).
   The function g is defined by \(\mathrm { g } ( x ) = - x ^ { 2 } - 1\) for \(x \leqslant - 1\).
3. Solve the equation \(\mathrm { fg } ( x ) - \mathrm { gf } ( x ) + 8 = 0\).

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 \(A\).

9 The functions f and g are defined for all real values of \(x\) by $$f ( x ) = ( 3 x - 2 ) ^ { 2 } + k \quad \text { and } \quad g ( x ) = 5 x - 1$$ where \(k\) is a constant.

1. Given that the range of the function gf is \(\mathrm { gf } ( x ) \geqslant 39\), find the value of \(k\).
2. For this value of \(k\), determine the range of the function fg .
3. The function h is defined for all real values of \(x\) and is such that \(\mathrm { gh } ( x ) = 35 x + 19\). Find an expression for \(\mathrm { g } ^ { - 1 } ( x )\) and hence, or otherwise, find an expression for \(\mathrm { h } ( x )\). \includegraphics[max width=\textwidth, alt={}, center]{b5eb378d-a9cb-40e0-9203-374b58f1dcf9-12_739_625_260_721} The diagram shows the circle with centre \(C ( - 4,5 )\) and radius \(\sqrt { 20 }\) units. The circle intersects the \(y\)-axis at the points \(A\) and \(B\). The size of angle \(A C B\) is \(\theta\) radians.

4 \includegraphics[max width=\textwidth, alt={}, center]{fdd6e942-b5bc-4369-8587-6de120459776-05_615_1169_260_488} In the diagram, the lower curve has equation \(y = \cos \theta\). The upper curve shows the result of applying a combination of transformations to \(y = \cos \theta\). Find, in terms of a cosine function, the equation of the upper curve.

11 A curve has equation \(y = 3 \cos 2 x + 2\) for \(0 \leqslant x \leqslant \pi\).

1. State the greatest and least values of \(y\).
2. Sketch the graph of \(y = 3 \cos 2 x + 2\) for \(0 \leqslant x \leqslant \pi\).
3. By considering the straight line \(y = k x\), where \(k\) is a constant, state the number of solutions of the equation \(3 \cos 2 x + 2 = k x\) for \(0 \leqslant x \leqslant \pi\) in each of the following cases.
   1. \(k = - 3\)
   2. \(k = 1\)
   3. \(k = 3\) Functions \(\mathrm { f } , \mathrm { g }\) and h are defined for \(x \in \mathbb { R }\) by $$\begin{aligned} & \mathrm { f } ( x ) = 3 \cos 2 x + 2 \\ & \mathrm {~g} ( x ) = \mathrm { f } ( 2 x ) + 4 \\ & \mathrm {~h} ( x ) = 2 \mathrm { f } \left( x + \frac { 1 } { 2 } \pi \right) \end{aligned}$$
4. Describe fully a sequence of transformations that maps the graph of \(y = \mathrm { f } ( x )\) on to \(y = \mathrm { g } ( x )\).
5. Describe fully a sequence of transformations that maps the graph of \(y = \mathrm { f } ( x )\) on to \(y = \mathrm { h } ( x )\). [2]
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

1

1. Express \(x ^ { 2 } + 6 x + 5\) in the form \(( x + a ) ^ { 2 } + b\), where \(a\) and \(b\) are constants.
2. The curve with equation \(y = x ^ { 2 }\) is transformed to the curve with equation \(y = x ^ { 2 } + 6 x + 5\). Describe fully the transformation(s) involved.

8

1. Express \(- 3 x ^ { 2 } + 12 x + 2\) in the form \(- 3 ( x - a ) ^ { 2 } + b\), where \(a\) and \(b\) are constants.
   The one-one function f is defined by \(\mathrm { f } : x \mapsto - 3 x ^ { 2 } + 12 x + 2\) for \(x \leqslant k\).
2. State the largest possible value of the constant \(k\).
   It is now given that \(k = - 1\).
3. State the range of f.
4. Find an expression for \(\mathrm { f } ^ { - 1 } ( x )\).
   The result of translating the graph of \(y = \mathrm { f } ( x )\) by \(\binom { - 3 } { 1 }\) is the graph of \(y = \mathrm { g } ( x )\).
5. Express \(\mathrm { g } ( x )\) in the form \(p x ^ { 2 } + q x + r\), where \(p , q\) and \(r\) are constants.

2 The graph of \(y = \mathrm { f } ( x )\) is transformed to the graph of \(y = \mathrm { f } ( 2 x ) - 3\).

1. Describe fully the two single transformations that have been combined to give the resulting transformation.
   The point \(P ( 5,6 )\) lies on the transformed curve \(y = \mathrm { f } ( 2 x ) - 3\).
2. State the coordinates of the corresponding point on the original curve \(y = \mathrm { f } ( x )\).

1 The graph of \(y = \mathrm { f } ( x )\) is transformed to the graph of \(y = 3 - \mathrm { f } ( x )\).
Describe fully, in the correct order, the two transformations that have been combined.

9 Functions f and g are both defined for \(x \in \mathbb { R }\) and are given by $$\begin{aligned} & \mathrm { f } ( x ) = x ^ { 2 } - 4 x + 9 \\ & \mathrm {~g} ( x ) = 2 x ^ { 2 } + 4 x + 12 \end{aligned}$$

1. Express \(\mathrm { f } ( x )\) in the form \(( x - a ) ^ { 2 } + b\).
2. Express \(\mathrm { g } ( x )\) in the form \(2 \left[ ( x + c ) ^ { 2 } + d \right]\).
3. Express \(\mathrm { g } ( x )\) in the form \(k \mathrm { f } ( x + h )\), where \(k\) and \(h\) are integers.
4. Describe fully the two transformations that have been combined to transform the graph of \(y = \mathrm { f } ( x )\) to the graph of \(y = \mathrm { g } ( x )\).

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