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

2 One of the diagrams below shows the graph of \(y = \sin \left( x + 90 ^ { \circ } \right)\) for \(0 ^ { \circ } \leq x \leq 360 ^ { \circ }\) Identify the correct graph. Tick ( \(\checkmark\) ) one box. \includegraphics[max width=\textwidth, alt={}, center]{f87d1b36-26db-4a0b-b9ec-d7d82a396aba-03_451_465_568_497} \includegraphics[max width=\textwidth, alt={}, center]{f87d1b36-26db-4a0b-b9ec-d7d82a396aba-03_124_154_724_1073} \includegraphics[max width=\textwidth, alt={}, center]{f87d1b36-26db-4a0b-b9ec-d7d82a396aba-03_458_472_1105_495}
□ \includegraphics[max width=\textwidth, alt={}, center]{f87d1b36-26db-4a0b-b9ec-d7d82a396aba-03_453_468_1647_497} \includegraphics[max width=\textwidth, alt={}, center]{f87d1b36-26db-4a0b-b9ec-d7d82a396aba-03_117_132_1809_1091} \includegraphics[max width=\textwidth, alt={}, center]{f87d1b36-26db-4a0b-b9ec-d7d82a396aba-03_461_479_2183_488}

2 The graph of \(y = 5 ^ { x }\) is transformed by a stretch in the \(y\)-direction, scale factor 5 State the equation of the transformed graph. Circle your answer.
[0pt] [1 mark] \(y = 5 \times 5 ^ { x }\) \(y = 5 ^ { \frac { x } { 5 } }\) \(y = \frac { 1 } { 5 } \times 5 ^ { x }\) \(y = 5 ^ { 5 x }\)

3 The curve $$y = \log _ { 4 } x$$ is transformed by a stretch, scale factor 2 , parallel to the \(y\)-axis.
State the equation of the curve after it has been transformed.
Circle your answer.
[0pt] [1 mark] $$y = \frac { 1 } { 2 } \log _ { 4 } x \quad y = 2 \log _ { 4 } x \quad y = \log _ { 4 } 2 x \quad y = \log _ { 8 } x$$

7 Sketch the graph of $$y = \cot \left( x - \frac { \pi } { 2 } \right)$$ for \(0 \leq x \leq 2 \pi\) [0pt] [3 marks] \includegraphics[max width=\textwidth, alt={}, center]{22ff390e-1360-43bd-8c7f-3d2b58627e91-08_1650_1226_587_408} \includegraphics[max width=\textwidth, alt={}, center]{22ff390e-1360-43bd-8c7f-3d2b58627e91-09_2488_1716_219_153}

3 The curve with equation \(y = \ln x\) is transformed by a stretch parallel to the \(x\)-axis with scale factor 2 Find the equation of the transformed curve.
Circle your answer. \(y = \frac { 1 } { 2 } \ln x \quad y = 2 \ln x \quad y = \ln \frac { x } { 2 } \quad y = \ln 2 x\)

8

1. Sketch the graph of \(y = \frac { 1 } { x ^ { 2 } }\) \includegraphics[max width=\textwidth, alt={}, center]{ad6590e8-6673-45ca-bef3-a14716978827-12_1273_1083_404_482} 8
2. The graph of \(y = \frac { 1 } { x ^ { 2 } }\) can be transformed onto the graph of \(y = \frac { 9 } { x ^ { 2 } }\) using a stretch in one direction. Beth thinks the stretch should be in the \(y\)-direction.
   Paul thinks the stretch should be in the \(x\)-direction.
   State, giving reasons for your answer, whether Beth is correct, Paul is correct, both are correct or neither is correct.
   [0pt] [3 marks]

2

1. Find the transformation which maps the curve \(y = x ^ { 2 }\) to the curve \(y = x ^ { 2 } + 8 x - 7\).
2. Write down the coordinates of the turning point of \(y = x ^ { 2 } + 8 x - 7\).

14 Curve \(C _ { 1 }\) has equation $$\frac { x ^ { 2 } } { 16 } + \frac { y ^ { 2 } } { 4 } = 1$$ 14

1. Curve \(C _ { 2 }\) is a reflection of \(C _ { 1 }\) in the line \(y = x\) Write down an equation of \(C _ { 2 }\) 14
2. Curve \(C _ { 3 }\) is a circle of radius 4 , centred at the origin.
   Describe a single transformation which maps \(C _ { 1 }\) onto \(C _ { 3 }\) 14
3. Curve \(C _ { 4 }\) is a translation of \(C _ { 1 }\) The positive \(x\)-axis and the positive \(y\)-axis are tangents to \(C _ { 4 }\) 14 (c) (i) Sketch the graphs of \(C _ { 1 }\) and \(C _ { 4 }\) on the axes opposite. Indicate the coordinates of the \(x\) and \(y\) intercepts on your graphs.
   [0pt] [2 marks]
   14 (c) (ii) Determine the translation vector.
   [0pt] [2 marks]
   14 (c) (iii) The line \(y = m x + c\) is a tangent to both \(C _ { 1 }\) and \(C _ { 4 }\) Find the value of \(m\)

5

1. Show that the equation of \(H _ { 1 }\) can be written in the form $$( x - 1 ) ^ { 2 } - \frac { y ^ { 2 } } { q } = r$$ where \(q\) and \(r\) are integers.
   5
2. \(\quad \mathrm { H } _ { 2 }\) is the hyperbola $$x ^ { 2 } - y ^ { 2 } = 4$$ Describe fully a sequence of two transformations which maps the graph of \(H _ { 2 }\) onto the graph of \(H _ { 1 }\) [0pt] [4 marks]

6 The ellipse \(E _ { 1 }\) has equation $$x ^ { 2 } + \frac { y ^ { 2 } } { 4 } = 1$$ \(E _ { 1 }\) is translated by the vector \(\left[ \begin{array} { l } 3 \\ 0 \end{array} \right]\) to give the ellipse \(E _ { 2 }\) 6

1. Write down the equation of \(E _ { 2 }\) 6
2. The ellipse \(E _ { 3 }\) has equation $$\frac { x ^ { 2 } } { 4 } + ( y - 3 ) ^ { 2 } = 1$$ Describe the transformation that maps \(E _ { 2 }\) to \(E _ { 3 }\) 6
3. Each of the lines \(L _ { A }\) and \(L _ { B }\) is a tangent to both \(E _ { 2 }\) and \(E _ { 3 }\) \(L _ { A }\) is closer to the origin than \(L _ { B }\) \(E _ { 2 }\) and \(E _ { 3 }\) both lie between \(L _ { A }\) and \(L _ { B }\) Sketch and label \(E _ { 2 } , E _ { 3 } , L _ { A }\) and \(L _ { B }\) on the axes below.
   You do not need to show the values of the axis intercepts for \(L _ { A }\) and \(L _ { B }\) \includegraphics[max width=\textwidth, alt={}, center]{13abb93f-2fef-465c-980c-3b412de06618-09_1095_1095_726_475} 6
4. Explain, without doing any calculations, why \(L _ { A }\) has an equation of the form $$x + y = c$$ where \(c\) is a constant.

10 The curve \(C _ { 1 }\) has equation $$\frac { x ^ { 2 } } { 25 } - \frac { y ^ { 2 } } { 4 } = 1$$ The curve \(C _ { 2 }\) has equation $$x ^ { 2 } - 25 y ^ { 2 } - 6 x - 200 y - 416 = 0$$ 10

1. Find a sequence of transformations that maps the graph of \(C _ { 1 }\) onto the graph of \(C _ { 2 }\) [4 marks]
   10
2. Find the equations of the asymptotes to \(C _ { 2 }\) Give your answers in the form \(a x + b y + c = 0\) where \(a , b\) and \(c\) are integers.

7. \begin{figure}[h] \end{figure} Figure 3 shows a plot of part of the curve \(C _ { 1 }\) with equation $$y = - 4 \cos x$$ where \(x\) is measured in radians.
Points \(P\) and \(Q\) lie on the curve and are shown in Figure 3.

1. State
   1. the coordinates of \(P\)
   2. the coordinates of \(Q\) The curve \(C _ { 2 }\) has equation \(y = - 4 \cos x + k\) where \(x\) is measured in radians and \(k\) is a constant. Given that \(C _ { 2 }\) has a maximum \(y\) value of 11
2. 1. state the value of \(k\)
   2. state the coordinates of the minimum point on \(C _ { 2 }\) with the smallest positive \(x\) coordinate. On the opposite page there is a copy of Figure 3 labelled Diagram 1.
3. Using Diagram 1, state the number of solutions of the equation $$- 4 \cos x = 5 - \frac { 10 } { \pi } x$$ giving a reason for your answer. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{c48e6503-9d26-4f55-bdca-feadfb1afb7c-23_860_1016_1676_529} \captionsetup{labelformat=empty} \caption{Diagram 1}
   \end{figure}

9. \begin{figure}[h] \end{figure} Figure 4 shows a sketch of the curve \(C\) with equation \(y = \mathrm { f } ( x )\), where $$f ( x ) = ( x + 5 ) \left( 3 x ^ { 2 } - 4 x + 20 \right)$$

1. Deduce the range of values of \(x\) for which \(\mathrm { f } ( x ) \geqslant 0\)
2. Find \(\mathrm { f } ^ { \prime } ( x )\) giving your answer in simplest form. The point \(R ( - 4,84 )\) lies on \(C\).
   Given that the tangent to \(C\) at the point \(P\) is parallel to the tangent to \(C\) at the point \(R\) (c) find the \(x\) coordinate of \(P\).
   (d) Find the point to which \(R\) is transformed when the curve with equation \(y = \mathrm { f } ( x )\) is transformed to the curve with equation,
   1. \(y = \mathrm { f } ( x - 3 )\)
   2. \(y = 4 \mathrm { f } ( x )\)

6

1. Sketch the graph of \(y = \cos 2 x\) for \(0 \leqslant x \leqslant 2 \pi\).
2. Describe the transformation which maps the graph of \(y = \cos x\) onto the graph of \(y = \cos 2 x\).

6 Describe fully the transformations which, when applied to the graph of \(y = \mathrm { f } ( x )\), will produce the graphs with equations given by

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

4 The graph of \(\mathrm { f } ( x )\) is shown below. \includegraphics[max width=\textwidth, alt={}, center]{69792771-6de6-4886-9c71-e794fcb7aaba-2_949_1127_1041_507} Draw the graphs of

1. \(\mathrm { f } ( x + 2 ) + 1\),
2. \(- \frac { 1 } { 2 } \mathrm { f } ( x )\).

4 The graph of \(\mathrm { f } ( x )\) is shown below. \includegraphics[max width=\textwidth, alt={}, center]{29e924de-bedf-4719-bbfe-f5e0d3191d59-2_949_1127_1041_507} Draw the graphs of

1. \(\mathrm { f } ( x + 2 ) + 1\),
2. \(- \frac { 1 } { 2 } \mathrm { f } ( x )\).

4 The graph of \(\mathrm { f } ( x )\) is shown below. \includegraphics[max width=\textwidth, alt={}, center]{0eb5bd24-e656-40f0-ad85-f21d3e2edf77-2_949_1127_1041_507} Draw the graphs of

1. \(\mathrm { f } ( x + 2 ) + 1\),
2. \(- \frac { 1 } { 2 } \mathrm { f } ( x )\).

7

1. Describe the transformation which maps the graph of \(y = \ln x\) onto the graph of \(y = \ln ( 1 + x )\).
2. By sketching the curves \(y = \ln ( 1 + x )\) and \(y = 4 - x\) on a single diagram, show that the equation $$\ln ( 1 + x ) = 4 - x$$ has exactly one root.
3. Use the Newton-Raphson method with \(x _ { 0 } = 2\) to find the root of the equation \(\ln ( 1 + x ) = 4 - x\) correct to 3 decimal places. Show the result of each iteration.

9 In this question, \(x\) denotes an angle measured in degrees.

1. Express \(4 \sin \left( 2 x + 30 ^ { \circ } \right) + 3 \cos 2 x\) in the form \(R \cos ( 2 x - \alpha )\), where \(R > 0\) and \(0 ^ { \circ } < \alpha < 90 ^ { \circ }\).
2. Give full details of the sequence of transformations which maps the graph of \(y = \cos x\) onto the graph of \(y = 4 \sin \left( 2 x + 30 ^ { \circ } \right) + 3 \cos 2 x\).
3. Find the smallest positive value of \(x\) that satisfies the equation \(4 \sin \left( 2 x + 30 ^ { \circ } \right) + 3 \cos 2 x = 6\).

The \(n\)th term of a number sequence is denoted by \(x _ { n }\). The \(( n + 1 )\) th term is defined by \(x _ { n + 1 } = 4 x _ { n } - 3\) and \(x _ { 3 } = 113\). a) Find the values of \(x _ { 2 }\) and \(x _ { 1 }\).
b) Determine whether the sequence is an arithmetic sequence, a geometric sequence or neither. Give reasons for your answer.
a) Express \(5 \sin x - 12 \cos x\) in the form \(R \sin ( x - \alpha )\), where \(R > 0\) and \(0 ^ { \circ } < \alpha < 90 ^ { \circ }\).
b) Find the minimum value of \(\frac { 4 } { 5 \sin x - 12 \cos x + 15 }\).
c) Solve the equation $$5 \sin x - 12 \cos x + 3 = 0$$ for values of \(x\) between \(0 ^ { \circ }\) and \(360 ^ { \circ }\). a) Find the range of values of \(x\) for which \(| 1 - 3 x | > 7\).
b) Sketch the graph of \(y = | 1 - 3 x | - 7\). Clearly label the minimum point and the points where the graph crosses the \(x\)-axis.

A curve \(C\) has parametric equations \(x = \sin \theta , y = \cos 2 \theta\). a) The equation of the tangent to the curve \(C\) at the point \(P\) where \(\theta = \frac { \pi } { 4 }\) is \(y = m x + c\). Find the exact values of \(m\) and \(c\).
b) Find the coordinates of the points of intersection of the curve \(C\) and the straight line \(x + y = 1\). The diagram below shows a sketch of the graph of \(y = f ( x )\). The graph crosses the \(y\)-axis at the point \(( 0 , - 2 )\), and the \(x\)-axis at the point \(( 8,0 )\). \includegraphics[max width=\textwidth, alt={}, center]{966abb82-ade0-4ca8-87a4-26e806d5add7-3_784_1080_1407_513}
a) Sketch the graph of \(y = - 4 f ( x + 3 )\). Indicate the coordinates of the point where the graph crosses the \(x\)-axis and the \(y\)-coordinate of the point where \(x = - 3\).
b) Sketch the graph of \(y = 3 + f ( 2 x )\). Indicate the \(y\)-coordinate of the point where \(x = 4\).

\includegraphics{figure_8} The diagram shows the graph of \(y = f(x)\) where the function \(f\) is defined by $$f(x) = 3 + 2\sin \frac{1}{4}x \text{ for } 0 \leqslant x \leqslant 2\pi.$$

1. On the diagram above, sketch the graph of \(y = f^{-1}(x)\). [2]
2. Find an expression for \(f^{-1}(x)\). [2]
3. \includegraphics{figure_8c} The diagram above shows part of the graph of the function \(g(x) = 3 + 2\sin \frac{1}{4}x\) for \(-2\pi \leqslant x \leqslant 2\pi\). Complete the sketch of the graph of \(g(x)\) on the diagram above and hence explain whether the function \(g\) has an inverse. [2]
4. Describe fully a sequence of three transformations which can be combined to transform the graph of \(y = \sin x\) for \(0 \leqslant x \leqslant \frac{1}{2}\pi\) to the graph of \(y = f(x)\), making clear the order in which the transformations are applied. [6]

\includegraphics{figure_2} The diagram shows two curves. One curve has equation \(y = \sin x\) and the other curve has equation \(y = \text{f}(x)\).

1. In order to transform the curve \(y = \sin x\) to the curve \(y = \text{f}(x)\), the curve \(y = \sin x\) is first reflected in the \(x\)-axis. Describe fully a sequence of two further transformations which are required. [4]
2. Find f\((x)\) in terms of \(\sin x\). [2]

\includegraphics{figure_6} The function f is defined by f\((x) = \frac{2}{x^2} + 4\) for \(x < 0\). The diagram shows the graph of \(y = \text{f}(x)\).

1. On this diagram, sketch the graph of \(y = \text{f}^{-1}(x)\). Show any relevant mirror line. [2]
2. Find an expression for f\(^{-1}(x)\). [3]
3. Solve the equation f\((x) = 4.5\). [1]
4. Explain why the equation f\(^{-1}(x) = \text{f}(x)\) has no solution. [1]