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

1

1. A curve has equation \(y = x ^ { 2 } - 4\). Find the \(x\)-coordinates of the points on the curve where \(y = 21\).
2. The curve \(y = x ^ { 2 } - 4\) is translated by \(\binom { 2 } { 0 }\). Write down an equation for the translated curve. You need not simplify your answer.

2 \begin{figure}[h] \end{figure} Fig. 2 shows graphs \(A\) and \(B\).

1. State the transformation which maps graph \(A\) onto graph \(B\).
2. The equation of graph \(A\) is \(y = \mathrm { f } ( x )\). Which one of the following is the equation of graph \(B\) ? $$\begin{aligned} & y = \mathrm { f } ( x ) + 2 \\ & y = 2 \mathrm { f } ( x ) \end{aligned}$$ $$\begin{aligned} & y = \mathrm { f } ( x ) - 2 \\ & y = \mathrm { f } ( x + 3 ) \end{aligned}$$ $$\begin{aligned} & y = \mathrm { f } ( x + 2 ) \\ & y = \mathrm { f } ( x - 3 ) \end{aligned}$$

3 You are given that \(\mathrm { f } ( x ) = ( x + 3 ) ( x - 2 ) ( x - 5 )\).

1. Sketch the curve \(y = \mathrm { f } ( x )\).
2. Show that \(\mathrm { f } ( x )\) may be written as \(x ^ { 3 } - 4 x ^ { 2 } - 11 x + 30\).
3. Describe fully the transformation that maps the graph of \(y = \mathrm { f } ( x )\) onto the graph of \(y = \mathrm { g } ( x )\), where \(\mathrm { g } ( x ) = x ^ { 3 } - 4 x ^ { 2 } - 11 x - 6\).
4. Show that \(\mathrm { g } ( - 1 ) = 0\). Hence factorise \(\mathrm { g } ( x )\) completely.

4

1. You are given that \(\mathrm { f } ( x ) = ( 2 x - 5 ) ( x - 1 ) ( x - 4 )\).
   (A) Sketch the graph of \(y = \mathrm { f } ( x )\).
   (B) Show that \(\mathrm { f } ( x ) = 2 x ^ { 3 } - 15 x ^ { 2 } + 33 x - 20\).
2. You are given that \(\mathrm { g } ( x ) = 2 x ^ { 3 } - 15 x ^ { 2 } + 33 x - 40\).
   (A) Show that \(\mathrm { g } ( 5 ) = 0\).
   (B) Express \(\mathrm { g } ( x )\) as the product of a linear and quadratic factor.
   (C) Hence show that the equation \(\mathrm { g } ( x ) = 0\) has only one real root.
3. Describe fully the transformation that maps \(y = \mathrm { f } ( x )\) onto \(y = \mathrm { g } ( x )\).

2 The functions \(\mathrm { f } ( x )\) and \(\mathrm { g } ( x )\) are defined for all real numbers \(x\) by $$\mathrm { f } ( x ) = x ^ { 2 } , \quad \mathrm {~g} ( x ) = x - 2$$

1. Find the composite functions \(\mathrm { fg } ( x )\) and \(\mathrm { gf } ( x )\).
2. Sketch the curves \(y = \mathrm { f } ( x ) , y = \mathrm { fg } ( x )\) and \(y = \mathrm { gf } ( x )\), indicating clearly which is which.

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

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 The functions \(\mathrm { f } ( x )\) and \(\mathrm { g } ( x )\) are defined for the domain \(x > 0\) as follows: $$\mathrm { f } ( x ) = \ln x , \quad \mathrm {~g} ( x ) = x ^ { 3 } .$$ Express the composite function \(\mathrm { fg } ( x )\) in terms of \(\ln x\).
State the transformation which maps the curve \(y = \mathrm { f } ( x )\) onto the curve \(y = \mathrm { fg } ( x )\).

4 \begin{figure}[h] \end{figure} Fig. 4 shows a sketch of the graph of \(y = \mathrm { f } ( x )\). On separate diagrams, sketch the graphs of the following, showing clearly the coordinates of the points corresponding to \(\mathrm { A } , \mathrm { B }\) and C .

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

8

1. Solve the equation \(\cos x = 0.4\) for \(0 ^ { \circ } \leqslant x \leqslant 360 ^ { \circ }\).
2. Describe the transformation which maps the graph of \(y = \cos x\) onto the graph of \(y = \cos 2 x\).

3 \begin{figure}[h] \end{figure} Fig. 3 shows sketches of three graphs, A, B and C. The equation of graph A is \(y = \mathrm { f } ( x )\). State the equation of

1. graph B ,
2. graph C.

5

1. On the same axes, sketch the graphs of \(y = \cos x\) and \(y = \cos 2 x\) for values of \(x\) from 0 to \(2 \pi\).
2. Describe the transformation which maps the graph of \(y = \cos x\) onto the graph of \(y = 3 \cos x\).

8 Draw two sketches of the graph of \(y = \sin x\) in the range \(0 ^ { \circ } \leq x \leq 360 ^ { \circ }\).

1. On the first sketch, draw also a sketch of \(y = \sin ( 2 x )\).
2. On the second sketch, draw also a sketch of \(y = 2 \sin x\).

2 The diagram shows the graph of \(y = \mathrm { f } ( x )\). The graph passes through the point with coordinates \(( 0,2 )\). \includegraphics[max width=\textwidth, alt={}, center]{1c52d6b5-84b4-455a-9620-c377ae457069-2_524_1350_775_346} On separate diagrams sketch the graphs of the following functions, indicating clearly the point of intersection with the \(y\) axis.

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

5 The diagram shows the curve \(y = \mathrm { f } ( x )\) where \(a\) is a positive constant. \includegraphics[max width=\textwidth, alt={}, center]{c55a5f04-3573-4f36-a12c-3755bdd4a45b-3_551_962_255_476} Sketch the following curves on separate diagrams, in each case stating the coordinates of points where they meet the \(x\) - and \(y\)-axes.

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

5. (i) Describe fully a single transformation that maps the graph of \(y = 3 ^ { x }\) onto the graph of \(y = \left( \frac { 1 } { 3 } \right) ^ { x }\).
(ii) Sketch on the same diagram the curves \(y = \left( \frac { 1 } { 3 } \right) ^ { x }\) and \(y = 2 \left( 3 ^ { x } \right)\), showing the coordinates of any points where each curve crosses the coordinate axes. The curves \(y = \left( \frac { 1 } { 3 } \right) ^ { x }\) and \(y = 2 \left( 3 ^ { x } \right)\) intersect at the point \(P\).
(iii) Find the \(x\)-coordinate of \(P\) to 2 decimal places and show that the \(y\)-coordinate of \(P\) is \(\sqrt { 2 }\).

1. (i) Sketch the curve \(y = \sin x ^ { \circ }\) for \(x\) in the interval \(- 180 \leq x \leq 180\).
   (ii) Sketch on the same diagram the curve \(y = \sin ( x - 30 ) ^ { \circ }\) for \(x\) in the interval \(- 180 \leq x \leq 180\).
   (iii) Use your diagram to solve the equation

$$\sin x ^ { \circ } = \sin ( x - 30 ) ^ { \circ }$$ for \(x\) in the interval \(- 180 \leq x \leq 180\).

1 The point \(\mathrm { R } ( 6 , - 3 )\) is on the curve \(y = \mathrm { f } ( x )\).

1. Find the coordinates of the image of R when the curve is transformed to \(y = \frac { 1 } { 2 } \mathrm { f } ( x )\).
2. Find the coordinates of the image of R when the curve is transformed to \(y = \mathrm { f } ( 3 x )\).

2 Fig. 8 shows the graph of \(y = \mathrm { g } ( x )\). \begin{figure}[h] \end{figure} Draw the graph of

1. \(y = \mathrm { g } ( 2 x )\),
2. \(y = 3 \mathrm {~g} ( x )\).

3 The point \(\mathrm { P } ( 6,3 )\) lies on the curve \(y = \mathrm { f } ( x )\). State the coordinates of the image of P after the transformation which maps \(y = \mathrm { f } ( x )\) onto

1. \(y = 3 \mathrm { f } ( x )\),
2. \(y = \mathrm { f } ( 4 x )\).

4 In this question, \(\mathrm { f } ( x ) = x ^ { 2 } - 5 x\). Fig. 4 shows a sketch of the graph of \(y = \mathrm { f } ( x )\). \begin{figure}[h] \end{figure} On separate diagrams, sketch the curves \(y = \mathrm { f } ( 2 x )\) and \(y = 3 \mathrm { f } ( x )\), labelling the coordinates of their intersections with the axes and their turning points.

5 State the transformation which maps the graph of \(y = x ^ { 2 } + 5\) onto the graph of \(y = 3 x ^ { 2 } + 15\).

6 \begin{figure}[h] \end{figure} Fig. 3 shows sketches of three graphs, A, B and C. The equation of graph A is \(y = \mathrm { f } ( x )\). State the equation of

1. graph B ,
2. graph C .

7

1. Solve the equation \(\cos x = 0.4\) for \(0 ^ { \circ } \leqslant x \leqslant 360 ^ { \circ }\).
2. Describe the transformation which maps the graph of \(y = \cos x\) onto the graph of \(y = \cos 2 x\).

8

1. The point \(\mathrm { P } ( 4 , - 2 )\) lies on the curve \(y = \mathrm { f } ( x )\). Find the coordinates of the image of P when the curve is transformed to \(y = \mathrm { f } ( 5 x )\).
2. Describe fully a single transformation which maps the curve \(y = \sin x ^ { \circ }\) onto the curve \(y = \sin ( x - 90 ) ^ { \circ }\).

9 Figs. 5.1 and 5.2 show the graph of \(y = \sin x\) for values of \(x\) from \(0 ^ { \circ }\) to \(360 ^ { \circ }\) and two transformations of this graph. State the equation of each graph after it has been transformed.

1. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{669be128-491c-4152-8f3a-e37a34dd9383-4_511_941_828_586} \captionsetup{labelformat=empty} \caption{Fig. 5.1}
   \end{figure}
2. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{669be128-491c-4152-8f3a-e37a34dd9383-4_517_937_1508_584} \captionsetup{labelformat=empty} \caption{Fig. 5.2}
   \end{figure}