Function Transformations

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

\includegraphics{figure_3} The diagram shows the graph of \(y = \text{g}(x)\). In the printed answer booklet, using the same scale as in this diagram, sketch the curves

1. \(y = \frac{3}{2}\text{g}(x)\), [2]
2. \(y = \text{g}\left(\frac{1}{2}x\right)\). [2]

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}

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

1. Sketch the curve \(y = -\frac{1}{x}\). [2]
2. The curve \(y = -\frac{1}{x}\) is translated by 2 units parallel to the x-axis in the positive direction. State the equation of the transformed curve. [2]
3. Describe a transformation that transforms the curve \(y = -\frac{1}{x}\) to the curve \(y = -\frac{1}{3x}\). [2]

8. (i) Describe fully a single transformation that maps the graph of \(y = \frac { 1 } { x }\) onto the graph of \(y = \frac { 3 } { x }\).
(ii) Sketch the graph of \(y = \frac { 3 } { x }\) and write down the equations of any asymptotes.
(iii) Find the values of the constant \(c\) for which the straight line \(y = c - 3 x\) is a tangent to the curve \(y = \frac { 3 } { x }\).

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. 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 \(C\).

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

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

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 \(\mathrm { f } ( x )\).

7 \includegraphics[max width=\textwidth, alt={}, center]{d858728a-3371-4755-880c-54f96c5e5156-3_465_748_1133_717} The diagram shows the curve with equation \(y = \cos ^ { - 1 } x\).

1. Sketch the curve with equation \(y = 3 \cos ^ { - 1 } ( x - 1 )\), showing the coordinates of the points where the curve meets the axes.
2. By drawing an appropriate straight line on your sketch in part (i), show that the equation \(3 \cos ^ { - 1 } ( x - 1 ) = x\) has exactly one root.
3. Show by calculation that the root of the equation \(3 \cos ^ { - 1 } ( x - 1 ) = x\) lies between 1.8 and 1.9 .
4. The sequence defined by $$x _ { 1 } = 2 , \quad x _ { n + 1 } = 1 + \cos \left( \frac { 1 } { 3 } x _ { n } \right)$$ converges to a number \(\alpha\). Find the value of \(\alpha\) correct to 2 decimal places and explain why \(\alpha\) is the root of the equation \(3 \cos ^ { - 1 } ( x - 1 ) = x\).

The point P \((5, 4)\) is on the curve \(y = f(x)\). State the coordinates of the image of P when the graph of \(y = f(x)\) is transformed to the graph of

1. \(y = f(x - 5)\), [2]
2. \(y = f(x) + 7\). [2]

2. The point \(P ( 2,3 )\) lies on the curve with equation \(y = \mathrm { f } ( x )\). State the coordinates of the image of \(P\) under the transformation represented by the curve with equation

1. \(y = \mathrm { f } ( x + 2 )\)
2. \(y = - \mathrm { f } ( x )\)
3. \(2 y = f ( x )\)
4. \(y = \mathrm { f } ( x ) - 4\) State the coordinates of the image of \(P\) under the transformation represented by the curve
   with equation (a) \(y = \mathrm { f } ( x + 2 )\)

9 Fig. 9 shows the curve \(y = \mathrm { f } ( x )\), which has a \(y\)-intercept at \(\mathrm { P } ( 0,3 )\), a minimum point at \(\mathrm { Q } ( 1,2 )\), and an asymptote \(x = - 1\). \begin{figure}[h] \end{figure}

1. Find the coordinates of the images of the points P and Q when the curve \(y = \mathrm { f } ( x )\) is transformed to
   (A) \(y = 2 \mathrm { f } ( x )\),
   (B) \(y = \mathrm { f } ( x + 1 ) + 2\). You are now given that \(\mathrm { f } ( x ) = \frac { x ^ { 2 } + 3 } { x + 1 } , x \neq - 1\).
2. Find \(\mathrm { f } ^ { \prime } ( x )\), and hence find the coordinates of the other turning point on the curve \(y = \mathrm { f } ( x )\).
3. Show that \(\mathrm { f } ( x - 1 ) = x - 2 + \frac { 4 } { x }\).
4. Find \(\int _ { a } ^ { b } \left( x - 2 + \frac { 4 } { x } \right) \mathrm { d } x\) in terms of \(a\) and \(b\). Hence, by choosing suitable values for \(a\) and \(b\), find the exact area enclosed by the curve \(y = \mathrm { f } ( x )\), the \(x\)-axis, the \(y\)-axis and the line \(x = 1\).

10 The curve \(y = \mathrm { f } ( x )\) has a minimum point at \(( 3,5 )\).
State the coordinates of the corresponding minimum point on the graph of

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

The curve C has equation \(y = f(x)\) C has a maximum point at P with coordinates \((a, 2b)\) as shown in the diagram below. \includegraphics{figure_7}

1. C is mapped by a single transformation onto curve \(C_1\) with equation \(y = f(x + 2)\) State the coordinates of the maximum point on curve \(C_1\) [1 mark]
2. C is mapped by a single transformation onto curve \(C_2\) with equation \(y = 4f(x)\) State the coordinates of the maximum point on curve \(C_2\) [1 mark]
3. C is mapped by a stretch in the \(x\)-direction onto curve \(C_3\) with equation \(y = f(3x)\) State the scale factor of the stretch. [1 mark]

8 \includegraphics[max width=\textwidth, alt={}, center]{88c7a3f3-e129-4e9c-acf8-8c96d2668d43-10_515_936_274_577} The diagram shows part of the graph of \(y = \sin ( a ( x + b ) )\), where \(a\) and \(b\) are positive constants.

1. State the value of \(a\) and one possible value of \(b\).
   Another curve, with equation \(y = \mathrm { f } ( x )\), has a single stationary point at the point \(( p , q )\), where \(p\) and \(q\) are constants. This curve is transformed to a curve with equation $$y = - 3 f \left( \frac { 1 } { 4 } ( x + 8 ) \right) .$$
2. For the transformed curve, find the coordinates of the stationary point, giving your answer in terms of \(p\) and \(q\).

The graph of \(y = f(x)\) is shown below. \includegraphics{figure_6}

1. Sketch the graph of \(y = f(-x)\) [2 marks]
2. Sketch the graph of \(y = 2f(x) - 4\) [2 marks]
3. Sketch the graph of \(y = f'(x)\) [3 marks]

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.

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)
   1. \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}
   2. \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}

Describe a sequence of transformations which maps the graph of $$y = |2x - 5|$$ onto the graph of $$y = |x|$$ [3 marks]

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.

A sequence of three transformations maps the curve \(y = \ln x\) to the curve \(y = \mathrm{e}^{3x} - 5\). Give details of these transformations. [4]

13 Prove by induction that, for all integers \(n \geq 1\) $$\sum _ { r = 1 } ^ { n } 2 ^ { - r } = 1 - 2 ^ { - n }$$ [4 marks]

Given that \(\text{f}(x) = x^3\) and \(\text{g}(x) = 2x^3 - 1\), describe a sequence of two transformations which maps the curve \(y = \text{f}(x)\) onto the curve \(y = \text{g}(x)\). [4]

2. Given that $$f ( x ) = \ln x , x > 0$$ Sketch on separate axes the graphs of
i) \(y = f ( x )\) ii) \(\quad y = f ( x - 4 )\) Show on each diagram, the point where the graph meets or crosses the \(x\)-axis. In each case, state the equation of the asymptote.

Figure 1 shows \(y = f(x)\). \includegraphics{figure_1} Which figure below shows \(y = f(2x)\)? Tick one box. \includegraphics{figure_2} \quad \includegraphics{figure_3} \quad \includegraphics{figure_4} \quad \includegraphics{figure_5} [1 mark]

Curve C has equation \(y = x^2\) C is translated by vector \(\begin{pmatrix} 3 \\ 0 \end{pmatrix}\) to give curve \(C_1\) Line L has equation \(y = x\) L is stretched by scale factor 2 parallel to the \(x\)-axis to give line \(L_1\) Find the exact distance between the two intersection points of \(C_1\) and \(L_1\) [6 marks]

3 The diagram shows the curve \(y = \mathrm { f } ( x )\). Points \(A , B , C\) and \(D\) on the curve have coordinates ( \(- 1,0 ) , ( 2,0 )\), \(( 5,0 )\) and \(( 0,2 )\) respectively. \includegraphics[max width=\textwidth, alt={}, center]{a31997f4-7890-42c1-9725-1b7058e8741f-2_593_1221_1041_406} On the copy of this diagram in the Printed Answer Book, sketch the curve \(y ^ { 2 } = \mathrm { f } ( x )\), giving the coordinates of the points where the curve crosses the axes.

The function \(f\) is defined by \(f(x) = \frac{8}{x^2}\).

1. Sketch the graph of \(y = f(x)\). [2]
2. On a separate set of axes, sketch the graph of \(y = f(x - 2)\). Indicate the vertical asymptote and the point where the curve crosses the \(y\)-axis. [3]
3. Sketch the graphs of \(y = \frac{8}{x}\) and \(y = \frac{8}{(x-2)^2}\) on the same set of axes. Hence state the number of roots of the equation \(\frac{8}{(x-2)^2} = \frac{8}{x}\). [2]

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.

Given that \(f(x) = \frac{1}{x}\), \(x \neq 0\),

1. sketch the graph of \(y = f(x) + 3\) and state the equations of the asymptotes. [4]
2. Find the coordinates of the point where \(y = f(x) + 3\) crosses a coordinate axis. [2]

2 The equation of a curve is \(y = e ^ { x }\). The curve is subject to a translation \(\binom { 3 } { 0 }\) and a stretch scale factor 2 parallel to the \(y\)-axis. Write down the equation of the new curve.

7 The diagram shows a part \(A B C\) of the curve \(y = 3 - 2 x ^ { 2 }\), together with its reflections in the lines \(y = x\), \(y = - x\) and \(y = 0\). \includegraphics[max width=\textwidth, alt={}, center]{65d9d34c-8c78-45fe-b9f0-dab071ae56bb-05_691_673_1957_678}

The curve \(y = \sqrt{2x - 1}\) is stretched by scale factor \(\frac{1}{4}\) parallel to the \(x\)-axis and by scale factor \(\frac{1}{2}\) parallel to the \(y\)-axis. Find the resulting equation of the curve, giving your answer in the form \(\sqrt{ax - b}\) where \(a\) and \(b\) are rational numbers. [3]

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

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

5. $$\mathrm { f } ( x ) \equiv 2 x ^ { 2 } + 4 x + 2 , \quad x \in \mathbb { R } , \quad x \geq - 1 .$$

1. Express \(\mathrm { f } ( x )\) in the form \(a ( x + b ) ^ { 2 } + c\).
2. Describe fully two transformations that would map the graph of \(y = x ^ { 2 } , x \geq 0\) onto the graph of \(y = \mathrm { f } ( x )\).
3. Find an expression for \(\mathrm { f } ^ { - 1 } ( x )\) and state its domain.
4. Sketch the graphs of \(y = \mathrm { f } ( x )\) and \(y = \mathrm { f } ^ { - 1 } ( x )\) on the same diagram and state the relationship between them.

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

On separate diagrams, sketch the graphs of

1. \(y = (x + 3)^2\), [3]
2. \(y = (x + 3)^2 + k\), where \(k\) is a positive constant. [2]

Show on each sketch the coordinates of each point at which the graph meets the axes.

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