Function Transformations

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

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

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. \(y = \mathrm { f } ( 4 x )\).

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

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

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

4 Fig. 4 shows the curve \(y = f ( x )\), where \(f ( x ) = \sqrt { 1 - 9 x ^ { 2 } } , - a \leqslant x \leqslant a\). \begin{figure}[h] \end{figure}

1. Find the value of \(a\).
2. Write down the range of \(\mathrm { f } ( x )\).
3. Sketch the curve \(y = \mathrm { f } \left( \frac { 1 } { 3 } x \right) - 1\).

3 The graph of the equation \(y = \frac { 1 } { x }\) is translated by the vector \(\left[ \begin{array} { l } 3
0 \end{array} \right]\)
3

1. Write down the equation of the transformed graph. 3
2. State the equations of the asymptotes of the transformed graph.

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.

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.

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

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.
\(y = \frac { 1 } { 2 } \ln x \quad y = 2 \ln x \quad y = \ln \frac { x } { 2 } \quad y = \ln 2 x\)

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

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.

3 The diagram below shows the graph of \(y = \mathrm { f } ( x )\).
\includegraphics[max width=\textwidth, alt={}, center]{2c30c315-c666-47d6-b05e-da71cb4decdd-05_755_1398_388_326}

1. On the diagram in the Printed Answer Booklet, draw the graph of \(y = \mathrm { f } \left( \frac { 1 } { 2 } x \right)\).
2. On the diagram in the Printed Answer Booklet, draw the graph of \(y = \mathrm { f } ( x - 2 ) + 1\).

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.

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

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}

3. Given that \(\quad \mathrm { f } ( x ) = \frac { 1 } { x } , \quad x \neq 0\),

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

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.

12 The mean value of the function f over the interval \(1 \leq x \leq 5\) is \(m\). The graph of \(y = \mathrm { g } ( x )\) is a reflection in the \(x\)-axis of \(y = \mathrm { f } ( x )\).
The graph of \(y = \mathrm { h } ( x )\) is a translation of \(y = \mathrm { g } ( x )\) by \(\left[ \begin{array} { l } 3
7 \end{array} \right]\)
Determine, in terms of \(m\), the mean value of the function h over the interval \(4 \leq x \leq 8\)

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