Multiple separate transformations (sketch-based, modulus involved)

Edexcel C3 2013 January Q3

9 marks Moderate -0.3

3. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{c78b0245-5c5a-407f-ad8a-602949a76e05-04_620_1095_223_420} \captionsetup{labelformat=empty} \caption{Figure 1}

\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.

Find the value of ff(-3). On separate diagrams, sketch the curve with equation
$y = \mathrm { f } ^ { - 1 } ( x )$,
$y = \mathrm { f } ( | x | ) - 2$,
$y = 2 \mathrm { f } \left( \frac { 1 } { 2 } x \right)$. Indicate clearly on each sketch the coordinates of the points at which the curve crosses or meets the axes.

Edexcel C3 2006 June Q3

9 marks Moderate -0.3

\begin{figure}[h]

\captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{f0f328ed-3550-4b8d-8b80-016df8773b21-04_568_881_312_504}

\end{figure} Figure 1 shows part of the curve with equation $y = \mathrm { f } ( x ) , x \in \mathbb { R }$, where f is an increasing function of $x$. The curve passes through the points $P ( 0 , - 2 )$ and $Q ( 3,0 )$ as shown. In separate diagrams, sketch the curve with equation

$y = | f ( x ) |$,
$y = \mathrm { f } ^ { - 1 } ( x )$,
$y = \frac { 1 } { 2 } \mathrm { f } ( 3 x )$. Indicate clearly on each sketch the coordinates of the points at which the curve crosses or meets the axes.

OCR C3 2008 June Q2

5 marks Moderate -0.3

2 \includegraphics[max width=\textwidth, alt={}, center]{5c501214-b41c-43a8-b9c6-986758e83e7d-2_529_855_397_646} The diagram shows the graph of $y = \mathrm { f } ( x )$. It is given that $\mathrm { f } ( - 3 ) = 0$ and $\mathrm { f } ( 0 ) = 2$. Sketch, on separate diagrams, the following graphs, indicating in each case the coordinates of the points where the graph crosses the axes:

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

AQA C3 2012 June Q7

11 marks Standard +0.3

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}

On Figure 2 on page 6, sketch the curve with equation $y = | \mathrm { f } ( x ) |$.
On Figure 3 on page 6, sketch the curve with equation $y = \mathrm { f } ( | x | )$.
Describe a sequence of two geometrical transformations that maps the graph of $y = \mathrm { f } ( x )$ onto the graph of $y = \frac { 1 } { 2 } \mathrm { f } ( x + 1 )$.
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}

AQA C3 2013 June Q7

9 marks Standard +0.3

7 The diagram shows a sketch of the curve with equation $y = \mathrm { f } ( x )$. \includegraphics[max width=\textwidth, alt={}, center]{063bbfa5-df49-44a1-8143-5e076397f63f-06_620_1216_356_422}

On Figure 1, below, sketch the curve with equation $y = - \mathrm { f } ( 3 x )$, indicating the values where the curve cuts the coordinate axes.
On Figure 2, on page 7, sketch the curve with equation $y = \mathrm { f } ( | x | )$, indicating the values where the curve cuts the coordinate axes.
Describe a sequence of two geometrical transformations that maps the graph of $y = \mathrm { f } ( x )$ onto the graph of $y = \mathrm { f } \left( - \frac { 1 } { 2 } x \right)$. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{063bbfa5-df49-44a1-8143-5e076397f63f-06_732_1237_1649_443}
\end{figure} \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 2} \includegraphics[alt={},max width=\textwidth]{063bbfa5-df49-44a1-8143-5e076397f63f-07_727_1211_340_466}
\end{figure}

AQA C3 2014 June Q4

11 marks Standard +0.3

4 The sketch shows part of the curve with equation $y = \mathrm { f } ( x )$. \includegraphics[max width=\textwidth, alt={}, center]{57412ec0-ad97-4418-8ba8-93f1f7d8aac1-08_536_1054_367_539}

On Figure 2 below, sketch the curve with equation $y = - | \mathrm { f } ( x ) |$.
On Figure 3 on the page opposite, sketch the curve with equation $y = \mathrm { f } ( | 2 x | )$.
1. Describe a sequence of two geometrical transformations that maps the graph of $y = \mathrm { f } ( x )$ onto the graph of $y = \mathrm { f } ( 2 x + 2 )$.
2. Find the coordinates of the image of the point $P ( 4 , - 3 )$ under the sequence of transformations given in part (c)(i). \begin{figure}[h]
  \captionsetup{labelformat=empty} \caption{Figure 3} \includegraphics[alt={},max width=\textwidth]{57412ec0-ad97-4418-8ba8-93f1f7d8aac1-09_778_1032_424_529}
  \end{figure}

Edexcel C3 Q7

14 marks Standard +0.2

7. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{a36989df-555f-4727-b6c6-e66362380011-4_481_808_248_424} \captionsetup{labelformat=empty} \caption{Figure 1}

\end{figure} Figure 1 shows the graph of $y = \mathrm { f } ( x )$ which meets the coordinate axes at the points $( a , 0 )$ and $( 0 , b )$, where $a$ and $b$ are constants.

Showing, in terms of $a$ and $b$, the coordinates of any points of intersection with the axes, sketch on separate diagrams the graphs of
1. $\quad y = \mathrm { f } ^ { - 1 } ( x )$,
2. $y = 2 \mathrm { f } ( 3 x )$. Given that $$\mathrm { f } ( x ) = 2 - \sqrt { x + 9 } , \quad x \in \mathbb { R } , \quad x \geq - 9 ,$$
find the values of $a$ and $b$,
find an expression for $\mathrm { f } ^ { - 1 } ( x )$ and state its domain.

Edexcel C3 Q27

10 marks Standard +0.3

\includegraphics{figure_1} Figure 1 shows a sketch of the curve with equation $y = f(x), -1 \leq x \leq 3$. The curve touches the $x$-axis at the origin $O$, crosses the $x$-axis at the point $A(2, 0)$ and has a maximum at the point $B(\frac{4}{3}, 1)$. In separate diagrams, show a sketch of the curve with equation

$y = f(x + 1)$, [3]
$y = |f(x)|$, [3]
$y = f(|x|)$, [4]

marking on each sketch the coordinates of points at which the curve

has a turning point,
meets the $x$-axis.

Edexcel C3 Q4

10 marks Standard +0.3

\includegraphics{figure_1} Figure 1 shows a sketch of the curve with equation $y = \text{f}(x)$, $-1 \leq x \leq 3$. The curve touches the $x$-axis at the origin $O$, crosses the $x$-axis at the point $A(2, 0)$ and has a maximum at the point $B(\frac{4}{3}, 1)$. In separate diagrams, show a sketch of the curve with equation

$y = \text{f}(x + 1)$, [3]
$y = |\text{f}(x)|$, [3]
$y = \text{f}(|x|)$, [4]

marking on each sketch the coordinates of points at which the curve

has a turning point,
meets the $x$-axis.

AQA Paper 3 2020 June Q6

7 marks Moderate -0.3

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

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

AQA Further Paper 1 2021 June Q7

7 marks Standard +0.3

The diagram below shows the graph of $y = \mathrm{f}(x)$ ($-4 \leq x \leq 4$) The graph meets the $x$-axis at $x = 1$ and $x = 3$ The graph meets the $y$-axis at $y = 2$ \includegraphics{figure_7}

Sketch the graph of $y = |\mathrm{f}(x)|$ on the axes below. Show any axis intercepts. [2 marks] \includegraphics{figure_7a}
Sketch the graph of $y = \frac{1}{\mathrm{f}(x)}$ on the axes below. Show any axis intercepts and asymptotes. [3 marks] \includegraphics{figure_7b}
Sketch the graph of $y = \mathrm{f}(|x|)$ on the axes below. Show any axis intercepts. [2 marks] \includegraphics{figure_7c}