1.02s Modulus graphs: sketch graph of |ax+b|

7

1. Sketch the graph of \(y = | 2 x |\).
2. On a separate diagram, sketch the graph of \(y = 4 - | 2 x |\), indicating the coordinates of the points where the graph crosses the coordinate axes.
3. Solve \(4 - | 2 x | = x\).
4. Hence, or otherwise, solve the inequality \(4 - | 2 x | > x\).

7

1. Describe a sequence of two geometrical transformations that maps the graph of \(y = x ^ { 2 }\) onto the graph of \(y = 4 x ^ { 2 } - 5\).
2. Sketch the graph of \(y = \left| 4 x ^ { 2 } - 5 \right|\), indicating the coordinates of the point where the curve crosses the \(y\)-axis.
3. 1. Solve the equation \(\left| 4 x ^ { 2 } - 5 \right| = 4\).
   2. Hence, or otherwise, solve the inequality \(\left| 4 x ^ { 2 } - 5 \right| \geqslant 4\).

5

1. Describe a sequence of two geometrical transformations that maps the graph of \(y = \ln x\) onto the graph of \(y = 4 \ln ( x - \mathrm { e } )\).
2. Sketch, on the axes given below, the graph of \(y = | 4 \ln ( x - \mathrm { e } ) |\), indicating the exact value of the \(x\)-coordinate where the curve meets the \(x\)-axis.
3. 1. Solve the equation \(| 4 \ln ( x - e ) | = 4\).
   2. Hence, or otherwise, solve the inequality \(| 4 \ln ( x - e ) | \geqslant 4\). \includegraphics[max width=\textwidth, alt={}, center]{7aa76d26-e3c4-4374-ae4f-8bb61e61b135-3_655_1428_2023_315}

4

1. Sketch and label on the same set of axes the graphs of:
   1. \(y = | x |\);
   2. \(y = | 2 x - 4 |\).
2. 1. Solve the equation \(| x | = | 2 x - 4 |\).
   2. Hence, or otherwise, solve the inequality \(| x | > | 2 x - 4 |\).

4

1. Sketch the graph of \(y = \left| 50 - x ^ { 2 } \right|\), indicating the coordinates of the point where the graph crosses the \(y\)-axis.
2. Solve the equation \(\left| 50 - x ^ { 2 } \right| = 14\).
3. Hence, or otherwise, solve the inequality \(\left| 50 - x ^ { 2 } \right| > 14\).
4. Describe a sequence of two geometrical transformations that maps the graph of \(y = x ^ { 2 }\) onto the graph of \(y = 50 - x ^ { 2 }\).

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}

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}

1. On Figure 2 below, sketch the curve with equation \(y = - | \mathrm { f } ( x ) |\).
2. On Figure 3 on the page opposite, sketch the curve with equation \(y = \mathrm { f } ( | 2 x | )\).
3. 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}

7. (a) Sketch on the same diagram the graphs of \(y = 4 a ^ { 2 } - x ^ { 2 }\) and \(y = | 2 x - a |\), where \(a\) is a positive constant. Show, in terms of \(a\), the coordinates of any points where each graph meets the coordinate axes.
(b) Find the exact solutions of the equation $$4 - x ^ { 2 } = | 2 x - 1 |$$

3

1. The diagram below shows the graphs of \(y = | 3 x - 2 |\) and \(y = | 2 x + 1 |\). \includegraphics[max width=\textwidth, alt={}, center]{e3942549-bfc0-432a-bf49-7d01d44af01a-4_423_682_1110_694} On the diagram in your Printed Answer Booklet, give the coordinates of the points of intersection of the graphs with the coordinate axes.
2. Solve the equation \(| 2 x + 1 | = | 3 x - 2 |\).

4

1. Sketch the graph of \(y = | 8 - 2 x |\).
2. Solve the equation \(| 8 - 2 x | = 4\).
3. Solve the inequality \(| 8 - 2 x | > 4\).

3

1. Solve the equation \(\operatorname { cosec } x = 2\), giving all values of \(x\) in the interval \(0 ^ { \circ } < x < 360 ^ { \circ }\).
   (2 marks)
2. The diagram shows the graph of \(y = \operatorname { cosec } x\) for \(0 ^ { \circ } < x < 360 ^ { \circ }\). \includegraphics[max width=\textwidth, alt={}, center]{9fd9fa54-b0e6-413d-8645-de34b99b859a-03_609_1045_559_479}
   1. The point \(A\) on the curve is where \(x = 90 ^ { \circ }\). State the \(y\)-coordinate of \(A\).
   2. Sketch the graph of \(y = | \operatorname { cosec } x |\) for \(0 ^ { \circ } < x < 360 ^ { \circ }\).
3. Solve the equation \(| \operatorname { cosec } x | = 2\), giving all values of \(x\) in the interval \(0 ^ { \circ } < x < 360 ^ { \circ }\).
   (2 marks)

2

1. Sketch, on the axes below, the curve with equation \(y = 4 - | 2 x + 1 |\), indicating the coordinates where the curve crosses the axes.
2. Solve the equation \(x = 4 - | 2 x + 1 |\).
3. Solve the inequality \(x < 4 - | 2 x + 1 |\).
4. Describe a sequence of two geometrical transformations that maps the graph of \(y = | 2 x + 1 |\) onto the graph of \(y = 4 - | 2 x + 1 |\).
   [0pt] [4 marks] \section*{Answer space for question 2}
   1. \includegraphics[max width=\textwidth, alt={}, center]{2df59047-3bfe-4b9c-a77f-142bc7506cbc-04_851_1459_1000_319}

4

1. Sketch the graph of \includegraphics[max width=\textwidth, alt={}, center]{08e1f291-7052-40a5-b7b2-13fd1d0137c2-04_933_1093_349_475} 4
2. Solve the inequality $$4 - | 2 x - 6 | > 2$$

4

1. On Figure 1 add a sketch of the graph of $$y = | 3 x - 6 |$$ 4
2. Find the coordinates of the points of intersection of the two graphs.
   Fully justify your answer. \includegraphics[max width=\textwidth, alt={}, center]{b7df05bf-f3fc-4705-a13c-6b562896fa9f-05_2488_1716_219_153}

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.

\includegraphics{figure_1} Figure 1 shows a sketch of the curve with equation \(y = e^{-x} - 1\).

1. Copy Fig. 1 and on the same axes sketch the graph of \(y = \frac{1}{2}|x - 1|\). Show the coordinates of the points where the graph meets the axes. [2]

The \(x\)-coordinate of the point of intersection of the graphs is \(\alpha\).

1. Show that \(x = \alpha\) is a root of the equation \(x + 2e^{-x} - 3 = 0\). [3]
2. Show that \(-1 < \alpha < 0\). [2]

The iterative formula \(x_{n+1} = -\ln[\frac{1}{2}(3 - x_n)]\) is used to solve the equation \(x + 2e^{-x} - 3 = 0\).

1. Starting with \(x_0 = -1\), find the values of \(x_1\) and \(x_2\). [2]
2. Show that, to 2 decimal places, \(\alpha = -0.58\). [2]

The function f is defined by $$f: x \mapsto |2x - a|, \quad x \in \mathbb{R}$$ where \(a\) is a positive constant.

1. Sketch the graph of \(y = f(x)\), showing the coordinates of the points where the graph cuts the axes. [2]
2. On a separate diagram, sketch the graph of \(y = f(2x)\), showing the coordinates of the points where the graph cuts the axes. [2]
3. Given that a solution of the equation f(x) = \(\frac{1}{2}x\) is \(x = 4\), find the two possible values of \(a\). [4]

By sketching the graphs of \(y = |2x + 1|\) and \(y = -x\) on the same axes, show that the equation \(|2x + 1| = -x\) has two roots. Find these roots. [4]