Modulus function

12. Given that \(k\) is a positive constant,

1. sketch the graph with equation $$y = 2 | x | - k$$ Show on your sketch the coordinates of each point at which the graph crosses the \(x\)-axis and the \(y\)-axis.
2. Find, in terms of \(k\), the values of \(x\) for which $$2 | x | - k = \frac { 1 } { 2 } x + \frac { 1 } { 4 } k$$

1 Identify the graph of \(y = 1 - | x + 2 |\) from the options below.
Tick ( \(\checkmark\) ) one box. \includegraphics[max width=\textwidth, alt={}, center]{838f0625-95e6-4ad4-b97b-3d3f77cc7f19-02_389_526_845_500}
B \includegraphics[max width=\textwidth, alt={}, center]{838f0625-95e6-4ad4-b97b-3d3f77cc7f19-02_362_442_1279_525} \includegraphics[max width=\textwidth, alt={}, center]{838f0625-95e6-4ad4-b97b-3d3f77cc7f19-02_113_116_977_1107}
C \includegraphics[max width=\textwidth, alt={}, center]{838f0625-95e6-4ad4-b97b-3d3f77cc7f19-02_496_704_1688_523}
D \includegraphics[max width=\textwidth, alt={}, center]{838f0625-95e6-4ad4-b97b-3d3f77cc7f19-02_474_686_2211_534}

8. \begin{figure}[h] \end{figure} Figure 2 shows a sketch of the graph with equation $$y = 2 | x + 4 | - 5$$ The vertex of the graph is at the point \(P\), shown in Figure 2.

1. Find the coordinates of \(P\).
2. Solve the equation $$3 x + 40 = 2 | x + 4 | - 5$$ A line \(l\) has equation \(y = a x\), where \(a\) is a constant.
   Given that \(l\) intersects \(y = 2 | x + 4 | - 5\) at least once,
3. find the range of possible values of \(a\), writing your answer in set notation.
   [0pt] [BLANK PAGE]

1 Solve the inequality \(| x + 1 | > | x |\).

1 Solve the inequality \(| x + 1 | > | x |\).

1 Solve the inequality \(| 5 x + 7 | > | 2 x - 3 |\).

2 Solve the inequality \(| 3 x + 2 | < | x |\).

3

1. In this question you must show detailed reasoning.
   Solve the inequality \(| x - 2 | \leqslant | 2 x - 6 |\).
2. Give full details of a sequence of two transformations needed to transform the graph of \(y = | x - 2 |\) to the graph of \(y = | 2 x - 6 |\).

1 Solve the inequality \(| 3 x - 7 | < | 4 x + 5 |\).

7 Solve the inequality \(| x - 1 | < 3\).

1 Solve the inequality \(| x - 1 | < 3\).

7 The function f is defined by $$f ( x ) = \left| \sin x + \frac { 1 } { 2 } \right| \quad ( 0 \leq x \leq 2 \pi )$$ Find the set of values of \(x\) for which $$f ( x ) \geq \frac { 1 } { 2 }$$ Give your answer in set notation.

1 Solve the equation \(| 0.4 x - 0.8 | = 2\).

1 Solve the equation \(| 0.4 x - 0.8 | = 2\).

6. Given that \(k\) is a positive constant,

1. on separate diagrams, sketch the graph with equation
   1. \(y = k - 2 | x |\)
   2. \(y = \left| 2 x - \frac { k } { 3 } \right|\) Show on each sketch the coordinates, in terms of \(k\), of each point where the graph meets or cuts the axes.
2. Hence find, in terms of \(k\), the values of \(x\) for which $$\left| 2 x - \frac { k } { 3 } \right| = k - 2 | x |$$ giving your answers in simplest form. \includegraphics[max width=\textwidth, alt={}, center]{76205772-5395-4ab2-96f9-ad9803b8388c-23_2647_1840_118_111}

1 Solve the inequality \(| x + 3 a | > 2 | x - 2 a |\), where \(a\) is a positive constant.

1 Solve the equation \(| x + a | = | 2 x - 5 a |\), giving \(x\) in terms of the positive constant \(a\).

1 Solve the inequality \(| x + 3 a | > 2 | x - 2 a |\), where \(a\) is a positive constant.

4 Sketch the graph of \(y = | 2 x - 3 |\).

4 Sketch the graph of \(y = | 2 x - 3 |\).

1．（a）Sketch the graph of the curve with equation $$y = | \ln ( 2 x + 5 ) | \quad x > - \frac { 5 } { 2 }$$ On your sketch you should clearly state the equations of any asymptotes and mark the coordinates of points where the curve meets the coordinate axes．
（b）Solve the equation \(| \ln ( 2 x + 5 ) | = \ln 9\)

1 Solve the inequality \(| x - 3 | > 2 | x + 1 |\).

1 Solve the inequality \(2 | x - 2 | > | 3 x + 1 |\).

1 Solve the inequality \(| x - 2 | > 3 | 2 x + 1 |\).

2 Solve the inequality \(2 x > | x - 1 |\).

1 Solve the inequality \(| 2 x - 5 | > x\).

2 Solve the inequality \(| x - 7 | > 4 x + 3\).

1 Solve the equation \(| 3 x + 2 | = 1\).

1 Solve the equation \(| 2 x - 3 | = 9\).

1. The functions \(f\) and \(g\) are defined by

$$\begin{aligned} & \mathrm { f } : x \rightarrow | 2 x - 5 | , \quad x \in \mathbb { R } , \\ & \mathrm {~g} : x \rightarrow \ln ( x + 3 ) , \quad x \in \mathbb { R } , \quad x > - 3 \end{aligned}$$

1. State the range of f .
2. Evaluate fg(-2).
3. Solve the equation $$\operatorname { fg } ( x ) = 3$$ giving your answers in exact form.
4. Show that the equation $$\mathrm { f } ( x ) = \mathrm { g } ( x )$$ has a root, \(\alpha\), in the interval [3,4].
5. Use the iterative formula $$x _ { n + 1 } = \frac { 1 } { 2 } \left[ 5 + \ln \left( x _ { n } + 3 \right) \right]$$ with \(x _ { 0 } = 3\), to find \(x _ { 1 } , x _ { 2 } , x _ { 3 }\) and \(x _ { 4 }\), giving your answers to 4 significant figures.
6. Show that your answer for \(x _ { 4 }\) is the value of \(\alpha\) correct to 4 significant figures.

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

4. \begin{figure}[h] \end{figure} Figure 1 shows a sketch of the curve with equation \(y = \mathrm { f } ( x )\).
The curve passes through the origin \(O\) and the points \(A ( 5,4 )\) and \(B ( - 5 , - 4 )\).
In separate diagrams, sketch the graph with equation

1. \(y = | f ( x ) |\),
2. \(y = \mathrm { f } ( | x | )\),
3. \(y = 2 f ( x + 1 )\). On each sketch, show the coordinates of the points corresponding to \(A\) and \(B\).

2．（a）On separate diagrams，sketch the curves with the following equations．On each sketch you should label the exact coordinates of the points where the curve meets the coordinate axes．
（i）\(y = 8 + 2 x - x ^ { 2 }\) （ii）\(y = 8 + 2 | x | - x ^ { 2 }\) （iii）\(y = 8 + x + | x | - x ^ { 2 }\) （b）Find the values of \(x\) for which $$\left| 8 + x + | x | - x ^ { 2 } \right| = 8 + 2 | x | - x ^ { 2 }$$

3 Solve $$x ^ { 2 } \geqslant | 5 x - 6 |$$ [5 marks]

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 \(\left| 4 x ^ { 2 } - 5 \right| \geqslant 4\).

1. In this question you must show all stages of your working.

Solutions relying entirely on calculator technology are not acceptable.
Use algebra to determine the set of values of \(x\) for which $$\frac { x ^ { 2 } - 9 } { | x + 8 | } > 6 - 2 x$$

2 Solve the equation \(2 \left| 3 ^ { x } - 1 \right| = 3 ^ { x }\), giving your answers correct to 3 significant figures.

1

1. Solve the equation \(| 3 x - 2 | = 5\).
2. Hence, using logarithms, solve the equation \(\left| 3 \times 5 ^ { y } - 2 \right| = 5\), giving the answer correct to 3 significant figures.

4

1. Sketch, on the same diagram, the graphs of \(y = | 3 x - 5 |\) and \(y = 2 x + 7\).
2. Solve the equation \(| 3 x - 5 | = 2 x + 7\).
3. Hence solve the equation \(\left| 3 ^ { y + 1 } - 5 \right| = 2 \times 3 ^ { y } + 7\), giving your answer correct to 3 significant figures.

3 It is g n th t \(a\) is a p itie co tan.

1. 1. Sketch sib ed ag am th g ad \(6 y = | 2 x - 3 a |\) ad \(y = | 2 x + 4 a |\).
   2. State th co dia tes

3 It is g n th t \(a\) is a p itie co tan.

1. 1. Sketch sib ed ag am th g ad \(6 y = | 2 x - 3 a |\) ad \(y = | 2 x + 4 a |\).
   2. State th co dia tes

6. (i) Sketch on the same diagram the graphs of \(y = | x | - a\) and \(y = | 3 x + 5 a |\), where \(a\) is a positive constant. Show on your diagram the coordinates of any points where each graph meets the coordinate axes.
(ii) Solve the equation $$| x | - a = | 3 x + 5 a |$$

1

1. Sketch, on the same diagram, the graphs of \(y = | 3 x - 5 |\) and \(y = x + 2\).
2. Solve the equation \(| 3 x - 5 | = x + 2\).

5

1. Sketch, on the same diagram, the graphs of \(y = | 2 x - 3 |\) and \(y = 3 x + 5\).
2. Solve the inequality \(3 x + 5 < | 2 x - 3 |\).

3

1. Sketch on the same diagram the graphs of \(y = | 3 x - 8 |\) and \(y = 5 - x\).
2. Solve the inequality \(| 3 x - 8 | < 5 - x\).
3. Hence determine the largest integer \(N\) satisfying the inequality \(\left| 3 e ^ { 0.1 N } - 8 \right| < 5 - e ^ { 0.1 N }\).

2. (a) Sketch, on the same axes,

1. \(y = | 2 x - 3 |\)
2. \(y = 4 - x ^ { 2 }\) (b) Find the set of values of \(x\) for which $$4 - x ^ { 2 } > | 2 x - 3 |$$

2

1. Sketch the graph of \(y = | 3 x - 7 |\), stating the coordinates of the points where the graph meets the axes.
2. Hence find the set of values of the constant \(k\) for which the equation \(| 3 \mathrm { x } - 7 | = \mathrm { k } ( \mathrm { x } - 4 )\) has exactly two real roots.

2 Given that \(x\) satisfies the equation \(| 2 x + 3 | = | 2 x - 1 |\), find the value of $$| 4 x - 3 | - | 6 x |$$

1 Solve the inequality \(\left| 2 ^ { x } - 8 \right| < 5\).

3. (a) Find the set of values of \(x\) for which $$x + 4 > \frac { 2 } { x + 3 }$$ (b) Deduce, or otherwise find, the values of \(x\) for which $$x + 4 > \frac { 2 } { | x + 3 | }$$

4

1. Sketch, on a single diagram, the following graphs.
   • \(y = | x - 1 |\)
   • \(y = \frac { k } { x }\), where \(k\) is a negative constant
   • Hence explain why the equation \(x | x - 1 | = k\) has exactly one real root for any negative value of \(k\).
   • Determine the real root of the equation \(x | x - 1 | = - 6\).

2 Express \(1 < x < 3\) im th \(\quad | x - a | < b\), where \(a\) and \(b\) are to be determined.