Sketch modulus functions involving quadratic or other non-linear

3. (a) Sketch, on the same axes, the graph of \(y = | ( x - 2 ) ( x - 4 ) |\), and the line with equation \(y = 6 - 2 x\).
(b) Find the exact values of \(x\) for which \(| ( x - 2 ) ( x - 4 ) | = 6 - 2 x\).
(c) Hence solve the inequality \(| ( x - 2 ) ( x - 4 ) | < 6 - 2 x\).
(2)(Total 11 marks)

6. (a) On the same diagram, sketch the graphs of \(y = \left| x ^ { 2 } - 4 \right|\) and \(y = | 2 x - 1 |\), showing the coordinates of the points where the graphs meet the axes.
(b) Solve \(\left| x ^ { 2 } - 4 \right| = | 2 x - 1 |\), giving your answers in surd form where appropriate.
(c) Hence, or otherwise, find the set of values of \(x\) for which \(\left| x ^ { 2 } - 4 \right| > | 2 x - 1 |\).
(3)(Total 12 marks)

3. (a) Use algebra to find the exact solutions of the equation $$\left| 2 x ^ { 2 } + x - 6 \right| = 6 - 3 x$$ (b) On the same diagram, sketch the curve with equation \(y = \left| 2 x ^ { 2 } + x - 6 \right|\) and the line with equation \(y = 6 - 3 x\).
(c) Find the set of values of \(x\) for which $$\left| 2 x ^ { 2 } + x - 6 \right| > 6 - 3 x$$ (3)(Total 12 marks)

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 $$4 - x ^ { 2 } > | 2 x - 3 |$$

1. (a) Use algebra to find the exact solutions of the equation

$$\left| 2 x ^ { 2 } + 6 x - 5 \right| = 5 - 2 x$$ (b) On the same diagram, sketch the curve with equation \(y = \left| 2 x ^ { 2 } + 6 x - 5 \right|\) and the line with equation \(y = 5 - 2 x\), showing the \(x\)-coordinates of the points where the line crosses the curve.
(c) Find the set of values of \(x\) for which $$\left| 2 x ^ { 2 } + 6 x - 5 \right| > 5 - 2 x$$

7. (i) 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.
(ii) Find the exact solutions of the equation $$4 - x ^ { 2 } = | 2 x - 1 |$$

7

1. On separate diagrams:
   1. sketch the curve with equation \(y = | 3 x + 3 |\);
   2. sketch the curve with equation \(y = \left| x ^ { 2 } - 1 \right|\).
2. 1. Solve the equation \(| 3 x + 3 | = \left| x ^ { 2 } - 1 \right|\).
   2. Hence solve the inequality \(| 3 x + 3 | < \left| x ^ { 2 } - 1 \right|\). \(8 \quad\) Use the substitution \(u = 1 + 2 \tan x\) to find $$\int \frac { 1 } { ( 1 + 2 \tan x ) ^ { 2 } \cos ^ { 2 } x } d x$$

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 $$4 - x ^ { 2 } = | 2 x - 1 |$$