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CAMBRIDGE IGCSETM AND O LEVEL
ADDITIONAL MATHEMATICS (0606/4037)
STUDENT’S BOOK SECOND EDITION
Student’s Book Second edition
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Using graphs to solve cubic inequalities
4 EQUATIONS, INEQUALITIES AND GRAPHS
Using graphs to solve cubic
inequalities
Solve the inequality 3(x + 2)(x − 1)(x − 7) −100 graphically.
Cubic graphs have distinctive shapes determined by the coefficient of x³.
Solution
Worked example
Because you are solving the inequality graphically, you will need to draw
the curve as accurately as possible on graph paper, so start by drawing up a
table of values.
3
3
Positive x term
Negative x term
y = 3(x + 2)(x − 1)(x − 7)
x
(x + 2)
The centre part of each of these curves may not have two distinct
turning points like those shown above, but may instead ‘flatten out’
to give a point of inflection. When the modulus of a cubic function is
required, any part of the curve below the x-axis is reflected in that axis.
(x − 1)
(x − 7)
x
−3
−2
View sample material
from our Student’s
Books
2
2
3
3
4
4
5
5
8
0
5
6
−7
−6
−5
−4
−3
−2
−1
0
1
48
42
0
−60
−120
−162
−168
−120
0
210
2
7
7
9
−1
−8
0
1
6
6
8
−2
−9
3
4
10
7
The solution is given by the values of x that correspond to the parts of the
curve on or below the line y = −100.
a Sketch the graph of y = 3(x + 2)(x − 1)(x − 7). Identify the points where the
You are asked for
curve cuts the axes.
a sketch graph,
b Sketch the graph of y = |3(x + 2)(x − 1)(x − 7)|.
so although it
must show the
Solution
main features,
a The curve crosses the x-axis at −2, 1 and 7. Notice that the distance
it does not need
betweenINEQUALITIES
consecutive points
is 3GRAPHS
and 6 units, respectively, so the y-axis is
4 EQUATIONS,
AND
to be absolutely
between the points −2 and 1 on the x-axis, but closer to the 1.
accurate. You
The curve crosses the y-axis when x = 0, i.e. when y = 3(2)(−1)(−7) = 42.
may find it easier
1
y
to draw the curve
5 a Use the substitution x = u 2 to solve the equation x 4 − 5 x 2 + 4 = 0.
b 42
Using the same substitution, show that the equation x 4 + 5 x 2 + 4 = 0
first, with the
has
positive x³ term
–2
1 no solution.
7 x
c Solve where possible:
determining the
4
2
4
2
y = 3(x + 2) (x – 1) (x – 7)
ii x 3 + 5 x 3 + 4 = 0
i x 3 − 5x 3 + 4 = 0
shape of the curve,
6 Sketch the following graphs, indicating the points where they cross
and then position
the x-axis:
the x-axis so that
a y = x(x – 2)(x + 2)
b y = |x(x – 2)(x + 2)|
b To obtain a sketch of the modulus curve, reflect any part of the curve that
the distance
c y = 3(2 x – 1)(x + 1)(x + 3)
d y = |3(2 x – 1)(x + 1)(x + 3)|
is below the x-axis in the x-axis.
between the
7 Solve the following equations graphically. You will need to use graph
y
first and second
paper.
b x(x + 2)(x − 3) −1
a x(x + 2)(x − 3) 1
intersections is
y = | 3(x + 2) (x – 1) (x – 7) |
c (x + 2)(x − 1)(x − 3) > 2
d (x + 2)(x − 1)(x − 3) < −2
about half that
8 Identify the following cubic graphs:
between the
42
second and third,
b
a
y
y
since these are
4
4
–2
1
7 x
3 and 6 units,
3
3
respectively.
2
2
1
y
50
–2.9
2.6
–1 O
–7 –6 –5 –4 –3 –2
1
2
3
6.2
4
5
6
7
8
x
Using graphs to solve cubic inequalities
–50
y = – 100
–100
y = 3(x + 2) (x – 1) (x – 7)
Past-paper questions
–150
1 (i)
Sketch the graph of y = |(2x + 3)(2x − 7)|.
(ii) How many values of x satisfy the equation
–200 |(2x + 3)(2x − 7)| = 2x?
Paper 23 Q6 November 2011
Remember: ( x )
means the positive
square root of x.
IGCSE
Additional Mathematics 0606
1 Where possible, use the substitution x = Cambridge
u ² to solve the
following
equations:
Paper 23 Q6 November 2011
a x − 4 x = −4 2 (i) On a grid like
b xthe
+ 2one
x below,
=8
sketch the graph of
c x − 2 x = 15
d +x3)|
+ 6forx−5=
−5x 4, and state the coordinates of
y = |(x − 2)(x
2
1
pointsthe
where
the curve
meets
x3 + 3
x 3 = the
4 . coordinate axes.
2 Use the substitution x = u 3the
to solve
equation
4
3
[4]
2
3 Use the substitution x = u 2 to solve the equation x 3 y− 10 x 3 = −9.
4 Using a suitable substitution, solve the following equations:
a x − 7 x = −12
c
2
b x −2 x +1= 0
1
x 3 + 3 x 3 = 10
1
5
O
–3 –2 –1
–1
1
2
3
x
–2
–2
–1
O
–1
1
x
–5
–4
–3
–2
–1
O
1
2
3
4
x
–2
–3
–3
9 Identify these graphs. (They are the moduli of cubic graphs.)
a
b
y
c
y
y
12
12
12
11
11
11
10
10
10
9
9
9
8
8
8
7
7
7
6
6
6
5
5
5
4
4
4
3
3
3
2
2
2
1
1
1
1
2
3
4
x
O
–5 –4 –3 –2 –1
–1
1
2
3
4
x –3 –2 –1 O
–1
–2
–2
–2
–3
–3
–3
6
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[4]
[2]
From the graph, the solution is x −2.9 or 2.6Cambridge
x 6.2. O Level Additional Mathematics 4037
Exercise 4.3
4
O
–3 –2 –1
–1
48
1
1
0
−3
Worked example
Look inside
0
−1
−1
−4
−10
−120
(ii) Find the coordinates of the stationary point on the curve
y = |(x − 2)(x + 3)|.
[2]
(iii) Given that k is a positive constant, state the set of values of
k for which |(x − 2)(x + 3)| = k has 2 solutions only.
[1]
Cambridge O Level Additional Mathematics 4037
Paper 12 Q8 November 2013
Cambridge IGCSE Additional Mathematics 0606
Paper 12 Q8 November 2013
3 Solve the inequality 9x 2 + 2x − 1 < (x + 1)2 .
[3]
Cambridge O Level Additional Mathematics 4037
Paper 22 Q2 November 2014
Cambridge IGCSE Additional Mathematics 0606
Paper 22 Q2 November 2014
1
2
3
4
x
7
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