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Squares and cubes
Quadratic graphs
Cubic graphs
Solving equations
Mastering Mathematics © Hodder and Stoughton 2014
Plotting quadratic and cubic graphs– Developing Understanding
Squares and cubes
You have already met square numbers and cube numbers.
32, or 3 squared, means 3 × 3 = 9.
There are 9 squares in the picture.
23, or 2 cubed, means 2 × 2 × 2 = 8.
There are 8 cubes in the picture.
You are going to draw graphs with terms in x2 and x3.
To do this you will need to use negative (-) values of x as
well as positive (+).
1. What is the value of (-8)2?
64 
-64 
2. What is the value of (-2)3?
8
-8 
Squaring means multiplying
4 means 2×2×2×2.
2a
number by itself.
25 means
72 =2×2×2×2×2.
7 × 7 = 49
Cubing
a number
Find
the value
of 24means
, (-2)4, 25
multiplying
and
(-2)5. three of the
numbers together.
43 = 4 × 4 × 4 = 64
Opinion  is correct. (-2)3 = (-2) × (-2) × (-2)
Opinion  is correct.
=
4
× (-2)
2
(-8) = (-8) × (-8) = +64
= -8
Remember
two negatives
makes a numbers
positive.
In this case multiplying
you are multiplying
three negative
Opinion
 answer
has a negative
answer
and so
is wrong.
and so the
is negative.
Opinion
 ithas
a positive
answer so it is wrong.
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Q1
Opinion 1 Opinion 2 Answer
Q2
Opinion 1 Opinion 2 Answer
Plotting quadratic and cubic graphs– Developing Understanding
Quadratic graphs
Complete the table for y = x2 + x.
Many equations do not produce straight
lines.
Here is a table showing values of x and y
when y = x2
A xquadratic
is one
the 3
-3 graph
-2 -1
0 where
1 2
highest power of x 2is x2.
2
x
1=x
0 1 4 9
x 9 4Area
x
-3
-2
-1
0
1
2
3
y = x2
9
4
1
0
1
4
9
y =Any
x2 +other
x 6terms are either in x or a 12
constant (a number).
Will y have any negative values?
x
1. How can you tell from the table that the
graph of y = x2 is not a straight line?
2. Does y = x2 ever have any negative values
of y?
When
the values
of x
No,
a square
is always
increase in equal steps,
positive or zero. 
the values of y do not. 
The
Yes,values
when of
x isy
decrease
between at
-1
first, and then
and 0. 
increase. 
Both
Opinions
correct. They
are saying
No. Opinion
are
is incorrect,
squaring
a
the
samenumber
thing in always
differentresults
ways. in
Opinion

negative
a positive
is
more precise.
number.
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Mastering Mathematics © Hodder and Stoughton 2014
Q1
Opinion 1 Opinion 2 Answer
Q2
Opinion 1 Opinion 2 Answer
Plotting quadratic and cubic graphs– Developing Understanding
Quadratic graphs
y
The graph shows y = x2. It is symmetrical.
16
14
1. Where is the line of symmetry of y = x2?
12
10
2. Here is the table for y = x2 + 3x – 3
8
x
-3
-2
-1
0
1
2
3
6
x2
9
4
1
0
1
4
9
4
+3x
-9
-6
-3
0
3
6
9
2
-3
-3
-3
-3
-3
-3
-3
-3
y = x2 +x - 3
-3
-5
-5
-3
1
7
15
x
0
-3
-2
-1
0
1
2
3
-2
Draw the graph of y = x2 + 3x – 3.
-4
-6
y =curve.
0
The purple

The
y-axis.
The
blue
curve. 
The correct
answer
is ythe
purple
It is the
y axis.
=0
is thecurve
Opinionequation
 incorrectly
of the assumes
x-axis. the curve is
symmetrical about the y-axis.
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Mastering Mathematics © Hodder and Stoughton 2014
Q1
The graphs of y = x2 and y = x2 + 3x – 3
are identical in shape.
Draw the graph of y = x2 – x – 2.
Is it the same shape?
Opinion 1 Opinion 2 Answer
Q2
Opinion 1 Opinion 2 Answer
Plotting quadratic and cubic graphs– Developing Understanding
Quadratic graphs
y
All quadratic graphs have the same U-shape.
2
For this table you
have to work out the
value of -x2.So 1when x is negative there
x
are three minuses,
making a minus
0
-2
-1
0
1
2
3
-1  made the mistake of
overall. Opinion
making the -x2-2terms positive. Opinion 
-3
is correct.
Look at the equation y = 2x – x2.
The x2 term is negative.
1. Complete the table for y = 2x – x2.
x
-2
-1
0
2x
-4
-2
0
-x2
-4
-1
y = 2x – x2
1
2
3
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0
-1
0
1
2
3
-4
-2
0
2
4
6
2x
0
-x2
-7-4
-1
0
-1
-4
-9
y = 2x – x2
-8-8
-3
0
1
0
-3
2. Here is the graph of y = 2x –
What can you say about its shape?
y = 2x – x2
-2
-5
0
x2.
It
shape
is the same
x
-2
as all the quadratic
2x it is -4
curves, but
upside down.
-x2
4
-4
x
-1 It0 is the
1 same
2
3
shape as all the
-2 quadratic
0
2 4 curves.
6

-6
Draw
theusual
graphU-shape
of y = x2but
– 2x.
does
It is the
hasWhere
been turned
2
this
curve
cross
y =is2xbecause
– x ? the x2 term is
upside
down.
This
negative instead of positive.
x
-2
-1
0
1
2
3
2x
-4
-2
0
2
4
6
1
0
1
4
9
-x2
-4
-1
0
1
4
9
-1
0
3
8
12
y = 2x – x2
-8
-3
0
3
8
15
Q1
Opinion 1 Opinion 2 Answer
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Mastering Mathematics © Hodder and Stoughton 2014
Q2
Opinion 1 Opinion 2 Answer
Plotting quadratic and cubic graphs– Developing Understanding
Cubic graphs
y
Cubic graphs have their own distinctive shape.
30
20
1. Compete this table of values for y = x3.
x
-3 -2 -1 0
1 2 3
y = x3
1
10
27
x
0
-3
-2
-1
0
1
2
3
-10
2. Describe the gradient of the graph
-20
 It decreases
x
-3at -2
first and
then
y = x3 27 8
increases.

-1 It 0is always
1
2
1 increasing.
0
1
8
3
-30
27
x
-3
-2
-1
0
1
2
3
y = x3
-27
-8
-1
0
1
8
27
3 look like?
What
willequation
the graph
of
The
gradient
of the
graph
isits
always
positive,
A
cubic
has
xy3 =as-x
highest
power
except
of
x. at x = 0 where it is 0. It decreases
until x = 0 and then increases.
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Mastering Mathematics © Hodder and Stoughton 2014
Q1
x
-3
-2
-1
0
1
2
3
y = x3
-27
-8
-1
0
1
8
27
Remember that when x is negative, x3 is
negative. Opinion  makes the mistake of
saying it is positive.
Opinion 1 Opinion 2 Answer
Q2
Opinion 1 Opinion 2 Answer
Plotting quadratic and cubic graphs– Developing Understanding
Cubic graphs
Look at the graph. There are three curves:
A: y = x3 + 4x
B: y = x3 – 4x C: y = x3 + 4
1. For each equation, find the value of y when
x = 0. What is the equation of the red curve?
2. For each equation, find the value of y when
x = -2. What is the equation of the green
curve?
3
A
+ -=84= -16
A yy==xx3 ++44xx ==0-8
+4
3
B
=0
B yy==xx3 --44xx ==0-8
– +4 8= -4
3
C
+ 4= 4= -4
C yy==xx3 ++44 ==0-8
+ 4
The green
curve
is y be
= x3 - 4x. 
red curve
could
y =These
x3 + 4three
x or curves
y = x3 +have
4. similar shapes
40y
The green
red curve
intercepts
the
curve
crosses
the y-axis
x-axis at
at
4. This
is where
Substituting
30 x =
xy = -2.
So substituting
x 0.
= -2
in the right
x = 0 gives
curve
will give y =200.
A y = x3 + y4x= =x3-8+ +4x-8 = -16 = 0 + 0
= 0= -8 10
B y = x3 - 4x
+8=0
3
3
B y = x + y4 == x-8 –+04x
=0–0 x
C
40 = -4 1
-3
-2
-1
2
3
0 is correct.
So Opinion= 
The
green
-10
3
3
C
=0+4 =4
curve
is B,yy==xx +-20
- 44x.
3 + 4.
So the red
must states
be C, ythat
= xthe
Opinion
 curve
mistakenly
-30
Opinion
makes anumber
mistakeiswith
curve
cube
of anegative
positive.
A. It states 4 × 0-40= 4.
but an important difference. What do you
notice about theAgradients
y = x3 + of
4xeach
= 0 curve
+0 =0
3
x + B4xy == 8
8x ==00 – 0 = 0
whenAxy==0?
x3+--4
B y = x3 - C4xy =
=8
x3++84 ==16
0+ 4 =4
3
C y = x + The
4 red
= 8 curve
+ 4 must
= 12 be
The green ycurve
= x3 is
+y
4.=x3 + 4x. 
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Mastering Mathematics © Hodder and Stoughton 2014
Q1
Opinion 1 Opinion 2 Answer
Q2
Opinion 1 Opinion 2 Answer
Plotting quadratic and cubic graphs– Developing Understanding
Solving equations
y
You can use a graph to solve an equation.
Here is the graph of y = x2 – 4.
6
5
4
1. What are the coordinates of the intercepts
on the x-axis?
3
2
1
2. What does this tell you about the value of
y when x = -2?
0
-3
-2
-1
The intercepts are
(-2, 0) and (2, 0). Opinion
 is correct. Opinion 
has the intercept on the
(0, -4) 
Use the graph to solvey-axis.
the equation
x2 – 3 = 1.
When x = -2, y
= 0
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0
1
2
3
x
-2
(-2, 0) and
(2, 0) 
When x = -2, y
does not exist.
-1
-3
-4
-5
When x = -2, y = 0.
Also, when x = 2, y = 0.
So the solution of the
equation x2 – 4 = 0 is
x = -2 or x = 2.
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Mastering Mathematics © Hodder and Stoughton 2014
Q1
Opinion 1 Opinion 2 Answer
Q2
Opinion 1 Opinion 2 Answer
Plotting quadratic and cubic graphs– Developing Understanding
Solving equations
25
You can use one graph to solve many
equations.
Here is the graph of y = x3 – 2x.
1. Write the equation
form x3 – 2x = ...
x3
y
20
15
10
– 2x – 5 = 0 in the
5
0
-3
-2
-1
0
1
2
x
3
-5
2. How can you use the graph to solve the
equation x3 – 2x – 5 = 0?
-10
-15
Draw the5graph
Subtract
from of
3
y = xside:
– 2x – 5 and
each
see
x3 where
– 2x =the
-5curve

crosses the x-axis. 
Add 5 to each side:
x3 – 2x = 5 
-20
-25
Opinion  is correct: x3 – 2x = 5. You must
How
could
you
usetothe
graph to
add
5 to
each
side
eliminate
thesolve
-5. the
Draw
the graph
of
equation
x3 – 3x?
Opinion
yis=wrong.
Subtracting
5 makes
y = 5 and see where it
the left hand side into:
3
x3 – crosses
2x – 10 y = x – 2x. 
and not:
x3 – 2x.
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Mastering Mathematics © Hodder and Stoughton 2014
Q1
x3 – 2x – 5 = 0 is the same as x3 – 2x = 5.
Draw the graph of y = 5 and find the value
of x where it crosses the curve. That is
where y = x3 – 2x and y = 5 have the same
value and so is the solution of the equation.
It is x = 2.1 (to 1 decimal place).
Opinion  said to draw y = x3 – 2x – 5 and
see where it crosses the x-axis. You could
do this but it would take you much longer.
Opinion 1 Opinion 2 Answer
Q2
Opinion 1 Opinion 2 Answer
Plotting quadratic and cubic graphs– Developing Understanding
Editable Teacher Template
Information
Vocabulary
1. Task – fixed
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2. Task – appears
Q1
Opinion 1
Q1
Opinion 2
Q1
Answer
Q2
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Q2
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Q2
Answer
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Plotting quadratic and cubic graphs– Developing Understanding