1. Steps Squares and cubes Quadratic graphs Cubic graphs
Solving equations

2. Squares and cubes
Squaring means multiplying a number by itself.
72 = 7 × 7 = 49
Cubing a number means multiplying three of the numbers together.
43 = 4 × 4 × 4 = 64
24 means 2×2×2×2.
25 means 2×2×2×2×2.
Find the value of 24, (-2)4, 25 and (-2)5.
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 
Opinion  is correct. (-2)3 = (-2) × (-2) × (-2)
= 4 × (-2)
= -8
In this case you are multiplying three negative numbers and so the answer is negative. Opinion  has a positive answer so it is wrong.
Opinion  is correct.
(-8)2 = (-8) × (-8) = +64
Remember multiplying two negatives makes a positive.
Opinion  has a negative answer and so it is wrong.
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3. Quadratic graphs x Area = x2 x
A quadratic graph is one where the highest power of x is x2.
Any other terms are either in x or a constant (a number).
Many equations do not produce straight lines.
Here is a table showing values of x and y when y = x2
Complete the table for y = x2 + x.
Will y have any negative values?
x Area = x2 x x -3 -2 -1 1 2 3 x2 9 4
y = x2 + x 6 12 x -3 -2 -1 1 2 3
y = x2 9 4
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 increase in equal steps, the values of y do not. 
The values of y decrease at first, and then increase. 
No, a square is always positive or zero. 
Yes, when x is between -1 and 0. 
Both Opinions are correct. They are saying the same thing in different ways. Opinion  is more precise.
No. Opinion  is incorrect, squaring a negative number always results in a positive number.
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4. Quadratic graphs y x The graph shows y = x2. It is symmetrical.
1. Where is the line of symmetry of y = x2?
2. Here is the table for y = x2 + 3x – 3
Draw the graph of y = x2 + 3x – 3. x -3 -2 -1 1 2 3 x2 9 4
+3x -9 -6 6
y = x2 +x - 3 -5 7 15 x
The purple curve. 
y = 0
The blue curve. 
The y-axis.
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?
The correct answer is the purple curve
Opinion  incorrectly assumes the curve is symmetrical about the y-axis.
It is the y axis. y = 0 is the equation of the x-axis.
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5. Quadratic graphs y x
For this table you have to work out the value of -x2.So when x is negative there are three minuses, making a minus overall. Opinion  made the mistake of making the -x2 terms positive. Opinion  is correct.
All quadratic graphs have the same U-shape.
Look at the equation y = 2x – x2.
The x2 term is negative.
Complete the table for y = 2x – x2. x -2 -1 1 2 3 2x -4
-x2
y = 2x – x2 x -2 -1 1 2 3 2x -4 4 6
-x2 -9
y = 2x – x2 -8 -3
2. Here is the graph of y = 2x – x2.
What can you say about its shape?
It is the usual U-shape but has been turned upside down. This is because the x2 term is negative instead of positive.
Draw the graph of y = x2 – 2x. Where does this curve cross y = 2x – x2?
It is the same shape as all the quadratic curves, but it is upside down. 
It is the same shape as all the quadratic curves.  x -2 -1 1 2 3 2x -4 4 6
-x2 9
y = 2x – x2 8 12 x -2 -1 1 2 3 2x -4 4 6
-x2 9
y = 2x – x2 -8 -3 8 15
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6. Cubic graphs y x Cubic graphs have their own distinctive shape.
1. Compete this table of values for y = x3. x -3 -2 -1 1 2 3
y = x3 27 x
2. Describe the gradient of the graph 
It decreases at first and then increases. x -3 -2 -1 1 2 3
y = x3 27 8
It is always increasing.
Remember that when x is negative, x3 is negative. Opinion  makes the mistake of saying it is positive. x -3 -2 -1 1 2 3
y = x3
-27 -8 8 27  x -3 -2 -1 1 2 3
y = x3
-27 -8 8 27
What will the graph of y = -x3 look like?
A cubic equation has x3 as its highest power of x.
The gradient of the graph is always positive, except at x = 0 where it is 0. It decreases until x = 0 and then increases.
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7. Cubic graphs
The green curve crosses the x-axis at x = -2. So substituting x = -2 in the right curve will give y = 0.
A y = x3 + 4x = = -16
B y = x3 - 4x = = 0
C y = x3 + 4 = = -4
So Opinion  is correct. The green curve is B, y = x3 - 4x.
Opinion  mistakenly states that the cube of a negative number is positive.
The red curve intercepts the y-axis at y = 4. This is where x = 0. Substituting x = 0 gives
A y = x3 + 4x = = 0
B y = x3 – 4x = 0 – 0 = 0
C y = x3 + 4 = = 4
So the red curve must be C, y = x3 + 4.
Opinion  makes a mistake with curve A. It states 4 × 0 = 4. y
Look at the graph. There are three curves:
A: y = x3 + 4x B: y = x3 – 4x C: y = x3 + 4
For each equation, find the value of y when x = 0. What is the equation of the red curve? x
2. For each equation, find the value of y when x = -2. What is the equation of the green curve?
A y = x3 + 4x = = 4
B y = x3 - 4x = 0 – 4 = -4
C y = x3 + 4 = = 4
The red curve could be
y = x3 + 4x or y = x 
A y = x3 + 4x = = -16
B y = x3 - 4x = = 0
C y = x3 + 4 = = -4
The green curve is y = x3 - 4x. 
These three curves have similar shapes but an important difference. What do you notice about the gradients of each curve when x = 0?
A y = x3 + 4x = = 0
B y = x3 - 4x = 0 – 0 = 0
C y = x3 + 4 = = 4
The red curve must be
y = x 
A y = x3 + 4x = = 0
B y = x3 - 4x = = 16
C y = x3 + 4 = = 12
The green curve is y = x3 + 4x. 
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8. Solving equations y x You can use a graph to solve an equation.
Here is the graph of y = x2 – 4.
1. What are the coordinates of the intercepts on the x-axis?
2. What does this tell you about the value of y when x = -2?
(-2, 0) and (2, 0) 
The intercepts are
(-2, 0) and (2, 0). Opinion  is correct. Opinion  has the intercept on the y-axis.
(0, -4) 
Use the graph to solve the equation x2 – 3 = 1.
When x = -2, y does not exist.
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.
When x = -2, y = 0
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9. Solving equations y x You can use one graph to solve many equations.
Here is the graph of y = x3 – 2x.
1. Write the equation x3 – 2x – 5 = 0 in the form x3 – 2x = ... x
2. How can you use the graph to solve the equation x3 – 2x – 5 = 0?
Subtract 5 from each side:
x3 – 2x = -5 
Draw the graph of
y = x3 – 2x – 5 and see where the curve crosses the x-axis. 
Add 5 to each side:
x3 – 2x = 5 
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  is correct: x3 – 2x = 5. You must add 5 to each side to eliminate the -5.
Opinion  is wrong. Subtracting 5 makes the left hand side into:
x3 – 2x – 10
and not:
x3 – 2x.
How could you use the graph to solve the equation y = x3 – 3x?
Draw the graph of
y = 5 and see where it crosses y = x3 – 2x. 
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