Algebra and functions
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Plotting and sketching graphs
Contents
Plotting and sketching graphs
Graphs of functions
Using graphs to solve equations
Transforming graphs of functions
Examination-style questions
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Plotting graphs
Suppose we wish to plot the graph of y = x3 – 7x + 2 for
–3 < x < 3.
We can find the coordinates of any number of points that
satisfy the equation using a table of values. For example:
x
–3
–2
–1
0
1
2
3
x3
–27
–8
–1
0
1
8
27
– 7x
+ 21
+14
+7
0
–7
– 14
– 21
+2
+2
+2
+2
+2
+2
+2
+2
y = x3 – 7x + 2
–4
8
8
2
–4
–4
8
These values of x and y correspond to the coordinates of
points that lie on the curve.
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Plotting graphs
x
–3
–2
–1
0
1
2
3
y = x3 – 7x + 2
–4
8
8
2
–4
–4
8
y
The points given in the table
are plotted …
10
8
… and the points are then
joined together with a smooth
curve.
The shape of this graph is
characteristic of a cubic
function.
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6
4
2
–3
–2
–1
0
–2
–4
1
2
3
x
Sketching graphs
When the general shape of a graph is known it is more usual to
sketch the graph.
To help us sketch a graph given its equation we can find:
Points where the curve intercepts the y-axis
These are found by putting x = 0 in the equation of the graph.
Points where the curve intercepts the x-axis
These are found by putting y = 0 in the equation of the graph.
The value of y when x is very large and positive
The value of y when x is very large and negative
Turning points
A turning point is a point where the gradient of a graph
changes from being positive to negative or vice versa.
It can be a maximum or a minimum.
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Sketching graphs
For example:
Sketch the curve of y = x3 + 2x2 – 3x.
When x = 0 we have
y = 03 + 2(0)2 – 3(0)
=0
So the curve passes through the point (0, 0).
When y = 0 we have
Factorizing gives
x3 + 2x2 – 3x = 0
x(x2 + 2x – 3) = 0
x(x + 3)(x – 1) = 0
x = 0, x = –3 or x = 1
So the curve also passes through the points (–3, 0) and (1, 0).
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Sketching graphs
We can plot these three points on our graph.
y
When x is very large and
positive, y is very large and
positive. We can write this as:
7
6
5
as x  , y  .
4
We can use this to sketch in this
part of the graph:
3
2
1
–4
–3
–2
–1 0
–1
–2
1
2
3
x
Also:
as x  , y  .
We can use this to sketch in this
part of the graph:
We can now produce a sketch of y = x3 + 2x2 – 3x.
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Graphs of functions
Contents
Plotting and sketching graphs
Graphs of functions
Using graphs to solve equations
Transforming graphs of functions
Examination-style questions
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© Boardworks Ltd 2005
Graphs of cubic functions
A cubic function in x can be written in the form:
y = ax3 + bx2 + cx + d (where a ≠ 0)
Graphs of cubic functions have a characteristic shape
depending on the values of the coefficients:
When the coefficient of x3 is positive the shape is
or
When the coefficient of x3 is negative the shape is
or
Cubic curves have rotational symmetry of order 2.
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Graphs of factorized cubic functions
In general:
When a cubic function is written in the form
y = a(x – p)(x – q)(x – r), it will cut the x-axis at
the points (p, 0), (q, 0) and (r, 0).
p, q and r are the roots of the cubic function.
To sketch the graph of a cubic function given in factorized form,
Find the roots of the function and plot these on the x-axis.
Find the y-intercept by putting x equal to 0 in the equation.
Look at the coefficient of x3 to decide whether the curve is
N -shaped or -shaped.
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The graphs of y = x2 and y = x3
You should be familiar with the graphs of y = x2 and y = x3:
y = x2
y
y = x3
y
0
0
This is a quadratic
function
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x
This is a cubic
function
x
Graphs of the form y = kxn
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The graph of y = 1/x
You should also be familiar with the graph of y = 1x .
1
y=
x
0
This is a reciprocal
function
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Notice that the curve gets
closer and closer to the x- and
y-axes but never touches
them.
y
x
The x- and y-axes form
asymptotes.
The graph of y = 1x is an
example of a discontinuous
function.
Graphs of the form y = kx–n
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The graph of y =
x
Another interesting graph is y = x .
This graph can only be drawn
for positive values of x.
y= x
This is because we cannot
find the square root of a
negative number.
y
0
x
y2 = x
Also, remember that y = x is
defined as the positive square
root of x.
The curve is therefore only
drawn in the first quadrant.
Compare this to the graph of
y2 = x.
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Graphs of the form y = k
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n
x
Contents
Using graphs to solve equations
Plotting and sketching graphs
Graphs of functions
Using graphs to solve equations
Transforming graphs of functions
Examination-style questions
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© Boardworks Ltd 2005
Using graphs to solve equations
By sketching an appropriate graph find the
solutions to the equation 2x2 – 5 = 3x.
We can do this by considering the left-hand side and the
right-hand side of the equation as two separate functions.
2x2 – 5 = 3x
y = 2x2 – 5
y = 3x
The points where these two functions intersect will give us the
solutions to the equation 2x2 – 5 = 3x.
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Using graphs to solve equations
10
The graphs of
y = 3x y = 2x2 – 5 and y = 3x
intersect at the points:
y = 2x2 – 5
8
(2.5, 7.5)
6
and (2.5, 7.5).
4
2
–4
–3
–2
–1 0
–2
(–1,–3)
–4
–6
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(–1, –3)
1
2
3
4
The x-values of these
coordinates give us the
solutions to the equation
2x2 – 5 = 3x as
x = –1
and
x = 2.5
Using graphs to solve equations
Alternatively, we can rearrange the equation so that all the
terms are on the left-hand side:
2x2 – 3x – 5 = 0
y = 2x2 – 3x – 5
y=0
The line y = 0 is the x-axis. This means that the solutions to
the equation 2x2 – 3x – 5 = 0 can be found where the function
y = 2x2 – 3x – 5 crosses the x-axis.
These points represent the roots of the function y = 2x2 – 3x – 5.
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Using graphs to solve equations
10
The graph of
y = 2x2 – 3x – 5 crosses
the x-axis at the points:
y = 2x2 – 3x – 5
8
(–1, 0)
6
and (2.5, 0).
4
2
(2.5, 0)
(–1,0)
–4
–3
–2
–1 0
–2
–4
–6
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1
2
y=0
3
4
The x-values of these
coordinates give us the
same solutions:
x = –1
and
x = 2.5
Using graphs to solve equations
Use a graph to solve the equation x3 – 3x = 1.
This equation does not have any rational solutions and so the
graph can only be used to find approximate solutions.
A cubic equation can have up to three solutions and so the
graph can also tell us how many solutions there are.
Again, we can consider the left-hand side and the right-hand
side of the equation as two separate functions and find the
x-coordinates of their points of intersection.
x3 – 3x = 1
y = x3 – 3x
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y=1
Using graphs to solve equations
10
y = x3 – 3x
The graphs of y = x3 – 3x
and y = 1 intersect at
three points.
8
This means that the
equation x3 – 3x = 1 has
three solutions.
6
4
2
–4
–3
–2
–1 0
–2
–4
–6
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y=1
1
2
3
4
Using the graph these
solutions are
approximately:
x = –1.5
x = –0.3
x = 1.9
Contents
Transforming graphs of functions
Plotting and sketching graphs
Graphs of functions
Using graphs to solve equations
Transforming graphs of functions
Examination-style questions
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© Boardworks Ltd 2005
Transforming graphs of functions
Graphs can be transformed by translating, reflecting, stretching
or rotating them.
The equation of the transformed graph will be related to the
equation of the original graph.
When investigating transformations it is most useful to express
functions using function notation.
For example, suppose we wish to investigate transformations
of the function f(x) = x2.
The equation of the graph of y = x2, can be written as y = f(x).
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Vertical translations
Here is the graph of y = x2.
This is the graph of y = x2 – 7.
y
What do you notice?
The graph of y = x2 has been
translated 7 units down.
x
If the original graph is written as y = f(x)
then the translated graph can be written
as y = f(x) – 7. In general:
The graph of y = f(x) + a is the graph
of y = f(x) translated by the vector 0 .
a
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Translating quadratic functions
vertically
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Translating cubic functions vertically
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Translating reciprocal functions
vertically
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Horizontal translations
Again, here is the graph of y = x2.
This is the graph of y = (x + 3)2.
y
What do you notice?
The graph of y = x2 has been
translated 3 units to the left.
x
If the original graph is written as y = f(x)
then the translated graph can be written
as y = f(x + 3). In general:
The graph of y = f(x + a) is the graph
of y = f(x) translated by the vector –a .
0
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Translating quadratic functions horizontally
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Translating cubic functions horizontally
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Translating reciprocal functions horizontally
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Reflections in the x‐axis
Here is the graph of y = x2 – 2x – 2.
This is the graph of y = –x2 + 2x + 2.
y
What do you notice?
The graph of y = x2 – 2x – 2 has been
reflected in the x-axis.
x
If the original graph is written as y = f(x)
then the translated graph can be written
as y = –f(x). In general:
The graph of y = –f(x) is the graph of
y = f(x) reflected in the x-axis.
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Reflecting quadratic functions in the x‐axis
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Reflecting cubic functions in the x‐axis
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Reflecting reciprocal functions in the x‐axis
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Reflections in the y‐axis
Here is the graph of y = x3 + 4x2 – 3.
This is the graph of y = (–x)3 + 4(–x)2 – 3.
y
What do you notice?
The graph of y = x3 + 4x2 – 3 has been
reflected in the y-axis.
x
If the original graph is written as y = f(x)
then the translated graph can be written
as y = f(–x). In general:
The graph of y = f(–x) is the graph of
y = f(x) reflected in the y-axis.
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Reflecting quadratic functions in the y‐axis
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Reflecting cubic functions in the y‐axis
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Reflecting reciprocal functions in the y‐axis
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Vertical stretches
Let’s start with the graph of y = x2 – 3 and add the graph of
y = 2x2 – 6.
We can produce the graph of y = 2x2 – 6
y
by doubling the y-coordinate of every
point on the original graph y = x2 – 3.
This has the effect of stretching the
graph in the vertical direction.
If the original graph is written as y = f(x)
then the translated graph can be written
as y = 2f(x). In general:
x
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The graph of y = af(x) is the graph of
y = f(x) stretched parallel to the y-axis
by scale factor a.
Stretching quadratic functions
vertically
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Stretching cubic functions vertically
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Stretching reciprocal functions
vertically
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Horizontal stretches
Let’s start with the graph of y = x2 + 3x – 4 and add the graph of
y = (2x)2 + 3(2x) – 4.
We can produce the second graph by
y
halving the x-coordinate of every point
on the original graph.
This has the effect of compressing the
graph in the horizontal direction.
x
If the original graph is written as y = f(x)
then the translated graph can be written
as y = f(2x). In general:
The graph of y = f(ax) is the graph of
y = f(x) stretched parallel to the x-axis
by scale factor 1a .
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Stretching quadratic functions horizontally
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Stretching cubic functions horizontally
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Stretching reciprocal functions horizontally
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Contents
Examination‐style questions
Plotting and sketching graphs
Graphs of functions
Using graphs to solve equations
Transforming graphs of functions
Examination-style questions
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© Boardworks Ltd 2005
Examination‐style question
This diagram shows the graph of y = f(x) which has a minimum
point at (2, –3).
y
a) Sketch the following graphs
on separate sets of axes,
indicating the turning point
in each case.
i) y = f(x + 4)
x
(2, –3)
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ii) y = f(2x)
b) Given that f(x) = ax2 + bx + 5
find the values of a and b.
Examination‐style question
ii) y = f(2x)
a) i) y = f(x + 4)
y
y
x
(–2, –3)
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x
(1, –3)
Examination‐style question
b) f(x) is quadratic and so it can be written in the form
a(x + p)2 + q where (–p, q) are the coordinates of the vertex.
The vertex is at the point (2, –3) so
f(x) = a(x – 2)2 – 3
= a(x2 – 4x + 4) – 3
= ax2 – 4ax + 4a – 3
But
So
ax2 – 4ax + 4a – 3 = ax2 + bx + 5
4a – 3 = 5
a=2
b = –8
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