1. A9 Graphs of non-linear functions
KS4 Mathematics
A9 Graphs of non-linear functions

2. A9.1 Plotting curved graphs
Contents
A9 Graphs of non-linear functions A
A9.1 Plotting curved graphs A
A9.2 Graphs of important non-linear functions A
A9.3 Using graphs to solve equations A
A9.4 Solving equations by trial and improvement A
A9.5 Function notation A
A9.6 Transforming graphs

3. Functions In maths, what do we mean by a function?
In maths, a function is a rule that maps one number called the input (x) onto an other number, the output (y).
There are many ways of expressing a function. For example, the function “multiply by 3 and subtract 1” can be written using:
a function machine, x
– 1
× 3 y
a mapping arrow,
x  3x – 1
Review the meaning of the word function.
Function notation is discussed in more detail later in the presentation.
y = 3x – 1
an equation,
or function notation.
f(x) = 3x – 1

4. Linear and non-linear functions
The simplest type of function is a linear function.
The equation of a linear function can always be arranged in the form y = mx + c, where m and c are constants.
The graph of a linear function will always be a straight line.
If a function cannot be arranged in the form y = mx + c then it is a non-linear function.
The graph of a non-linear function is usually curved.

5. Non-linear functions Examples of non-linear functions include,
y = x2 + 1
y = 7x3 – 3x
y = 2x + x8
y = – 6 5
x – 2
y = 3 + 2x
Ask pupils to identify why each function shown is not linear. For example, because x is raised to a power other than 1 or x appears as a power (exponent) or in the denominator.
We can plot the graphs of non-linear functions using a graphics calculator or a computer.
We can also use a table of values.

6. Using a table of values Plot the graph of y = x2 – 3
for values of x between –3 and 3.
We can use a table of values to generate coordinates that lie on the graph as follows: x
y = x2 – 3 –3 –2 –1 1 2 3
Talk through the substitution for each value of x to give the corresponding value of y. 6 1 –2 –3 –2 1 6
(–3, 6)
(–2, 1)
(–1, –2)
(0, –3)
(1, –2)
(2, 1)
(3, 6)

7. Using a table of values x y = x2 – 3 –3 –2 –1 1 2 3 61 2 3 6
The points given in the table are plotted … x –2 –1 –3 1 2 3 4 5 y
… and the points are then joined together with a smooth curve.
The shape of this graph is called a parabola.
You may like to point out that a parabola is always symmetrical with a minimum or maximum point.
It is characteristic of a quadratic function.

8. Using a table of values Plot the graph of y = x3 – 7x + 2
for values of x between –3 and 3.
This function is more complex and so it is helpful to include more rows in the table to show each stage in the substitution. x x3
– 7x
+ 2
y = x3 – 7x + 2 –3 –2 –1 1 2 3
–27 –8 –1 1 8 27
The blue shaded cells in the table show the intermediate stages in the substitution.
+ 21
+14
+ 7
+ 0
– 7
– 14
– 21
+ 2
+ 2
+ 2
+ 2
+ 2
+ 2
+ 2 –4 8 8 2 –4 –4 8

9. Using a table of values x –3 –2 –1 1 2 3 y = x3 – 7x + 2 –4 8 x y1 2 3
y = x3 – 7x + 2 –4 8 x –2 –1 –3 1 2 3 –4 4 6 8 10 y
The points given in the table are plotted …
… and the points are then joined together with a smooth curve.
Tell pupils that cubic curve have rotational symmetry of order 2 and ask pupils to give you the centre of rotation for this curve.
The shape of this graph is characteristic of a cubic function.

10. A9.2 Graphs of important non-linear functions
Contents
A9 Graphs of non-linear functions A
A9.1 Plotting curved graphs A
A9.2 Graphs of important non-linear functions A
A9.3 Using graphs to solve equations A
A9.4 Solving equations by trial and improvement A
A9.5 Function notation A
A9.6 Transforming graphs

11. Quadratic functions
A quadratic function always contains a term in x2. It can also contain terms in x or a constant.
Here are examples of three quadratic functions:
y = x2
y = x2 – 3x
y = –3x2
Draw pupils attention to the characteristic shape of each of these quadratic functions and to the fact that they are symmetrical.
The characteristic shape of a quadratic function is called a parabola.

12. Exploring quadratic graphs
Point out the basic features of the quadratic graph. Draw particular attention to its symmetry and to the fact that it will always have a maximum or minimum value.
Explore the effect of changing the coefficient of x2. In particular, notice that if the coefficient of x2 is is positive the curve has a minimum point and if it is negative it has a maximum point.
Changing the value of c translates the curve vertically. This is discussed in more detail in A9.6 Translating graphs.
The coordinates of the x- and y-intercepts and the maximum or minimum value can be revealed if required. Discuss how the values of these points can be found by putting x or y equal to 0 in the function and solving the corresponding equation.

13. Cubic functions
A cubic function always contains a term in x3. It can also contain terms in x2 or x or a constant.
Here are examples of three cubic functions:
y = x3 – 4x
y = x3 + 2x2
Draw pupils’ attention to the characteristic shape of each of these cubic functions and to the fact that they have rotational symmetry of order 2.
y = -3x2 – x3

14. Exploring cubic graphs
Point out the basic features of the cubic graph. Draw particular attention to its rotational symmetry and to the position of its turning points.
Explore the effect of changing the coefficients of x3, x2 and x and the constant value.
The coordinates of the x- and y-intercepts can be revealed if required. Discuss how the values of these points can be found by putting x or y equal to 0 in the function and solving the corresponding equation.

15. Reciprocal functions
A reciprocal function always contains a fraction with a term in x in the denominator.
Here are examples of three simple reciprocal functions:
y = 3 x
y = 1 x
y = –4 x
Draw pupils attention to the characteristic shape of each of these reciprocal functions and to the fact that they each has two asymptotes.
The shape of this graph is called a hyperbola.
In each of these examples the axes form asymptotes. The curve never touches these lines.

16. Exploring reciprocal graphs
Point out the basic features of the reciprocal graph. Draw particular attention to its asymptotes formed by the x- and y-axes.
Explore the effect of changing the constant.
By considering very large and very small positive and negative values of x, discuss why the curve approaches, but doesn’t touch, the x- and y-axes.

17. Exponential functions
An exponential function is a function in the form y = ax, where a is a positive constant.
Here are examples of three exponential functions:
y = 2x
y = 3x
y = 0.25x
Tell pupils that exponent is another word for index.
Explain that these graphs are typical of of problems that involve repeated proportional change.
In each of these examples, the x-axis forms an asymptote.

18. Exploring exponential graphs
Modify the graph and observe what happens to the curve.
Note that all of the graph will always pass through the point (0, 1) and ask pupil to give you the reason for this.
Establish that when a is greater than 0 the graph shows an exponential increase for positive values of x. When a is between 0 and 1, the graph shows an exponential increase for positive values of x. Ask pupils to explain why the graph will never touch the x-axis.

19. The equation of a circle
One more graph that you should recognize is the graph of a circle centred on the origin. x y
We can find the relationship between the x and y-coordinates on this graph using Pythagoras’ theorem.
(x, y) r y
Let’s call the radius of the circle r. x
We can form a right angled triangle with length y, height x and radius r for any point on the circle.
You may wish to explain that the equation of a circle is not a function, but a locus. This is because values of x do not map onto unique values of y. Each value of x between r and –r maps onto two values of y (except when x = ±r).
Using Pythagoras’ theorem this gives us the equation of the circle as:
x2 + y2 = r2

20. Exploring the graph of a circle
Explore the graphs of circles of various radii. Establish that the equation holds for every point on the circle using Pythagoras’ Theorem.
Note that the table of values gives two values for y. This means that x2 + y2 = r2 is not a function, since functions may only map each value of x onto a unique value of y.
The relationship x2 + y2 = r2 is called a locus since it describes the points on a given path, in this case a circle.

21. Matching graphs with equations
Ask pupils for the names of the functions as they are matched to their corresponding graphs.

22. A9.3 Using graphs to solve equations
Contents
A9 Graphs of non-linear functions A
A9.1 Plotting curved graphs A
A9.2 Graphs of important non-linear functions A
A9.3 Using graphs to solve equations A
A9.4 Solving equations by trial and improvement A
A9.5 Function notation A
A9.6 Transforming graphs

23. Using graphs to solve equations
Solve the equation 2x2 – 5 = 3x using graphs.
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
This can be compared with solving simultaneous equations. The difference is that we are only interested in the x-value in the coordinates of the crossing points.
The points where these two functions intersect will give us the solutions to the equations.

24. Using graphs to solve equations
The graphs of
y = 2x2 – 5 and y = 3x intersect at the points: –1 –2 –3 –4 1 2 3 4 –6 6 8 10
y = 2x2 – 5
y = 3x
(2.5, 7.5)
(–1, –3)
and (2.5, 7.5).
The x-value of these coordinates give us the solution to the equation 2x2 – 5 = 3x as
Stress again that, unlike simultaneous equations where we want the values of both x and y, when solving an equation in x we are only interested in the x-values in the coordinates.
(–1,–3)
x = –1
and x = 2.5

25. Using graphs to solve equations
Solve the equation 2x2 – 5 = 3x using graphs.
Alternatively, we can rearrange the equation so that all the terms are on the right-hand side,
2x2 – 3x – 5 = 0
y = 2x2 – 3x – 5
y = 0
Ask pupils to tell you what is special about the line y = 0 before revealing that it is the x-axis.
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 intersects with the x-axis.

26. Using graphs to solve equations–1 –2 –3 –4 1 2 3 4 –6 6 8 10
y = 2x2 – 3x – 5
The graphs of
y = 2x2 – 3x – 5 and y = 0 intersect at the points:
(–1, 0)
and (2.5, 0).
The x-value of these coordinates give us the same solutions
(–1,0)
(2.5, 0)
y = 0
Finding graphical solutions to quadratic equations is also explored in A6.1.
x = –1
and x = 2.5

27. Using graphs to solve equations
Solve the equation x3 – 3x = 1 using graphs.
This equation does not have any exact 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.
Discuss the limitations of this method for equations that do not have exact solutions (that is, when the solutions are irrational numbers).
The main benefit in using a graph is to find out how many solutions there are and what values they lie between. We can then find more accurate solutions using an other method such as trial and improvement.
x3 – 3x = 1
y = x3 – 3x
y = 1

28. Using graphs to solve equations
y = x3 – 3x
The graphs of y = x3 – 3x and y = 1 intersect at three points: –1 –2 –3 –4 1 2 3 4 –6 6 8 10
This means that the equation x3 – 3x = 1 has three solutions.
y = 1
Using the graph these solutions are approximately:
The solutions are given by the x-coordinate of each point of intersection.
The solutions given by the graph can be found to any number of decimal places using trial and improvement. This is demonstrated for the value between x = 1.8 and 1.9 in the next section.
x = –1.5
x = –0.3
x = 1.9

29. A9.4 Solving equations by trial and improvement
Contents
A9 Graphs of non-linear functions A
A9.1 Plotting curved graphs A
A9.2 Graphs of important non-linear functions A
A9.3 Using graphs to solve equations A
A9.4 Solving equations by trial and improvement A
A9.5 Function notation A
A9.6 Transforming graphs

30. Solving equations by trial and improvement
The equation x3 – 3x = 1 has a solution when x is approximately equal to 1.9.
Find this solution to 3 decimal places.
The value 1.9 was found using a graph. We can improve the accuracy of this answer by substituting 1.9 into the equation and noting whether it is too high or too low.
We then substitute a better value and continue the process until we have a solution to the required degree of accuracy.
This solution was found using a graph as shown in the previous section.
This method of finding a solution is called trial and improvement.

31. Solving equations by trial and improvement
The equation x3 – 3x = 1 has a solution when x is approximately equal to 1.9.
Find this solution to 3 decimal places.
Set up a table as follows, x
x3 – 3x
comment
1.9
1.159
too high
1.8
0.432
too low
This tells us that the solution is between 1.8 and 1.9
Stress that we are trying to find a value for x that will give x3 – 3x as close to 1 as possible.
1.85
too low
1.87
too low
1.88
too high

32. Solving equations by trial and improvement
The equation x3 – 3x = 1 has a solution when x is approximately equal to 1.9.
Find this solution to 3 decimal places.
The solution is between 1.87 and 1.88, so try next, x
x3 – 3x
comment
1.875
too low
1.878
too low
1.879
too low
Explain that we need to find out whether the solution is closer to or to see whether we need to round up or round down.
Since is too high, we know that the solution is closer to This is therefore our solution to 3 decimal places.
As an extension activity discuss how this method could be carried out using a spreadsheet.
The solution is between and
1.8795
too high
The solution is to 3 decimal places.

33. Solving equations by trial and improvement
Use this activity to practice finding solutions by trial and improvement.

34. A9 Graphs of non-linear functions
Contents
A9 Graphs of non-linear functions A
A9.1 Plotting curved graphs A
A9.2 Graphs of important non-linear functions A
A9.3 Using graphs to solve equations A
A9.4 Solving equations by trial and improvement A
A9.5 Function notation A
A9.6 Transforming graphs

35. Functions
Remember, a function is a rule that maps one number called the input (x) onto an other number, the output (y).
For example, the function “square” can be written using,
a function machine, x
square y
a mapping arrow,
x  x2
y = x2
an equation, or
One more way of expressing a function is to use function notation. For example,
f(x) = x2
“f of x equals x squared”

36. Function notation We write f(x) = x2 to define the function f.
The function f can then act on any number, term or expression that is in the brackets. For example,
f(5) = 52 = 25
f(–2) = (–2)2 = 4
f(a) = a2
For each example ask pupils to identify which number is the input and which is the output.
f(x + 4) = (x + 4)2 =
x2 + 8x + 16
f(–x) = (–x)2 = x2

37. Function notation Suppose g(x) = 2x – 5 Find g(4) = 2 × 4 – 5 = 3
2 × 1.5 – 5 = –2
g(a) =
2a – 5
g(x + 3) =
2(x + 3) – 5
= 2x + 6 – 5
= 2x + 1
Explain that we can use different letters, for example g, to stand for different functions.
g(x) + 3 =
2x – 5 + 3
= 2x – 2
g(2x) =
2 × 2x – 5 =
4x – 5
2g(x) =
2(2x – 5) =
4x – 10

38. A9 Graphs of non-linear functions
Contents
A9 Graphs of non-linear functions A
A9.1 Plotting curved graphs A
A9.2 Graphs of important non-linear functions A
A9.3 Using graphs to solve equations A
A9.4 Solving equations by trial and improvement A
A9.5 Function notation A
A9.6 Transforming graphs

39. 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).

40. Vertical translations
Here is the graph of y = x2, where y = f(x). y
This is the graph of y = f(x) + 1 x
and this is the graph of y = f(x) + 4.
What do you notice?
This is the graph of y = f(x) – 3
and this is the graph of y = f(x) – 7.
What do you notice?
Establish that for y = f(x) + a, if a is positive the curve y = f(x) is translated a units upwards. If a is negative, the curve y = f(x) is translated a units downwards.
This can be investigated for other graphs and functions using the activities at the end of this section.
The graph of y = f(x) + a is the graph of y = f(x) translated by the vector a

41. Horizontal translations
Here is the graph of y = x2 – 3, where y = f(x). y
This is the graph of y = f(x – 1), x
and this is the graph of y = f(x – 4).
What do you notice?
This is the graph of y = f(x + 2),
and this is the graph of y = f(x + 3).
What do you notice?
Establish that for f(x + a), if a is negative the curve is translated a units to the right (in the positive horizontal direction). If a is positive, the curve is translated a units to the left (in the negative horizontal direction).
This can be investigated for other graphs and functions using the activities at the end of this section.
The graph of y = f(x + a ) is the graph of y = f(x) translated by the vector –a

42. Reflections in the x-axis
Here is the graph of y = x2 –2x – 2, where y = f(x). y x
This is the graph of y = –f(x).
What do you notice?
Establish that the graph of y = –f(x ), is a reflection of y = f(x) in the x-axis.
This can be investigated for other graphs and functions using the activities at the end of this section.
The graph of y = –f(x) is the graph of y = f(x) reflected in the x-axis.

43. Reflections in the y-axis
Here is the graph of y = x3 + 4x2 – 3 where y = f(x). y x
This is the graph of y = f(–x).
What do you notice?
Establish that the graph of y = f(–x ), is a reflection of y = f(x) in the y-axis.
This can be investigated for other graphs and functions using the activities at the end of this section.
The graphs of some functions remain unchanged when reflected in the y-axis. For example, the graph of y = x2. These functions are called even functions.
The graph of y = f(–x) is the graph of y = f(x) reflected in the y-axis.

44. Stretches in the y-direction
Here is the graph of y = x2, where y = f(x).
This is the graph of y = 2f(x). y
What do you notice?
This graph is is produced by doubling the y-coordinate of every point on the original graph y = f(x).
This has the effect of stretching the graph in the vertical direction.
Demonstrate that the distance from the x-axis to the curve y = f(2x) is always double the distance from the x-axis to the curve y = f(x).
For example, the point (2,4) becomes (2,8) and the point (–1, 1) becomes (–1, 2). The x-coordinate stays the same in each case and the y-coordinate doubles.
This can be investigated for other graphs and functions using the activities at the end of this section. x
The graph of y = af(x) is the graph of
y = f(x) stretched parallel to the y-axis by scale factor a.

45. Stretches in the x-direction
Here is the graph of y = x2 + 3x – 4, where y = f(x).
This is the graph of y = f(2x). y x
What do you notice?
This graph is is produced by halving the x-coordinate of every point on the original graph y = f(x).
This has the effect of compressing the graph in the horizontal direction.
Pupils should notice that the intersection on the y-axis has not changed and that the graph has been compressed (or squashed) horizontally.
Demonstrate that the distance from the y-axis to the curve y = f(2x) is always half the distance from the y-axis to the curve y = f(x).
Ask pupils to predict what the graph of y = f(½x) would look like.
This is probably the most difficult transformation to visualize. Using the activities on the next few slides will help.
The graph of y = f(ax) is the graph of
y = f(x) stretched parallel to the x-axis by scale factor . a 1

46. Transforming linear functions
Use this activity to investigate a variety of transformations applied to linear functions.
Once a transformation has been chosen the equation for the original graph can be modified to explore the effect of the chosen transformation.
To explore the transformations given by –f(x) and f(–x) choose af(x) or f(ax) and set the value of a to –1.
Translations of trigonometric functions are examined in S4 Further trigonometry.

47. Transforming quadratic functions
Use this activity to investigate a variety of transformations applied to quadratic functions.

48. Transforming cubic functions
Use this activity to investigate a variety of transformations applied to cubic functions.