1. 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

2. 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.

3. 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.

4. 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

5. 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.

6. 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.

7. 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.

8. 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.

9. 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.

10. 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

11. 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.

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

13. 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

14. 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).

15. 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

16. 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

17. 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.

18. 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.

19. 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.

20. 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

21. 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.

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

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