ks4 mathematics

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A9 Graphs of non-linear functions

KS4 Mathematics


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A9.1 Plotting curved graphs

Contents


A9.3 Using graphs to solve equations

A9.4 Solving equations by trial and improvement

A9.5 Function notation

A9.6 Transforming graphs

A9.2 Graphs of important non-linear functions


Functions

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:

In maths, what do we mean by a function?

a function machine,

x – 1× 3 y

y = 3x – 1

an equation, or function notation.

f(x) = 3x – 1

a mapping arrow,

x 3x – 1


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.


Non-linear functions

Examples of non-linear functions include,

y = x2 + 1

y = 3 + 2x

y = 7x3 – 3x

y = – 65

x – 2

y = 2x + x8

We can plot the graphs of non-linear functions using a graphics calculator or a computer.

We can also use a table of values.


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 0 1 2 3

6

(–3, 6)

1 –2 –3 –2 1 6

(–2, 1)(–1, –2)(0, –3) (1, –2) (2, 1) (3, 6)



x

y = x2 – 3

–3 –2 –1 0 1 2 3

6 1 –2 –3 –2 1 6

The points given in the table are plotted …

x0–2 –1–3 1 2 3

–1

–2

1

2

3

4

5

yy

… and the points are then joined together with a smooth curve.

The shape of this graph is called a parabola.

It is characteristic of a quadratic function.



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 0 1 2 3

–27 –8 –1 0 1 8 27

+ 21 +14 + 7 + 0 – 7 – 14 – 21

+ 2 + 2 + 2 + 2 + 2 + 2 + 2

–4 8 8 2 –4 –4 8



The points given in the table are plotted …

x0–2 –1–3 1 2 3

–2

–4

2

4

6

8

10

yy

… and the points are then joined together with a smooth curve.

The shape of this graph is characteristic of a cubic function.

x –3 –2 –1 0 1 2 3

y = x3 – 7x + 2 –4 8 8 2 –4 –4 8


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Quadratic functions

A quadratic function always contains a term in x2. It can also contain terms in x or a constant.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:

The characteristic shape of a quadratic function is called a parabola.

y = x2 y = x2 – 3x y = –3x2


Exploring quadratic graphs


y = x3 – 4x

Cubic functions

A cubic function always contains a term in x3. It can also contain terms in x2 or x or a constant.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 + 2x2 y = -3x2 – x3


Exploring cubic graphs


Reciprocal functions

A reciprocal function always contains a fraction with a term in x in the denominator.A reciprocal function always contains a fraction with a term in x in the denominator.

Here are examples of three simple reciprocal functions:

In each of these examples the axes form asymptotes. The curve never touches these lines.

y = 3x y = 1x y = –4x


Exploring reciprocal graphs


Exponential functions

An exponential function is a function in the form y = ax, where a is a positive constant.An exponential function is a function in the form y = ax, where a is a positive constant.

Here are examples of three exponential functions:

In each of these examples, the x-axis forms an asymptote.

y = 2x y = 3x y = 0.25x


Exploring exponential graphs


The equation of a circle

One more graph that you should recognize is the graph of a circle centred on the origin.

Using Pythagoras’ theorem this gives us the equation of the circle as:

x2 + y2 = r2x2 + y2 = r2

x

y

0

We can find the relationship between the x and y-coordinates on this graph using Pythagoras’ theorem.

Let’s call the radius of the circle r.

We can form a right angled triangle with length y, height x and radius r for any point on the circle.

r

(x, y)

x

y


Exploring the graph of a circle


Matching graphs with equations


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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

The points where these two functions intersect will give us the solutions to the equations.



–1–2–3–4 0 1 2 3 4–2

–4

–6

2

4

6

8

10 y = 2x2 – 5y = 3x

(–1,–3)

(2.5, 7.5)

The graphs of y = 2x2 – 5 and y = 3x intersect at the points:

The x-value of these coordinates give us the solution to the equation 2x2 – 5 = 3x as

(–1, –3)

and (2.5, 7.5).

x = –1

and x = 2.5



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,

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.

2x2 – 3x – 5 = 0

y = 2x2 – 3x – 5 y = 0



–1–2–3–4 0 1 2 3 4–2

–4

–6

2

4

6

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10 y = 2x2 – 3x – 5

y = 0(–1,0) (2.5, 0)

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

x = –1

and x = 2.5



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.

x3 – 3x = 1

y = x3 – 3x y = 1


–1–2–3–4 0 1 2 3 4–2

–4

–6

2

4

6

8

10


y = x3 – 3x

y = 1

The graphs of y = x3 – 3x and y = 1 intersect at three points:

This means that the equation x3 – 3x = 1 has three solutions.

Using the graph these solutions are approximately:

x = –1.5

x = –0.3

x = 1.9


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Solving equations by trial and improvement

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.

The equation x3 – 3x = 1 has a solution when x is approximately equal to 1.9. Find this solution to 3 decimal places.

We then substitute a better value and continue the process until we have a solution to the required degree of accuracy.

This method of finding a solution is called trial and improvement.



Set up a table as follows,


1.8

This tells us that the solution is between 1.8 and 1.9

1.87

1.85 0.781625 too low

0.929203 too low

0.432 too low

x x3 – 3x comment

1.9 1.159 too high

1.88 1.004672 too high



The solution is between 1.87 and 1.88, so try 1.875 next,


1.878

The solution is between 1.879 and 1.880.

0.9894882 too low

x x3 – 3x comment

1.875 0.9667969 too low

1.879 0.9970744 too low

1.8795 1.0008718 too high

The solution is 1.879 to 3 decimal places.The solution is 1.879 to 3 decimal places.


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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,

f(x) = x2

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 of x equals x squared”


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

f(x + 4) = (x + 4)2 = x2 + 8x + 16

f(–x) = (–x)2 = x2


Function notation

Findg(4) = 2 × 4 – 5 =

g(a) =

g(x) + 3 = 2x – 5 + 3

g(2x) = 4x – 5

Suppose g(x) = 2x – 5

3

g(1.5) = 2 × 1.5 – 5 = –2

2a – 5

g(x + 3) = 2(x + 3) – 5 = 2x + 6 – 5 = 2x + 1

= 2x – 2

2g(x) = 4x – 10

2 × 2x – 5 =

2(2x – 5) =


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AA9.6 Transforming graphs

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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).


x

Vertical translations

This is the graph of y = f(x) + 1

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?

Here is the graph of y = x2, where y = f(x).

y

The graph of y = f(x) + a is the graph

of y = f(x) translated by the vector .0a


x

Horizontal translations

This is the graph of y = f(x – 1),

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?

The graph of y = f(x + a ) is the graph

of y = f(x) translated by the vector .–a0

y

Here is the graph of y = x2 – 3, where y = f(x).


x

Reflections in the x-axis

This is the graph of y = –f(x).

What do you notice?

The graph of y = –f(x) is the graph of

y = f(x) reflected in the x-axis.

Here is the graph of y = x2 –2x – 2, where y = f(x).

y


x

Reflections in the y-axis

Here is the graph of y = x3 + 4x2 – 3 where y = f(x).

y

This is the graph of y = f(–x).

What do you notice?

The graph of y = f(–x) is the graph of

y = f(x) reflected in the y-axis.


x

Stretches in the y-direction

This is the graph of y = 2f(x).

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.

Here is the graph of y = x2, where y = f(x).

y

The graph of y = af(x) is the graph of y = f(x) stretched parallel to the y-axis by scale factor a.


x

Stretches in the x-direction

Here is the graph of y = x2 + 3x – 4, where y = f(x).

y

The graph of y = f(ax) is the graph of y = f(x) stretched parallel to the x-axis by scale factor .a

1

This is the graph of y = f(2x).

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.


Transforming linear functions


Transforming quadratic functions


Transforming cubic functions

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