45: The graph of 45: The graph of xay

© Christine Crisp

““Teach A Level Maths”Teach A Level Maths”

Vol. 1: AS Core Vol. 1: AS Core ModulesModules

xay

"Certain images and/or photos on this presentation are the copyrighted property of JupiterImages and are being used with permission under license. These images and/or photos may not be copied or downloaded without permission from JupiterImages"

Module C2

xay

xy 2e.g.

Functions of this type, with a > 1, are called growth functions.

The reason for the name can be seen from the graph of xy 2

In this example, a is greater than

1.

xay The function

xay

1

Every time x increases by 1, the y value “grows” by doubling. e.g.

12,1 yx

e.g.

xy 2

21

02,0 yx

,1x 2y

,2x 4y

,3x 8y

xay

Every time x increases by 1, the y value “grows” by doubling. e.g.

12,1 yx

e.g.

xy 2

21

02,0 yx

,1x 2y

,2x 4y

,3x 8y

1

xay

xy 2

However far left we go, the graph never reaches the x-axis, even though it looks as though it does on a calculator or computer!

e.g. yx 100

Although is very small, it isn’t zero! 1002

1 100100

2

12 yor

xay

xy 2

Tip: When sketching exaggerate the gap between the curve and the x-axis.

xay

You don’t need to show the values on the x-axis. It’s the shape that’s important and the intercept on the y-axis.

)1,0(

xay

xy 2

xy 3

Sketch the graph of and, on the same axes, sketch the graph of

xy 2xy 3

Solution

You can omit these

Exercise

)1,0(

xay

xy 2

xy 3xy 4

As the value of a increases in the graph gets steeper e.g.

xay

All the curves pass through since for all values of a.

10 a)1,0(

)1,0(

xay

xy 53 xy 2

xy 51

a need not be an integer e.g.

Also, if a = 1, 11 yyay xx

xy 1)1,0(

xay

xy 50

Instead of growing, the function decays

e.g.

450

12

y

1501 yx

2502 yxe.g.

250

1

y etc

.

Values of a between 0 and 1

x

x

)1,0(

xay

xy 50

xy 70

ExerciseSketch the graph of and, on the same axes, sketch the graph of

xy 50xy 70

Solution

)1,0(

xay

Values of a between 0 and 1 e.g.

N.B. It is not possible for a to be less than zero.

xy 30

xy 70

xy 10xy 50

)1,0(

xay

The function is defined for a > 0 and all values of x .

xay SUMMARY

For a > 1, the function grows.

a is never zero or negative .

For ,the function decays.

10 a

All the curves pass through since for all values of a.

10 a)1,0(

)1,0(

)1,0(

xay )1( a

xay )10( a

xay

xay

The following slide contains repeats of information on earlier slides, shown without colour, so that it can be printed and photocopied.

xay

The function is defined for a > 0 and all values of x .

xay SUMMARY

For a > 1, the function grows.

a is never zero or negative .

For ,the function decays.

10 a

All the curves pass through since for all values of a.

10 a)1,0(

)1,0(

)1,0(

xay )1( a

xay )10( a