45: the graph of © christine crisp “teach a level maths” vol. 1: as core modules
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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
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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
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