1. “Teach A Level Maths” Vol. 2: A2 Core Modules
4: The function
© Christine Crisp

2. Module C3
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3. We’ve already met the function
e.g.
growth
Functions of this type, with a > 1, are called
functions.
Autograph demo

4. We will now investigate the gradient of e.g.
Notice first that as x increases, y increases x x x x x x
Autograph demo

5. We will now investigate the gradient of e.g.
Notice first that as x increases, y increases
. . . and the
also increases x x x x x x x x x x x x
The gradient function
looks
It the same as
but . . .

6. We will now investigate the gradient of e.g.
Notice first that as x increases, y increases
. . . and the
also increases
e.g. x
The gradient function x

7. Putting the 2 graphs on the same axes . . .
What do you think will happen if we repeat the process for ?
Well, goes up more steeply than
so we get a similar result but the gradient function is above the curve.

8. It can be shown that
So,
The 1st gradient graph is under the original curve . . .
and the 2nd is above the curve . . .
suggesting that there is a value of a between 2 and 3 where the gradient of is equal to .

9. gradient of equals
The value of a where the
is an irrational number, written as e, where

10. gradient of equals
The value of a where the
is an irrational number, written as e, where
Using a letter for an irrational number isn’t a new idea to you.
You have used p ( the Greek p ) for

11. gradient of equals
The value of a where the
is an irrational number, written as e, where

12. More Indices and Logs
We know that
( since an index is a log )
The function contains the index x, so x is a log.
BUT the base of the log is e not 10, so
Logs with a base e are called natural logs
We write as ( n for natural ) so,

13. is a one-to-one function so has an inverse function.
The Inverse of
is a one-to-one function so has an inverse function.
We can sketch the inverse by reflecting in y = x.
Finding the equation of the inverse function is easy!
Forwards x e it  = f(x)
Backwards opposite of e it is ln it
Autograph demo
So, So
N.B. The domain is

14. SUMMARY
is a growth function.
(3 d.p.)
At every point on , the gradient equals y:
The inverse of is
( log with base e )
is defined for x > 0 only

15. Can you suggest equations for the unlabelled graphs below?
HINT:
Both graphs are stretches of

16. This . . .
is a stretch with scale factor 2 parallel to the y-axis.
The equation is

17. is a stretch with scale factor parallel to the x-axis.
This . . .
The equation is

19. The following slides contain repeats of information on earlier slides, shown without colour, so that they can be printed and photocopied.
For most purposes the slides can be printed as “Handouts” with up to 6 slides per sheet.

20. More Indices and Logs
The function contains the index x, so x is a log.
BUT the base of the log is e not 10, so
We know that
( since an index is a log )
We write as ( n for natural ) so,
Logs with a base e are called natural logs

21. SUMMARY
is a growth function.
(3 d.p.)
At every point on , the gradient equals y:
The inverse of is
( log with base e )
is defined for x > 0 only