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

Module C3 "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"

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

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

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 . . .

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

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.

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 .

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

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

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

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,

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 .

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

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

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

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

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.

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

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