C2: Exponential Functions Learning Objective: to be able to recognise a function in the form of f(x) = a x

Exponential functions So far in this course we have looked at many functions involving terms in x n. In an exponential function, however, the variable is in the index. For example: The general form of an exponential function to the base a is: y = a x where a > 0 and a ≠1. You have probably heard of exponential increase and decrease or exponential growth and decay. A quantity that changes exponentially either increases or decreases more and more rapidly as time goes on. y = 2 x y = 5 x y = 0.1 x y = 3 – x y = 7 x +1

Graphs of exponential functions

Exponential functions In both cases the graph passes through (0, 1) and (1, a ). This is because: a 0 = 1 and a 1 = a for all a > 0. When 0 < a < 1 the graph of y = a x has the following shape: y x 1 1 When a > 1 the graph of y = a x has the following shape: y x (1, a )

If we were to plot the graphs y = 1 x y = 2 x y = 3 x y = 4 x etc we would find that all these curves pass through (0,1) because y = a 0 = 1 There is a special curve that passes through (0,1) which has a gradient of 1, this is The Exponential Function, called y = e x, which we shall meet later.

Task 1 : Draw the graph of y = (0.6) x, for -4≤ x ≤ 4. Use your graph to solve the equation (0.6) x = 2.