I’m professor Turner, and today we’re going to learn about exponential functions. 

·      A function is a relationship or expression involving one or more variables:

o   One you may already be familiar with is the linear function:

§  Y = mx + b

o   The exponential function typically takes the form of:

§  Q = f(t) (pronounced “f of t”)= ab^t

§  For our purpose, let’s consider “t” as the time step

·      So Q is equal to f at time “t”

§  a is the initial value or y-intercept

·      where the line crosses the y-axis

o   changing the value of a shifts (horizontally) the line along the y-axis

§  b is the growth factor or base

·      when b > 1 – the slope is positive or the line is increasing exponentially

·      when 0 < b < 1 – the slope is negative or the line is decreasing

·      having a larger positive base > 1 will make the slope steeper

o   and bring it closer to the y-axis (horizontal compression)

o   Put it all together:

§  Lets suppose that at every time-step (one day), we double (b) our initial (a) value or amount of songs in our itunes library – what might this look like in the form of an exponential function?

·      Our initial (on day 1 – t = 0) amount of songs is 3 (a= 3)

·      Each time-step (t) is one day

·      We will double (b = 2) our songs everyday for a month (30 days)

·      So: how many songs will we have on day 2, 3, 4, 15 and 30?

o   Our initial equation looks like:

§  No time step has occurred

§  f(0) = 3*2⁰ = 3*1 = 3

o   Our first time step:

§  One day (time-step) of doubling has occurred

§  f(1) = 3*2¹ = 3*2 = 6

o   Second time step:

§  We’ve doubled the amount from the day before

§  f(2) = 3*2²  = 3*(2*2) = 3*4 = 12

o   Third

§  f(3) = 3*2³ = 3 * (2*2*2) = 3*8 = 24

o   Fourth

§  f(4) = 3*2⁴ = 3 * (2*2*2*2) = 3* 16 = 48

·      We can see that the product of the base combined with its exponent doubles at every time-step

o   E.g. 1 , 2, 4, 8, 16