Exponential Function Lecture

Good afternoon everyone. My name is Professor Shahad. Today we are going to be talking about exponential functions. This might seem like a daunting concept, but it is actually really simple. So I will try to explain this as concisely as I can to avoid confusion. If you could bear with me for 10 minutes while I explain. Then we will follow with an example and group activity.

All right. Firstly an exponential function is shown as f(t)= ab^t, where

·      a = initial value , a≠0 (also known as the y-intercept)

·      b = growth rate, b>0

A few things to keep in mind regarding exponential functions

1.    they always increase from left to right

2.    there is a horizontal asymptote of y=0

3.   the graph is concave up and is continuous

Let me explain what that all means for a second:

 On a graph the exponential function starts from left to right  so as the graph moves from the left to the right the values increase. That would suggest that the graph is concave up.

2.    As the graph approaches the line y=0 when the value of t moves between -∞ and ∞ the line never reaches the line y=0, making it a horizontal asymptote about the line y=0

3.    The graph intercepts the y-axis at (0,a), a being the initial value and a≠0.

Now that you know what the value of ‘a’ is. Let’s focus on the value of ‘b’.

As I said so far the value of b is the growth value. The growth value can either be negative or positive. So technically it could be both a decreasing or increasing rate. However b>0.

Basically the growth factor is shown by 1+r with

·      1 representing 100% of the initial value

·      r representing either the percentage increase or decrease.

So for example if you have and allowance of $10 a week and your parents decided to give you a 10% increase it would look like this:

You have 100% of your old allowance + 10% increase so in total you have 110% of your old allowance.

If you change it into a decimal you have 100÷100 =1 and 10÷100 =.10

 So that is 1+.10, which if you notice is in the form 1+r.

Therefore the growth rate is 1.10

Keep in mind that this could also be a decay rate. Yet the same logical steps apply.

           Suppose your parents reduce your initial allowance by 10%

So you would have,

100% of your old allowance – 10% = 90% of your old allowance left.

As before just turn this into a decimal by dividing by 100, 90÷100 = 0.90 is your decay rate.

In general remember this rule to know whether or not it is a decay or growth rate.

When:

• ·       r>0 and b>1, then 1+r = growth rate
• ·       r<0 and 0<b<1 then 1+r = decay rate

Finally the value of ‘t’ is simply the amount of times that you will multiply your initial value ‘a’ by your growth factor ‘b’.

So let’s complete one example to tie this all together.

Assume we have $1,000 in your wallet and you wish to invest it. You could increase the amount you have by 10% every year for 3 years. So, firstly you should decide what value is what.

So you initially have $1,000 that would mean your initial value is 1000.

            So a = 1000

Secondly, you increase by 10%. Therefore calculate the growth rate.

You have 100% of the initial value + 10%

= 100% + 10% convert this into decimal form

= 1 +.10

= 1.10

So your growth rate is 1.10

            b= 1.10

Finally, t is the amount of times you will multiply a by b. So that would be 3 years.

            t= 3.

Now all that is left to do is to plug these values into the formula.

F(t) = ab^t

       = 1000 x 1.10³

          =1331

So by the end of 3 years you should have $1331 in the bank account.

I think we should stop here for today in terms of lecture. Please come collect the group activity that you will complete. Let me know if you have any questions.