Exponential function A linear relationship is one in which there is a fixed rate of change (slope). An exponential relationship is one in which for a.

Exponential function

A linear relationship is one in which there is a fixed rate of change (slope).

An exponential relationship is one in which for a fixed change in x, there is a fixed percent change in y.

Percent ChangeLet’s say I have 2 Lamborghinis and

someone gives me 1 Lamborghini. By what percent has Lamborghini collection grown?3 2

.5 or 50%2

New Value Reference Value%

Reference Value

New Old - Change

Old

If I have 3 Lamborghinis and were to decrease my collection by 33% I would be left with:

3*(1 0.33) 2

*(1 % )New value old value change

x y

0 192

1 96

2 48

3 24

Looking at this data set, is there a fixed percent increase or decrease in our y variable?

Exponential Function

For linear functions once we found that the slope stayed the same (that it was a linear function) we then were able to find the y intercept and create a linear model.

Same for exponential functions. Once we know that there is a fixed percent increase or decrease in the y variable, we can then create an exponential model of our data.

An exponential function is just repeating a percent increase or decrease

For example: Let’s say we throw a party and invite 100ppl. They all show up at 7pm, but, since the party is raging, they all invite some of their friends. The party increases by 10% after the first hour. So at 8pm we have:

100*(1 + 0.1) = 110

Exponential Function

At 7pm we had 100ppl and by 8pm we have 110ppl.

Let’s say after another hour our party increases an additional 10%, so this time it’s 10% increase on 110ppl.

At 9pm (after 2 hours) we’ll have:110 * (1 + 0.1) = 121Since 100*(1 + 0.1) = 110, I can write:100* (1 + 0.1) * (1 + 0.1) = 121What if from 9pm to 10pm our 121 ppl increases by another 10%?

121 * (1 + 0.1) = 133.1 ppl50 * (1 + 0.1) * (1 + 0.1) * (1 + 0.1) =

133.1pplTo recap:100 * (1 + 0.1)0 = 100ppl (after 0 hours)100 * (1 + 0.1)1 = 110ppl (after 1 hour)100 * (1 + 0.1)2 = 121ppl(after 2 hours)100 * (1 + 0.1)3 = 133ppl(after 3 hours)Can we come up with a formula for the

number of people at the party, where y is the number of people and x is the number of hours after the party has started?

General Exponential Formula

Where y is the new value, P is the initial value, r is the percent increase (+) or decrease (-) in decimal form, and x is the number of repetitions.

How many people will be at the party after 10 hours assuming a 10% increase in people every hour?

*(1 )xy P r 100*(1 0.1)xy

*(1 )xy P r

Two ways to find how many people will be at the party after 10 hours. First is with the formula

Second is with excel…

10100*(1 0.1) 259.4 259y ppl

x y

0 192

1 96

2 48

3 24

Express this data set as an exponential formula.

y = 192 * (1-.5)x

Exponential? If so, can we put it into a exponential

model?x y

0 0.5

5 1.5

10 4.5

15 13.5

Example: A bacteria population is at 100 bacteria and is growing by 5% per minute.

How many bacteria cells are present after one hour?

Do this first by setting up an excel spreadsheet.

Second using our formula.See if you get the same results.How long does it take for there to be

10000 bacteria?

Proof that our two expressions are essentially one in the same

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