Activity 1A helicopter takes off from the roof of

a building that is 200 feet about the ground.  The altitude of the helicopter increases by 150 feet each minute.  Express the height of the helicopter as a function of minutes after it starts rising.

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 10 apples. If someone gives me 1 apple then I’ve

increased my apples to 11 apples. Old value = 10, New Value = 11 I can also say I’ve increased my apple

supply by what percent?

11 10.1 or 10%

10

Let’s say I have $10 and someone steals $2. By what percent has my dollar amount changed?

Can we come up with a general formula for percent change?

new old new reference

ercent Changeold reference

P

8 10 2.2 or 20%

10 10

What if I said I had 10 apples and I increased my number of apples by 10%? Can we come up with a formula that gives the desired result of 11 apples?

What if I said I had $10 and I decreased my dollars by 20%? Can we come up with a formula that gives the desired result of $8?

11 10*(1 0.1)

8 10*(1 0.2) *(1 % )New value reference value change

Proof that our two expressions are essentially one in the same

*(1 % )New value reference value change

(1 % )

New valuechange

reference value

1 %

New valuechange

reference value

%

New value reference valuechange

reference value reference value

%

New value reference valuechange

reference value

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 50ppl. 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:

50*(1 + 0.1) = 55

Exponential Function

At 7pm we had 50ppl and by 8pm we have 55ppl.

Let’s say after another hour our party increases an additional 10%, so this time it’s 10% of 55ppl.

At 9pm (after 2 hours) we’ll have:55 * (1 + 0.1) = 60.5Since 50*(1 + 0.1) = 55, I can write:50 * (1 + 0.1) * (1 + 0.1) = 60.5What if from 9pm to 10pm our 60.5 ppl increases by another 10%?

60.5 * (1 + 0.1) = 66.55 ppl50 * (1 + 0.1) * (1 + 0.1) * (1 + 0.1) =

66.55pplTo recap:50 * (1 + 0.1)0 = 50ppl (after 0 hours)50 * (1 + 0.1)1 = 55ppl (after 1 hour)50 * (1 + 0.1)2 = 60.5ppl (after 2 hours)50 * (1 + 0.1)3 = 66.55ppl (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 50*(1 0.1)xy

*(1 )xy P r 1050*(1 0.1) 129.7 130y ppl

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

Second is with excel…

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?