Exponential Functions

The “I’m going to lie to you a bit” version

Exponential functions measure steady growth

• If you really want to know what that means exactly, take differential equations (after Calculus)

• Here’s the basic (lying) version• An exponential growth happens when

something is making more of itself (in a “steady” way)

• People, money, bacteria, etc…

Example

• One dollar makes one dollar every year.

$1

Year 0

$1

Year 1

Example

• One dollar makes one dollar every year.

$1

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Year 1

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Example

• One dollar makes one dollar every year.

$1

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Example

• One dollar makes one dollar every year.

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Example

• One dollar makes one dollar every year.

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Example

• One dollar makes one dollar every year.

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Example

• Every year I keep what I have and add what I have.

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Example

• Every year I double my money

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My dollars over time

Example

• Every year I double my money

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y=1(2x)y=# of $

x=# of yrs

Approximation function ƒ(x)=2x

Standard form(s) of an exponential

• y=abx

a is the initial value (y-intercept) b is the growth factor

• y=aerx

a is the initial value (y-intercept) r is the “per-capita rate” also called “exponential

growth rate”• Conversion

b=er, r=ln(b)

When to use an exponential model

• Things making more things– People making more people, bacteria making

more bacteria, money making more money (interest)

• Lots of identical events happening at random times– Radioactive decay (atoms decay at random times),

Heat transfer (atoms bump into each other randomly)

Asymptotes

I have to lie to you until you take Calculus edition

Asymptote

• An asymptote is a line that an equation gets close to but never reaches.

• Every exponential y=abx has a horizontal asymptote at y=0.

Growth and Decay

I’m still lying to you a bit

Simple Version

• A growth function is an exponential function where y gets bigger when x gets bigger.

• A decay function is an exponential function where y gets smaller when x gets bigger.

• The easy way to tell is to graph it!

Growth and Decay

Exponential Growth Exponential Decay

Non-calculator examples

• y=3(4)x.– When x=0, y=3. When x=1, y=12.– When x gets bigger, y gets bigger.– Exponential Growth

• y=2(1/3)x

– When x=0, y=2. When x=1, y=2/3.– When x gets bigger, y gets smaller.– Exponential Decay

Non-calculator examples

• Y=7(2)-x.– When x=0, y=7. When x=1, y=7(2)-1=7/2.– When x gets bigger, y gets smaller.– Exponential Decay

• Y=1(1/2)-x

– When x=0, y=1. When x=1, y=(1/2)-1=2.– When x gets bigger, y gets bigger.– Exponential Growth

1) y=3x growth. 2) y=3-x decay.

3) y=(1/2)x decay.4) y=(1/2)-x growth.

d) 2&3 represent decay

Logarithms

Seriously, take calculus, please

Example

• Every year I double my money

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Example• If I know what time it is and want to know how much money I have

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a=1(2t)a=# of $

t=# of yrs

Example

• What if I know money and want to know time?

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Example

• How long would it take me to get $1,000,000?

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Flashback!

I start 10 miles away from my house and drive away from my house at 30 mph. If I know how far I am from my house, how long have I been driving?d=number of miles away from my houset=number of hours I’ve been drivingd=30t+10(d-10)/30=t

Flashback!

I start 10 miles away from my house and drive away from my house at 30 mph. If I know how far I am from my house, how long have I been driving?d=number of miles away from my houset=number of hours I’ve been drivingd=30t+10(d-10)/30=t I need an inverse!

Example

• What’s the log27?• Get log(7) in your calculator ≈

0.845098• Get log(2) in your calculator ≈

0.30103• Log27 = Log(7)/Log(2) ≈

0.845098/0.30103• Log27≈2.80735

Example problem

• Find the domain of 2log7(4x-3)+7x-9

Whatever is inside the log has to be >0.I can find an answer whenever 4x-3>0x>3/4