How to Deal with Exponential Functions

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GENERAL EXPONENTIAL FUNCTIONS:

These are used to show the growth/decay of a value over time using exponents and the initial value. 

Formula: 

f(x) = a ⋅ bˣ  

a = the value you start with

x = amount of time

b = the growth/decay factor

EXPONENTIAL GROWTH:

In here, b will ALWAYS be GREATER than 0.  

Key words on identifying exponential growth problems:

Tripled ==> b = 3

Doubled ==> b = 2

Example Problem:

Minnie Mouse had 70 bows. Over the course of 4 years, the amount of bows kept tripling each year. How much does she have now?

 As you can probably see, I have put key words/numbers in bold. 

a = 70 (starting amount)

x = 4 (amount of years)

b = 3 (growth factor)

Now just set up the formula!

f(x) = 70 ⋅ 3⁴

3⁴ = 81 

70 ⋅ 81 = 5,670

In total, Minnie Mouse has 5,670 bows now. 

EXPONENTIAL DECAY: 

In here, b will ALWAYS be BETWEEN 0 and 1.

Key words on identifying exponential decay problems:

Halved ==> b = 0.5 OR 1/2

Cut down to a third ==> b = 1/3

Example Problem:

Mickey Mouse had 128 cookies at his clubhouse for a party. Every hour, the amount of cookies halved. It has been 6 hours, how many cookies are left now?

a = 128 (original amount)

b = 1/2  (decay factor)

x = 6 (amount of hours) 

f(x) = 128 ⋅ (1/2)⁶

(1/2)⁶ = 1/64 

128/64 = 2

There are 2 cookies left.

EXPONENTIAL GROWTH WITH PERCENTAGES:

Formula: f(x) = a ⋅ (1 + r)ˣ

Example Problem:

Daisy Duck wants to buy a Labubu that costs sixty dollars. However, its price keeps increasing by 16% each year. It's been five years, how much should it cost now?

Just to make life easier...

Please use the formula that was given under:

"EXPONENTIAL GROWTH WITH PERCENTAGES"

Convert the percentage into a decimal or a fraction. In this case, a decimal.

16% = 0.16

a = 60

r = 0.16?

NO! 

Since this is a problem that has percentages, we will use the other formula.

r = 1 + 0.16 = 1.16

x = 5

f(x) = 60 ⋅ (1.16)⁵

PLEASEEEEE USE A CALCULATOR! 

60 ⋅ 2.10034166 = 126.0204996

YOU MUST ROUND TO THE NEAREST HUNDREDTH!

The Labubu costs $126.02 now.

EXPONENTIAL DECAY WITH PERCENTAGES:

Formula: f(x) = a ⋅ (1 - r)ˣ

Similarly, always convert b (percentage) into a decimal or fraction. 

Example Problem:

Donald Duck bought a Lamborghini for seven dollars. (It was on sale, duh!) Over the course of two years, its value depreciated by 3%. How much does it cost now?

a = 7 (original amount)

r = 0.97 (decay factor)

3% = 0.03

1 - 0.03 = 0.97

x = 2 (amount of years)

7 ⋅ (0.97)² 

7 ⋅ 0.9409 = 6.5863

His Lamborghini costs $6.59 now.

...Just to let you know, that is not true in reality.

VERY IMPORTANT SIDE NOTE!

I only put problems in here that don't have "grouped timelines". 

For example, my friend gave me fifty candy bars. Every hour, the amount tripled, and it's been seven hours. 

That is NOT a grouped timeline problem, that's because it's "every" hour, meaning the hours I'm accounting for are individual. Therefore, the math formula will still remain normal. 

But on the other hand, I have seventy candy bars, and the amount doubles every decade, and it's been five decades so far. I want to know the individual years stuff.

That IS a grouped timeline, because we're talking about decades, 10 year periods. My formula can change into:

Formula: f(x) = a ⋅ bᵗ/ᵏ

There's a fraction in the exponents because we're accounting for individual years. 

So in this case:

70 ⋅ 2⁵⁰/¹⁰ = 70 ⋅ 2⁵ = 2,240

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