Table of Contents

• Part I: Introduction
• 1: Word of the Day
• 2: Two bank accounts
• 3: Graphing your money
• 4: Graphing fruit fly populations
• 5: "Exponential" is not always "fast"
• Part II: Exponential Growth and Shrinkage
• 6: How to calculate next year's population
• 7: Using exponents
• 8: Problems using exponents
• 9: Negative rates = Shrinkage
• 10: Shortcut for exponential shrinkage
• 11: A quick review of exponential models
• Part III: Practice Problems
• 12: Drug dosage
• 13: Caffeine withdrawal
• 14: Growth of the computer industry
• 15: Viral growth
• 16: Wind energy
• 17: Ecology
• 18: Human population growth

Graphing your money

If you save $50 per year

Remember that the slope of a line tells you how much y changes as x changes – in this case, how much your balance changes with each passing year.  If you're depositing 50 bucks every year, then the slope of the line representing your savings account is simply .  It stays the same over the years, and therefore we have a straight line.

So here’s the line:

Of course if you saved more every year, the slope would look less pathetically shallow. So cut out your NetFlix subscription, save $100 per year, and the line will go up twice as steeply. Cut out the text messaging, save $1000 per year, and the line will go up even more steeply. But, no matter how much you save, if its the same amount every year, the graph will always look like a straight line.

What about 5% per year

 

When you get interest on your savings, you do NOT get the same amount every year. Let's say you start with a "nest egg" of $1000 and never add another penny. The bank adds interest, so in the second year you have more money than in the first, and the second year's interest is higher than the first year's. Each year this account grows, and therefore it gets more interest than it did the year before.  What would a graph of this look like? 

  Answer these questions first:

1. Will it be a straight line?

Yes
No

2. How does the slope change over time?

Increases
Decreases
Stays the same
Changes unpredictably

3. Where should the line start? 
4. Now try drawing the line on the screen.  Then click on the button to see how close you got.

 

 

Bottom line: since your balance is growing, the interest added each year grows as well, and the line curves upward. The curving effect on this graph is very slight (put a piece of paper along the curve at year 1 if you need to be convinced that there IS a curve), but the curve is still there.

Of course if you find a better bank that will offer you 8% interest, the line will curve up even more. Or if you lend it to your brother-in-law at 1% upward, it will curve less. But, assuming you're getting an interest rate greater than 0%, it will always curve up.