Investigation 2: Exponential Functions: Growth & Decay

Onto a real world situation with an exponential function.

Suppose you've saved up $1,000.

You are pondering two choices.

The first

placing it in a bank that's offering 6.5% percent compound

interest each year.

Or the second, loaning it to a friend who's offering you 10%

simple interest each year for borrowing the money until he

pays you back.

The question is which is the best deal for you, and why?

A little prep work before you turn to the TI Inspire.

Simple interest is simple.

It means that each year you earn 10% of $1,000, which is

$100, until your friend decide to reimburse the low loan.

When interest is compounded, it means that each year, the

interest is added on to your principal

$1,000

which makes for a larger principle the following year.

Here we go.

Press the Home key for a new document.

Save or delete the open document and then select the

Lists and Spreadsheet page.

Move to the top of column a and type "year" for

the first list name.

Press Enter to move down.

You're now on row two, called the header or formula row.

Press Control Menu and select Generate Sequence.

u of n is the nth term of the sequence.

In this case, the n's year.

Each year it's one more than the previous year.

As usual, we use Tab to scroll down.

Enter 0 for the starting value.

Then tab down and click OK.

You've generated the year numbers.

Use the Nav Pad to move to the top of column b, and type

"bank" for the bank offer.

Press Enter.

Press Control Menu, and again select Generate Sequence.

You will then see amount of savings you'll have the end of

the year n.

It equals the amount from the previous year plus 6.5% of

that amount.

0.065 is the decimal form of 6.5%.

You begin with $1,000, so enter 1,000 next for the

initial term.

You only need one initial term, because we define each u

of n using the previous term only.

Tab down to click OK.

You can see your total savings at the end of years one

through four if you put your money in the bank.

Before scrolling down, let's complete the third column.

At the top of column c, type "friend" for

your friend's offer.

Press Enter to move down.

Press Control Menu one last time and select Generate

Sequence again.

Your friend is offering to pay you a flat 10% of the loan

amount, or $100, at the end of each year, until the

$1,000 is paid back.

The $100 is added to the amount from the previous year.

This formula will calculate simple interest.

Next, enter took 1,000 again for the initial

amount, same as before.

Even if the money is loaned, it's still yours.

It's like when you rent out a home.

You make money when someone else uses it, but

you're still the owner.

And tab down to click OK.

Now you can also see your total savings at the end of

years one through four if you accepted your friend's offer.

Notice that the first few rows of data suggest that your

friend's offer seems more advantageous.

Before scrolling down, take a moment to

reflect on these questions.

Is your friend's simple interest offer better than the

bank's compound interest plan?

If you're thinking it depends, what do you

think it depends on?

Pause a moment to jot down your conjectures.

Now let's go back to the TI Inspire to continue our

investigation.

Starting at 1,000, the values in column c increase at a

constant rate of 100 per year, like a staircase whose steps

have a common rise of 100 units.

We are familiar with such relation.

Each cell of yearly savings y is a linear function of the

year number x, with a slope of 100 and a

y-intercept of 1,000.

y equals 1,000 plus x as a straight line graph.

Notice that up until the 14th year, your friend's offer

would be better for you.

But at the end of the 14th year and beyond, the bank's

offer begins to take the lead.

So your decision would depend on how long your friend takes

before reimbursing your $1,000.

Now I want to focus on column b, as it's more complex.

And it's the topic of our lesson.

Let's begin by creating a scatter plot of the points

with x-coordinates in column a and y-coordinates in column b.

Use the Nav Pad to move to the year cell at top of column a.

Press the up arrow once again to select the entire column.

Hold down the Shift key as you use the right arrow to

move to column b.

Now both columns are selected.

Press Control Menu and select Quick Graph.

The scatter plot of points whose ordered pairs are a, b

or year, bank

are now plotted at the right.

To widen the right part of the screen, press Control Home and

under Page Layout, select Custom Split.

Use the left arrow to drag the mid-line all

the way to the left.

Press Enter.

Let's use the exponential regression feature of the TI

Inspire to find the equation for this function.

Press Menu.

And under Analyze, select Regression.

Then select Exponential.

The TI Inspire, using complex computation beyond the scope

of beginning algebra, finds the equation of the function

that passes through all the points.

Jot down the equation for later discussion.

Then we'll do one more thing.

Press Menu.

And under Analyze, select Plot Function.

Type in the equation of your friend's offer

1,000 plus 100x

then press Enter.

This confirms what we already saw.

The linear function graph is above the exponential graph

for the first 14 years or so, which makes it the better deal

during that time.

But then the exponential graph passes it and becomes the

better deal for you thereafter.

So indeed, the answer depends.

It depends on how long your friend takes to reimburse the

$1,000 he borrowed.

Because at that time, you'd stop earning simple interest

on the loan.

Now I'd like to unpack the algebra behind the equation we

found for the bank's compound interest investment account.

You begin with the principle of 1,000.

After one year, you have the original 1,000 plus 6.5% of

that 1,000.

Factoring out the common factors gives 1,000 times

1.065, or 1,000 times 1.065 to the first power.

This is your new principal.

After two years, you have the principle of the end of year 1

plus 6.5% of that.

This time, factoring out the common factors from both

terms, yields 1,000 times 1.065 to the second power.

This is your new principal after year two.

After three years, you have 1,000 times 1.065 to the

second power plus 6.5% of that.

Factoring out the common factors again in both terms

gives 1,000 times 1.065 to the third power.

So there's a pattern, and the exponent corresponds to the

number of years elapsed.

So after x years, you would have y equals 1,000 times the

factor 1.065 to the x power dollars in the bank.

And this would continue to grow exponentially as long as

you leave the money in your account.

As you can see, exponential functions can be helpful as

you begin to explore personal finances.

I hope you'll use the TI Inspire to further explore

these and other types of exponential function

applications.