ExponentialFunctions--FullProgram.mp4

Algebra Nspirations

Exponents and Exponential Functions

The algebraic formula describing the shape of the

Saint Louis Gateway Arch.

The growth patterns of aesthetically pleasing

geometric shapes called fractals.

The rapid spread of disease among populations of humans,

animals, and plants.

The process called carbon dating archaeologists use to

estimate the age of artifacts.

And the disappearance of pattern of African elephants

due to loss of natural grazing land.

All use exponents in their algebraic expression or

functional equation.

Indeed, exponents and exponential functions are used

to describe these and other interesting objects and

processes in fields such as demography, biology,

chemistry, physics, economics, and finance.

We'll delve into some real world applications, but first

we'll look at exponents themselves by examining

notation, properties, and graphs.

Let's begin with some fun numbers.

A 31-year-old has lived over one billion seconds.

The number of oxygen atoms in an average sized

thimble is 1 octillion.

And a googol

not the internet search engine

is 1 followed by 100 zeros.

Now for some very small numbers.

A braid of 20 strands of fair hair is about

one millimeter thick.

A germ is typically one micrometer long.

And a water molecule is less than one

nanometer in diameter.

Multiplying an integer by 10 results in adding a zero.

So one billion is 10 multiplied by 10 nine times.

You may already know that mathematics has a shorthand

notation for this

10 to the ninth power

where the exponent denotes the number of factors.

Since 1 octillion has 27 zeroes, we

write 10 to the 27th.

And one googol is 10 the 100th power.

These are called powers of 10.

10 is the base, and the numbers at top

right are the exponents.

Here's the general notation for a power, any base, b.

Zero and negative exponents are easy to understand if we

look at the following patterns.

Notice from line to line, we're dividing by 10, thus

obtaining smaller and smaller powers of 10.

If you keep dividing by 10, we get 10 over 10, which is 1.

So 10 to the 0 is 1.

Dividing further, one over 10 is 10 to the negative 1.

1 over 10 times 10, or one over 10 squared, is 10 to the

negative 2.

One over 10 times 10 times 10, or one over 10 cubed, is 10 to

the negative 3.

And so on.

We can now use negative exponents to express a very

small number.

One millimeter is a thousandth of a meters.

1 over 1,000 is 10 to the negative 3.

Likewise, one micron or micrometer is a

millionth of a meter.

One over 10 to the sixth is 10 to the negative 6.

And one nanometer, used in nanotechnology, is one

billionth of a meter.

One over 10 to the ninth is 10 to the negative 9.

Scientific notation, a compact method of writing large and

small numbers favored by mathematicians, scientists,

and engineers, uses powers of 10.

For example, $10.71 trillion dollars, the total United

States federal debt as of February, 2009, can be written

in many equivalent ways.

The last line is in scientific notation, because the number

multiplying the power of 10 is between 1 and 10.

The wavelength of blue light, which is within the visible

light spectrum for humans, is about 475 nanometers.

These are all equivalent, but again, the last line is

scientific notation, because the number multiplying the

power of 10 is between 1 and 10.

In general, a number in scientific notation has the

form a times 10 to the n, where a real

number and n an integer.

If a is positive, a is greater than or equal to 1 but less

than 10, or greater than negative 10 and less than or

equal to negative 1 if a is negative.

Using an absolute value, we can combine both conditions

into one and write it like this.

This form allows for easy comparison of two numbers at

the same time, as the exponent gives

their order of magnitude.

Here's a fun problem.

I'm sure you'll quit your friends on it.

Take a sheet of paper this is a standard sheet of 20 pound,

recycled paper about 0.0038 inches thick.

Translating that into something you can visualize,

300 sheets of these stand about 1 inch tall.

Fold it once.

I have two layers.

Twice.

I have four layers.

Thrice.

I have eight.

Four times

16.

And after five times, I have 32 layers.

How high will this stack be after 50 folds?

Imagine a huge sheet of paper.

You'd have to cut and stack after a while, because folding

will become impossible.

Our doubling process here can be modeled with powers of 2.

n folds produces 2 the n layers.

So 50 folds would give a stack of 2 to the 50th layers, or

sheets of paper.

With 300 sheets equal 1 inch, 12 inches equal 1 foot, 3 feet

equals 1 yard, and 1,760 yards equal 1 mile, I'll let you do

the computation.

After just 50 folds, to do with that would be about 60

million miles high.

Yes, you heard it right.

That's more than 2/3 of the way from the Earth to the sun.

Mind boggling.

We've seen powers of 10 or 10 to the x, and powers of

2 or 2 to the x.

Let's turn now to the TI Inspire and use function

graphs to visualize the patterns of change of the

simplest exponential function of the form y equals b to the

x for a variety of base values b.

When the pattern increases, we have exponential growth.

When it decreases, it's exponential decay.

Finally, notice that we use b to the n for integer exponents

and b to the x for real exponents.

Turn on the TI Inspired.

Press the Home key for a new document.

If a document is open, you'll be prompted to save it.

After deciding, select 2 to create a graph

and geometry page.

To define f1 of x, enter the letter b for the base,

followed by the power or carrot key.

And then x for the exponent.

Press Enter to graph.

No graph appears because we haven't defined values for b.

To do so, let's insert a slider.

Press Menu, and under Action, select Insert Slider.

Type b to replace the default variable, z1.

Press Escape or Tab and the graph will appear.

Notice the graph lies above the x-axis, meaning that the y

values of all points on the graph are positive.

Let's therefore modify the window settings.

Press Menu and under Window, select Window Setting.

Use the Tab key to scroll down from one line to the next.

Enter negative 8 and 8 for x.

Skip Auto, then negative 3 and 12 for y.

Tab down to OK.

Next, use the Nav Pad to move the pointer over the slide

bar, also called the thumb of the slider.

Press Control click to grab the thumb.

You should now see a closed hand.

Use the Nav Pad to slide the thumb along the slider track,

noting the changing values of b and how they affect the

shape of the exponential graph.

Since the tick marks on the track represent 2, 4, 6, and 8

along the continuous interval, 0 to 10, we couldn't see very

well what happened when b took on small values

between 0 and 1.

To better observe this, let's modify the slider setting.

Press Control, Menu to access the Slider Context Menu, and

select Settings.

Here again, use Tab to scroll down.

Skip b, then enter 0, negative 0.1, and 3.1.

For step size, select Enter Value by

pressing the down arrow.

Type 0.2 and then tab down to OK.

Again, slide the bar slowly along the

interval from 0 to 2.

Carefully observe what happens at b equals 0 and b equal 1.

And notice when the graph switches from a falling graph

to a rising graph

or from descending to ascending.

Stop at the graph corresponding to b equal 3.

One more property of exponential function graph

before summarizing our findings.

Press Menu, and under Points and Lines, select Point On.

Use the Nav Pad to drag the pointer to quadrant one.

Click to place one point at the top of the graph, another

in the middle, and a third lower down.

For each point, xy 3 to the x equals y.

Press Menu again, and this time, under Transformation,

select Reflection.

To reflect each point about the y-axis, drag the blinking

pencil to the y-axis first.

Click, then click on the point.

Repeat for each point.

Press Escape to exit this selection mode.

Notice the short decimal expansion of

numbers in this document.

You can do the same by pressing Home, then selecting

Document Settings.

Change your settings as you wish.

Next, press Control G to reveal the function entry

line, and enter the following for f2.

The left paren.

Control division for a fraction placeholder.

The fraction 1/3.

The right arrow.

The right paren.

They carrot key.

And finally x.

This is the exponential function in base 1/3.

Press enter to graph.

f2 of x passes through all three reflected points.

Notice how f1 of x equals 3 to the x, and f2 of x equals 1/3

to the x are reflections

or mirror images of each other

about the y-axis.

Can you figure out why?

Lastly, notice the intersection

point of the two graphs.

If y-coordinate is 1, and since it's on the y-axis, its

x-coordinate is 0.

In fact, all exponential functions of the form b to the

x pass through 0, 1.

Let's generalize what we've found in the form of

fundamental properties for all exponential functions of the

form b to the x.

One, exponential functions of the form b to the x are

defined for positive base values only.

We run into problems if b is negative.

We also exclude b equals 1, since 1 to the x

always equals 1.

Two, the domain is the set of all real numbers, since b to

the x is defined for all x.

Three, the range of the set of positive real numbers, since y

equals b to the x is always positive.

The graphs were all above the x-axis.

Four, all exponential functions b to the x have a

y-intercept of one.

Indeed, their graphs intersect the y-axis at the

point 0 comma 1.

Five, if b is greater than one, the functions increase.

And if b is strictly between 0 and 1, the functions decrease.

Six, if the bases are reciprocals of each other,

like 3 and 1/3, the graphs are symmetric about the y-axis.

The last result is explained by the following facts.

1 over b to the x equals b to the negative x, and x and

negative x are symmetric about the y-axis.

Reflect on that for a moment.

And if you need to, pause the video.

Before we investigative a finance application of

exponential functions, try this.

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 back.

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.

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