Video Captions: Algebra
Applications: Exponential
Functions

Video Captions: Algebra
Applications: Exponential
Functions

[Music]

Narrator: In August 2008, China
hosted the Olympics to

great fanfare.

But only a few months earlier
China was host to an unwelcome

guest, a massive earthquake
that devastated the

Sichuan Province.

As a result of the earthquake
of May 8, 2008 nearly seventy

thousand people died and
millions were left homeless.

The magnitude of the earthquake
at its epicenter

was seven point nine on the
Richter scale.

As the name suggests, the
epicenter is where the most

damage occurs.

As you get farther away from
the epicenter there is

less damage.

In fact, the amount of damage
decreases dramatically.

This suggests that the
magnitude of an earthquake

changes in a non-linear fashion.

This color-coded map from the
US Geological Survey shows the

intensity of the earthquake
near the epicenter.

The red areas are where the
most intense earthquake

activity occurred.

The green area showed moderate
activity.

The distance from the red area
to the green area is about one

hundred and fifty kilometers,
or about one hundred miles, a

very short distance compared to
the overall size of

the earthquake.

Because of the dramatic changes
in intensity from one

point to another earthquakes
are an ideal example of

exponential functions.

Unlike a linear function, which
changes in value in a

constant way, exponential
functions change value in

big leaps.

If earthquake intensity could
be modeled using a linear

function then the map would
show a gradual change from red

to green, so that a larger area
would be affected by

the earthquake.

Instead a non-linear function,
in this case an exponential

function, shows the worst of
the earthquake damage

localized to a relatively small
area.

Exponential functions and their
inverses provide ideal

models for us to better
understand earthquakes, but

before applying these
mathematical models let's

first get a better
understanding of what an
earthquake is.

An earthquake is a vibration in
the Earth's crust.

This vibration causes buildings
near the epicenter

of the quake to shake.

Depending on the intensity of
the earthquake, a shaking

building can collapse from such
an intense vibration.

This table shows the amount of
shaking measured in the

different regions near the
Sichuan earthquake of 2008.

The first column shows the
distance from the epicenter of

the earthquake.

The second column shows the
acceleration of buildings and

other structures in the area.

Let's use the TI-Nspire to
analyze this data set.

Turn on the TI-Nspire.

Create a new document.

You may need to save a previous
document.

Create a spreadsheet window.

Move to the top of column A and
add this column heading.

Press TAB to go to the top of
column B and add this

column heading.

Move to cell A1.

Input the data in the table and
columns A and B.

Pause the video to input the
data.

Create a scatter plot.

Press HOME to create a
statistics window.

Use the nav pad to move the
pointer to the x-axis.

When you see the words 'click
to change variable' press the

CLICK button once.

Then use the DOWN ARROW to
select distance.

Press click again.

Use the nav pad again to move
the pointer to the

vertical axis.

Once again look for the words
'click to change variable'.

Press CLICK and select
ACCELERATION.

Your graph should look
something like this.

If it doesn't go back to the
spreadsheet to make sure you

have input the data correctly.

Click CONTROL and the LEFT
ARROW to see

the spreadsheet.

The scatter plot shows a
non-linear pattern.

In fact, if you overlay a
curve, like the one shown,

you'll see how the data are in
the shape of an

exponential function.

To find the actual equation of
the exponential curve modeled

by this data set use the
regression capabilities of

the Nspire.

First make sure that the
statistics window is active.

Press MENU, and under ANALYZE
select REGRESSION, and under

that sub-menu select
EXPONENTIAL.

Your graph should look
something like this.

You may need to move a label
for the equations so that it

doesn't overlap the graph and
data points.

This is an example of a
decreasing

exponential function.

All exponential functions are
of the form of A times

B to the X.

These functions are either
increasing or decreasing.

This is an example of an
increasing graph.

As X increases in value so does
Y.

The data from the Sichuan
earthquake, on the other hand,

is a decreasing function.

The farther away from the
epicenter the less the

earthquake can be felt.

A decreasing function slopes
downward for increasing

values of X.

There are two ways to generate
such a function.

If b is greater than one then a
decreasing function requires

that X have a negative
coefficient.

For example, the graphs of two
to the minus X and ten to the

minus X are shown.

Another way to generate a
decreasing exponential

function, and this applies to
the Sichuan earthquake data,

is for b to be less than one.

For example, one half to the X
and one tenth to the X

are shown here.

In fact, notice that the graphs
of two to the minus X

and one half to the X are the
same, as are the graphs of ten

to the minus X and one tenth to
the X, which brings us to a

more general way of writing an
exponential function, Y equals

A times B to the CX.

This general form has certain
properties that apply to all

exponential functions.

When X equals zero Y is equal
to A, which is the

y-intercept.

In the case of the Sichuan
earthquake X equals zero is

the epicenter of the earthquake.

It is the highest value, which
means the values of X less

than zero are not defined.

As you can see with the Sichuan
earthquake, when B is

less than one and C is greater
than or equal to one then the

exponential function is a
decreasing function.

This table summarizes the
possible combinations of B and

C and their effects on the
graph.

In particular, the form of the
equation where C is greater

than zero and B is less than
one allows us to compare the

intensities of different
earthquakes.

So, now that we know what an
earthquake is let's take a

closer look at the difference
between earthquake intensity

and magnitude.

The magnitude of an earthquake
indicates how much damage

it caused.

So an earthquake of magnitude
seven point nine causes much

more damage than one of
magnitude five point zero.

But how much more?

The magnitude of an earthquake
is related to its intensity

according to this exponential
function.

Let's use this equation to
compare two earthquakes that

occurred in the Sichuan
Province.

In March 2009, a magnitude five
point zero earthquake

struck the same region.

The ratio of the 2008 and 2009
earthquakes is shown.

The law of exponents allows us
to express the ratio of the

powers of ten as shown.

Simplifying shows that the
intensity of the 2008

earthquake is nearly eight
hundred times larger than the

2009 earthquake.

So even though the magnitudes
differ by three the

intensities differ by a
significantly larger amount.

So when you hear something
described as growing

exponentially now you can see
how dramatic this

growth can be.

Magnitude, on the other hand,
does not grow exponentially.

We can see this using the
TI-Nspire.

Create a new document.

You may want to save your
previous work, otherwise press

the TAB key and press ENTER.

Graph the function Y equals ten
to the X.

This is the general form of the
intensity equation

for an earthquake.

Now add three points to the
graph.

Select the point on feature,
then use the nav pad to add

three points to the graph.

We want to reflect the graph
and the points across the line

of Y equals X.

To do so activate the line tool.

Use the nav pad to move the
pointer to the origin.

Click once.

Then move the pointer to
coordinate one comma one,

knowing that for now this is an
estimate.

To adjust the line so that it
is the graph of Y equals X

display the equation.

Press MENU and under ACTIONS
select COORDINATES AND

EQUATIONS and then click twice
on the graph.

You will see the equation of
the line, which will likely

not be Y equals X.

To change the equation press
ESCAPE, then move the pointer

to the line.

Press and hold the CLICK key
then adjust the line up or

down until you see Y equals X.

We are now ready to reflect the
exponential graph across

the line Y equals X.

Why do this?

The reflected graph is the
inverse of the exponential.

Press MENU and under
TRANSFORMATION

select REFLECTION.

Use the nav pad to click on one
of the points on the

exponential graph.

Then click on the graph of Y
equals X.

You'll see the reflective point
below the graph of

Y equals X.

Repeat with the other two
points.

Try to get your screen to look
like this.

You can see the graph of the
inverse taking shape.

To see the entire graph that
fits the three points use the

locus of points tool.

Click on MENU, and under
CONSTRUCTION select LOCUS.

Use the nav pad to move the
pointer above one of the

points on the exponential graph.

Click to select it.

Then use the nav pad to select
the corresponding

reflected point.

When you do you will see the
inverse graph.

Try to get your screen to look
like this.

You can now see that the graph
of the inverse is a mirror

image of the exponential graph,
if you consider the

line Y equals X as the mirror.

Furthermore, if you display the
three coordinates on the

inverse you'll see that they
are mirror images of the

coordinates of the exponential
function.

In fact, the inverse of ten to
the X is log base 10 of X,

usually just written as log X.

As with all inverses, F inverse
of F of X is

equal to X.

In other words the log of ten
to the X is X.

What this means is that the
exponential graph gives you

the intensity of the earthquake
while the

logarithmic graph gives you the
magnitude.

The exponential graph increases
rapidly the way that

earthquake intensity does,
while the logarithmic graph

increases slowly the way that
earthquake magnitude does.

Now that you've seen the
difference between intensity

and magnitude, here is
something else to consider.

When an earthquake occurs what
is felt is the intensity, but

what is measured is the
magnitude, and so this brings

up the question: How is
magnitude measured?

An earthquake is a vibration in
the Earth's crust.

All vibrations are waves that
can be modeled a trigonometric

function, like the one shown
here.

The height of the wave above
the x-axis is called

the amplitude.

When an earthquake occurs a
seismic wave is produced.

An instrument called a
seismograph is used to detect

the seismic wave.

The measure of the amplitude is
used in this formula to

find the magnitude of the
earthquake.

Notice that this is a
logarithmic function.

Once the magnitude is measured,
using the

exponential function ten to the
X gives the intensity of

the earthquake.

To see how dramatic the changes
are in the amplitude

of a seismic wave we can link
the coordinates of the

exponential function to the
amplitude.

Turn on the TI-Nspire.

Create a new document.

You may want to save your
previous work, otherwise press

the TAB key and press ENTER.

Graph the function Y equals ten
to the X.

Place a point on the graph of
the exponential function, then

press ESCAPE.

Move the pointer to the Y
coordinate of the point.

Recall that this is the value
of the earthquake intensity.

Press the variable key then one.

You will be storing the value
of this coordinate in the

variable A.

Now graph the sine curve, but
link the value of the

amplitude to the variable A.

We now need to change the
window settings.

The x-axis should go from zero
to ten to approximate the

range of values for the
magnitude.

The y-axis, on the other hand,
should extend to at least

one billion.

So, press MENU, and under
WINDOW ZOOM select

WINDOW SETTINGS.

Change xMin to negative one,
press the TAB key, and change

xMax to ten.

Tab to yMin and change it to
negative one billion.

Press the TAB key and change
yMax to positive one billion,

then tab your way to OK and
press ENTER.

Now use the nav pad to hover
over the point on the

exponential graph.

Click and hold to grasp the
point.

Use the RIGHT ARROW to move the
point to higher

values of X.

Although the sine curve is
there you can't see it until

the value of X in the
exponential function is

greater than five.

This corresponds to a low
rumbling but not very

powerful earthquake.

But as X approaches ten the
sine curve explodes into view.

This shows just how rapidly the
earthquake becomes

powerful and extremely damaging.

For example, move the point
from an X coordinate of five

to an X coordinate of nine.

The sine curve grows
dramatically in size and shows

that a magnitude eight
earthquake is ten thousand

times more powerful than one of
magnitude five.

[Music]