Video Captions: Algebra Applications: What Is an Earthquake?

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.