Algebra Applications: Linear Functions

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Title: Algebra Applications: Linear Functions

Narrator: A lot of your study OF algebra will involve

studying functions.

When two variables are connected in such a way that

the value of one variable depends on the value of the

other variable then there is a functional relationship.

In the case of linear functions, one variable, x ,

changes and the other variable, Y, changes by a

consistent amount.

A staircase is a good model of a linear function.

In this staircase as you take one step in a horizontal

direction you take a consistent amount in the

vertical direction.

The graph of a linear function is a line.

The equation of a linear function is usually written in

slope-intercept form, where mx is the slope, or steepness of

the line, and b is the y-coordinate of the point

where the line intersects the y-axis.

In this program you will explore different applications

of linear functions.

In the first segment you will explore the slope in the

context of cycling.

Cyclists refer to the steepness of the hill by

its grade.

You will learn what the relationship is between slope

and grade.

You'll also use the slope formula to calculate the

steepness of a hill in Tuscany, as well as determine

how far the cyclist travel on their way up the hill.

In the next segment we investigate a data set that

can be modeled by a linear function.

The data for the US usage of oil is graphed and a linear

function is identified.

This function is used to predict future usage.

Finally, the possible use of oil from Alaska's ANWR region

is explored relative to future use of oil.

In the last segment you will explore a linear model for

heart rate during aerobic exercise.

This linear equation is of the form y equals mx plus b, and

you will use this equation to generate an exercise

data chart.

This program with cover the following concepts in linear

functions: the slope formula, the slope-intercept form of a

linear function, linear regression.

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Tuscany is a region in the northern part of Italy.

Known for its ancient beauty, its old-world charm, and for

its surprisingly steep mountains.

But less well known is how popular cycling is.

The city of Cortona is on a hill and is five hundred

meters above sea level.

It is a city of steep, narrow roads, but that doesn't stop

these cyclists.

Cyclists refer to the steepness of a hill by noting

its percent grade.

For example, a hill with a six percent grade means that for

every one hundred meters you move horizontally you increase

your altitude by six meters.

How does this relate to slope?

In a coordinate system, suppose there are two points

labeled x1, y1, and x2, y2, and suppose there

is a line connecting the two points.

The slope of the line is defined as the rise over the

run, or the ratio of the difference in the y

coordinates over the difference in the

x coordinates.

This is summarized in the slope formula shown here.

So if a hill has a six percent grade then traveling one

hundred meters horizontally corresponds to a six-meter

vertical rise.

We can use these facts to identify two coordinates.

The first coordinate is (0, 0) and the second

coordinate is (100, 6).

Let's use the slope formula to find the slope.

Input the values for y1, y2, x1, and x2.

Let's designate the coordinate 0, 0 as x1-y1, and

let's designate 100, 6 as x2-y2.

Plug in the coordinates into their appropriate slots

and simplify.

The slope of the line connecting the two points

is 0.06

Notice that this is the decimal form of six percent.

This means that the grade of a hill is the same as the slope

written as a percent.

Also, for practical purposes, the grade will never be

greater than a hundred percent since such a grade would be

much too steep.

We can use the y=mx form of a linear equation to

explore different grades.

For example, the graph of y=0.06x is a model of a hill

of grade 6.

Now let's use a TI-Nspire to explore slope.

Turn on the TI-Nspire.

Create a new document.

You may need to save a previous document.

Open a graph window.

Show the background grid by selecting MENU, and under view

selecting SHOW GRID.

Place two points on the grid.

Press MENU, and under POINTS AND LINES select POINT.

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

press CLICK to place a point.

Do the same with a point in the first quadrant.

Connect the points of the line by selecting MENU, and under

POINTS AND LINES selecting LINE.

Move the pointer so that it is on top of each point and

press ENTER over each point.

Change the range for the x-axis by moving the pointer

over the axis, holding the SHIFT key, and pressing the

LEFT button.

Choose a max of about a hundred and ten.

Display the coordinates of the points by pressing MENU, and

under ACTION selecting coordinates and equations.

Place the pointer over one of the points and press ENTER

once to see the coordinates, and ENTER again to place the

coordinates in a specific part of the screen.

Move the coordinates to the upper part of the screen.

Repeat the previous two steps with the other point.

Now display the slope of the line by selecting MENU, and

under MEASUREMENT selecting slope.

Move the pointer over the line and click on it.

Then move the label for the slope to the same area of the

screen as you did with the coordinates.

Then press ESCAPE.

Change the coordinates of the two points to 0, 0

and 100, 6.

Move the pointer to highlight the x coordinate of one

of the points.

Press ENTER twice to make the coordinate editable and change

the value.

Repeat with the y coordinate.

Then repeat with the other set of coordinates.

You'll see that the grade is the same as the slope

expressed as a percent.

The five hundred meter rise to Cortona is at a four

percent grade.

How far do these bikers have to travel?

We can use the slope formula to solve this problem.

A four percent grade is the same as the slope of 0.04.

The rise is five hundred meters, so we need to solve

for the run.

The distance the cyclists need to travel is twelve thousand

five hundred meters.

This long distance is made up of a long spiraling road

up to Cortona.

But once you get to the top the view is magnificent.

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The Arctic National Wildlife Reserve, ANWR, is in

a corner of Alaska.

It is a national park the size of South Carolina.

It is a vast area with rivers, mountains, and wildlife.

Large herds of porcupine caribou roam the valleys.

Polar bears search for food, and in the spring flowers and

birds make their brief appearance.

But this national park is also a national treasure.

Geologists estimate that there could be as much as fifteen

billion barrels of oil in the coastal region.

This should not be too surprising since Prudhoe Bay,

North America's largest source of oil, is a few miles down

the coast.

How long will fifteen billion barrels of oil last?

At one million barrels a day, and using scientific notation,

we get one point five ten to he fourth days.

Converting that to years we find that there is enough oil

for approximately forty-one years.

But how does this compare to our current usage of oil?

The US currently uses twenty-one million barrels of

oil per day.

So an additional one million barrels represents

five percent.

This is not a huge amount, but here is where ANWR can make a

big difference, and for that we'll need the TI-Nspire.

For this activity you will be graphing data for the daily US

consumption of oil from 1980 to 2007.

Turn on the TI-Nspire.

Create a new document.

You may need to say a previous document.

Create a new list and spreadsheet window.

Enter the years in column A, pressing ENTER after

each entry.

When you're done move to column B where you'll enter

the daily oil usage data.

The usage data is in thousands of barrels, so each number of

the second column represents millions of barrels.

Pause the video to enter your data.

Be sure to include the column headings for each data set and

make a note of these column headings.

We want to test if this data set is linear.

To do that we perform a linear regression, which is a way of

testing to see how closely the graph of the data aligns with

the linear function.

Before performing a linear regression you must select

the data.

Move the cursor to the top of column B.

Beside the letter B, press the UP ARROW once more to select

the column.

It should now be highlighted.

Then press and hold the SHIFT key while holding the

LEFT ARROW.

Now both columns should be selected.

Next press MENU, and under statistics select STAT

CALCULATIONS, and under that select LINEAR REGRESSION.

Press tab to move all the way down to OK and press ENTER,

or click.

Your cursor lands on entry one of column D.

To widen the column press MENU, and under ACTION select

RESIZE, and under that MENU select RESIZE COLUMN followed

by the RIGHT ARROW a few times until the width

is satisfactory.

Then press ENTER.

Notice the coefficients r2 and r are both very close to

1, which would represent a good fit.

To plot the data points and their regression line together

press CONTROL and I and select GRAPHS to insert a new

graph page.

Press MENU, and under GRAPH TYPE select SCATTER PLOT.

Input the name of the column heading of column A in your

spreadsheet, press the DOWN ARROW, then input the name

of the column heading of column B in your spreadsheet.

Press ENTER again.

To see the data press MENU, and under ZOOM, select

ZOOM DATA.

To plot the regression line over the points press MENU,

and under GRAPH TYPES select FUNCTION.

Press the UP ARROW to access F1, and finally press ENTER.

Lastly, press control and T for the function table.

Press MENU, and under tables select edit table settings.

Change the table start value to 1980.

Tab down to select OK.

Suppose that ANWR oil production started in 2010 and

took five years before the oil became available.

By 2015 the expected increase in demand from 2007 is what

ANWR could mostly take care of.

That, combined with more fuel efficiency would keep the US

from becoming more dependent on foreign oil supplies.

This would be a small step in the direction of reducing our

dependence on oil and looking for alternatives.

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Narrator: Everyone knows the value of exercise.

In particular aerobic exercise is great for your metabolism,

and for your heart.

Whether jogging, cycling, or swimming, aerobic exercise

exerts your heart muscle and helps your metabolism.

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Narrator: But as you get older the amount of aerobic

exercise changes.

The exertion on the heart needs to decrease with age.

For a twenty year old the maximum number of heartbeats

should not exceed two hundred beats per minute.

For a sixty year old, the maximum number of heartbeats

should not exceed one hundred sixty beats per minute.

There is a linear function that relates heartbeat to age,

y=220 - x , where y is the maximum number of

heartbeats per minute, and x is the age of the person.

Suppose that your health teacher wants you to create an

exercise chart that shows the maximum heart rate for people

aged fifteen to sixty-five.

Turn on the TI-Nspire.

Create a new document.

You may need to save a previous document.

Create a new graph window.

Input the function 220 - x and press ENTER.

Notice that there is no equation on screen.

This is because the y intercept, 220 is off the screen.

Change the WINDOW SETTINGS by pressing MENU, and under

WINDOW ZOOM select WINDOW SETTINGS.

Change x-min to 0, x-max to 70, y-min to 150, y-max to 250.

Now you can see the graph.

What we're interested in is generating a data table.

To do that click on CONTROL and T.

Change the table settings, selecting MENU, and under

TABLE select edit TABLE SETTINGS.

Change the table start value to 15 and tab your way to

the OK button and click on it.

You can now take this data table and bring it into a

spreadsheet or word processing program.

Simply highlight the data table and copy the data to

the clipboard.

Paste the data into your spreadsheet or word processing

program, format the table and print it.

You now have an aerobic exercise chart for people

from ages fifteen to sixty-five.

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