LinearFunctions--FullProgram.mp4

Algebra Nspirations

Linear Function, Equations, and
Graphs: The Geometry of Graphs

What is a linear relationship
between two variables?

What is the difference between
the slope-intercept form and

the standard form of an
equation?

How can we visualize the
concepts of slope and

y-intercept?

The TI-Nspire technology will
help us answer all these

questions, and more.

Hello, everyone.

My name is Monica Neagoy, and
I will be your guide as you

explore the linear world

functions, equations, and
graphs.

In our first investigation,
we'll examine the simple

linear equations related to
takeoffs and landings.

And in our second, we'll
derive a linear function from

a data set, and predict future
carbon dioxide levels in the

Earth's atmosphere.

Let's get started.

I'm sure you've watched
airplanes taking off.

And you've probably seen space
shuttle landings on

television.

And perhaps you've even seen
documentaries of submarines

dashing up or down to meet a
target.

These vehicles in motion
provide opportunities to

examine linear equations.

Let's take a passenger plane,
for example.

The average cruising altitude
for long-distance flights

ranges from 30,000 to about
40,000 feet.

At an air speed of about 370
miles per hour, suppose an

airplane climbs at a steady
rate of 1,500 feet per minute

after takeoff.

Namely, the plane's altitude
increases by

1,500 feet each minute.

Let the time x equal 0 at
takeoff.

The altitude y is also 0 at
takeoff.

For any time x greater than or
equal to 0, the equation

expressing the plane's
altitude over time is y equals

1,500x, where x is the time in
minutes since takeoff and y is

the plane's altitude in feet.

Since the altitude depends on
the time, we say the altitude

is a function of time.

And in algebra, we write y
equals f of x.

So y equals 1,500x and f of x
equals 1,500x are equivalent.

Let's create a graph
representing the plane's

altitude over time, from
takeoff until it reaches its

cruising altitude of 36,000
feet.

Remember, the x-axis will
denote time in minutes.

And since we're dealing with
very large numbers, the y-axis

will denote altitude in
thousands of feet.

Here are some guiding
questions to jot down.

One, how high will the plane
be after 5 minutes?

And two, when will the
aircraft reach its cruising

altitude of 36,000 feet?

Turn on the TI-Nspire.

Press the Home key to open a
new document.

A previous document may be
open.

If so, a prompt will ask you
if you

wish to save the document.

Click to choose Yes or press
Tab, then click to choose No.

Select 2 to create a Graphs
and Geometry Page.

The blinking cursor is on the
function entry

line by f1 of x.

Type in 1.5x.

Here, 1.5 means 1.5 thousand,
or 1,500.

Press Enter to graph.

Since we want a representation
of altitude over time after

takeoff, only the first
quadrant is of interest.

Move the pointer arrow to a
clear area of quadrant I. Then

press the Click button and
hold it down until the pointer

changes to a grasping hand.

Use the navigation pad to drag
the origin to the bottom-left

corner of the monitor, such
that both the x and y-units

are visible.

Press Escape to exit the grab
and drag feature.

To increase the y-max to about
45 for 45,000 feet, move the

pointer to a tick mark on the
y-axis.

The tick marks on both axes
start blinking and the pointer

becomes an open hand.

Press and hold the Click
button until it becomes a

closed hand.

Pressing and holding the Shift
key while pressing the Down

Arrow on the nav pad changes
the y-scale, leaving the

x-scale intact.

Select a y-max of about 45.

Press Escape to exit grab and
drag.

Now we're ready to answer the
two questions.

One asks how high, so we're
solving for y, the altitude,

given x equals 5 minutes.

And the other asks when.

This means we're solving for
x, the time, given y equals

36, the 36,000-foot cruising
altitude.

Let's solve using our graph.

Press Menu and under Trace,
select Graph Trace.

The x-shaped cursor is on 0,
0.

Namely, 0 minutes and 0 feet
for the

coordinates at takeoff.

Pressing the Right Arrow on
the nav pad traces the graph

and displays the coordinates
of the points along the way.

To skip ahead to x equals 5,
press 5, then Enter, and you

obtain y equals 7.5.

Press Escape to exit the trace
mode.

0.5, 7.5 tells us that after
five minutes into flight, the

aircraft has reached an
altitude of 7.5 thousand feet,

or 7,500 feet.

Let's go back and answer the
second question, this time

using the function table.

Press Control-P to bring up
the function

table on the same screen.

Scroll down the f1 of x column
using the nav

pad and stop at 36.

Notice that the corresponding
x-value is 24.

From the point 24, 36, we
learned that it would take

this aircraft 24 minutes to
reach its cruising altitude of

36,000 feet.

If you wanted to solve this
algebraically without the help

of the handheld, you would set
y equal to 36 and solve for x.

Dividing both sides of the
equation by 1.5 yields x

equals 24 minutes.

Did you realize that's roughly
one mile higher than Mount

Everest, which stands at about
29,000 feet?

So for a plane flying near the
Himalayan Mountains, its

cruising altitude would have
to be much

greater than 36,000 feet.

You may already know the

slope-intercept form of a
line.

It's general equation is y
equals mx plus b, where m is

the slope and b is the
y-intercept.

By rotating and translating
our graph, let's learn more

about these important
algebraic concepts.

Press Control and Escape to
exit the split screens and

return to the graph.

Let's begin by dragging the
graph's equation to the

top-left corner of the screen.

Hover over the equation with
the pointer.

Click and hold until the open
hand becomes a closed hand.

Use the Left and Up Arrows to
drag the equation to the

top-left corner.

Then press Escape.

Next, move the pointer over
the graph of the line near the

top of the screen.

The pointer changes to two
curved arrows.

This feature allows us to
rotate the line around the

y-intercept.

Click and hold until the arrow
turns into a closed hand.

Use the arrows on the nav pad
to rotate the line.

Do so slowly to see the
changes in the slope value.

Notice that as the line
becomes less steep but is

still rising from left to
right,

the slope value decreases.

As the line comes closer to
the horizontal, its slope

approaches 0.

A perfectly horizontal line,
such as the

x-axis, has a 0 slope.

And as the line falls from
left to right,

the slope is negative.

This equation, for example, f1
of x equals negative 1x, would

be a model for a submarine
descending from sea level at a

sink rate of 1,000 feet per
minute.

Let's stop the rotation of the
line at a slope value of about

negative 2.

Here's what we've learned
while varying the

slope of our line.

Rising lines from left to
right have positive slopes.

Horizontal lines have 0
slopes.

And falling lines have
negative slopes.

Note that vertical lines
cannot be written in

slope-intercept form because
the slope is undefined.

That is why there is a
standard form ax plus by

equals c for all lines.

Next, let's keep the slope
constant and vary the

y-intercept.

The y-intercept is the
y-coordinate of the point

where the line intersects the
y-axis.

Press Escape to exit graph
rotation mode.

Move the pointer over the
graph near the origin until

the pointer changes to a cross
of arrows.

This feature allows us to
translate the line and create

parallel lines.

Click and hold until it turns
into a closed hand.

Now, use the Right Arrow on
the nav pad to shift the line

slowly to the right.

Notice that the slope remains
unchanged, while the

y-intercept changes.

Stop at about f1 of x equals
negative 2x plus 40.

Suppose you begin monitoring
the shuttle's altitude at 40

miles above the Earth's
surface.

You'd know that it descends in
a steady glide and is losing

altitude at a rate of 2 miles
per minute, a dozen times

faster than an airplane.

If our axes now represent
minutes and miles, f1 of x

equals negative 2x plus 40
would be a good model for our

hypothetical space shuttle.

The slope of negative 2
represents the steady drop in

altitude of 2 miles per
minute.

And the positive y-intercept
represents the altitude when

you began monitoring.

To figure out when the shuttle
lands, set the altitude y

equal to 0 and solve for x.

It's a simple linear function,
so we can see that when x

equals 20, y equals 0.

So the shuttle would land in
20 minutes.

Well, we've learned a lot
about lines and their

equations, their slopes, and
their y-intercepts.

The next investigation will
help us delve deeper and

uncover more pieces of the
puzzle on linear

functions and equations.

I'm sure you've heard recent
reports about global warming.

Let's examine the issue a
little closer and use a linear

function to make some
predictions about the future

of this phenomenon.

Global warming is the steady
increase in the average

temperature of the Earth's
near-surface air and oceans in

recent decades.

It is caused by the emission
of gases that trap the sun's

heat in the Earth's
atmosphere.

Closely related is the natural
process called

the greenhouse effect.

The various gases in the
atmosphere absorb the heat

lost by the Earth at night and
radiate it back for the

survival of life on Earth.

The problem though is that one
of these gases

carbon dioxide

is on the rise.

Since it is primarily the
product of fossil fuel

combustion, such as gasoline
and coal, many people believe

its increase is the result of
pollution.

The concern is that higher
global temperatures entail

weather-related disasters such
as heat waves, droughts,

floods, hurricanes, and more.

Global levels of carbon
dioxide, or CO2, began an

upward spike around 1960.

This table shows approximate
CO2 levels in the Earth's

atmosphere in 1960 and in
2007.

CO2 is measured in parts per
million, or PPM.

So for example, 380 PPM means
that there are 380 molecules

of CO2 for every million
molecules of dry air.

Since CO2 levels have risen
steadily from 1960 onwards,

we'll set 1960 as our starting
point, x equals 0, and assume

linear growth from then on.

Let's find a linear equation
for CO2 levels, y, as a

function of the year, x.

First, calculate the slope
using the slope formula and

the two given points.

Then use the slope-intercept
form, y equals mx plus b,

since the y-intercept is
given.

The y-intercept is the y-value
when x equals 0.

The slope m equals 380 minus
318 over 47 minus 0.

Or, 62 divided by 47, which is
approximately 1.32.

So the linear equation y
equals 1.32x plus 318 gives

the global level of CO2 for
any year x since 1960.

To predict the amount of
carbon dioxide in the Earth's

atmosphere in the year 2025,
assuming the constant rise

will continue, you would set x
equal to 65, because 2025 is

65 years after 1960, and
obtain y equals 1.32

times 55 plus 318.

Equals 85.8 plus 318, which is
403.8 parts per million.

A very alarming level indeed.

As a final exercise, let's use
a powerful feature of the

TI-Nspire called linear
regression to predict CO2

levels in 2025 and 2050 based
on the following data.

The TI-Nspire finds the line
that would

best fit these points.

It's called the line of best
fit.

Turn on the TI-Nspire.

Press the Home key to open a
new document.

Choose whether or not to save
your open document.

Then start the list and
spreadsheet application.

Enter the years in column A,
pressing

Enter after each entry.

When you're done, use the nav
pad to move to column B, where

you'll enter the CO2 level.

Pause the video to enter your
data.

Press the Up Arrow once more
to select column B. It should

now be highlighted.

Then press and hold the Shift
key while

pressing the Left Arrow.

Now both columns should be
selected.

Next, press Menu, 4, 1, and 3
to select Linear Regression.

Press Tab to move all the way
down to OK and

press Enter, or click.

Your cursor lands on entry 1
of column D.

To widen the column, press
Menu, 1, 2, and 1, followed by

the Right Arrow a few times
until the width is

satisfactory.

Then press Enter.

Notice in this column that the
slope m is pretty close to

what we found with just two
points.

And the coefficients, r square
and r below it, are both very

close to 1, which would
represent a perfect fit.

To plot the six data points
and the regression line

together, press Control and I,
then 2 to

insert a new graph page.

Press Menu, 3, and 4 for a
scatter plot.

Press the Down Arrow to select
stat.xreg.

Press on the Click button.

Next, select stat.yreg.

So let's zoom data by pressing
Menu, 4, and 9.

To plot the regression line
over the points, press

Menu, 3, and 1.

Press the Up Arrow to access
f1.

And finally, press Enter.

Lastly, press Control-T for
the function table and Menu,

5, and 5 to change the table
start value to 2025.

Tab down to select OK.

You can see that in 2025, the
CO2 levels will be about 402

PPM, very close to what we
found.

Use the nav pad to scroll down
to 2050 and find 435.085 PPM.

We'll need to find creative
ways to limit greenhouse gases

if we want future generations
to live in harmony with the

environment.

In closing, let's summarize
the meaning of a linear

relationship between two
variables, x and y.

Geometrically, it means the
graph is a straight line.

Algebraically, it means the
exponents of x and y are both

equal to 1.

And mathematically, it means
that y increases

or decreases

proportionally with x.

I hope you'll investigate
linear relationships further

on your own.

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