LinearFunctions--Segment01.mp4

Algebra Nspiration

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.