FunctionsRelations--FullProgram.mp4

Algebra Applications: Functions and Relations

[Music]

Title: Functions and Relations

Title: A Voyage to Mars

Narrator: The year is 2030 and the US is venturing on its

first trip to Mars.

Over sixty years separates its first moon mission to this

manned mission to Mars.

The Ares rocket, descendent of the Saturn rocket that

propelled astronauts to the moon, carries its crew.

It will take nine months to reach Mars.

As it ascends over the surface of the earth the rocket goes

in a curved orbit that is parabolic in shape.

A parabola is known as a conic section.

There are four conic sections: the circle, the ellipse, the

parabola, and the hyperbola.

Scientists rely on these curves when launching rockets

into space.

This trip to Mars allows us to investigate these

conic sections.

A rocket is launched straight up, as you can see from the

launch pad.

SO why does it end up going in a curved path?

Why doesn't it go straight up?

And of all paths, why a parabolic path?

It depends on your frame of reference.

From the point of view of the spaceship it is a straight

path up.

But from the point of view of Earth it is a

curved path.

The Earth is rotating on its axis and so is the rocket.

But once the rocket lifts off it is no longer rotating with

the Earth.

This diagram shows that the rocket is moving in two

directions that are perpendicular to each other.

It is moving sideways at the constant speed of the Earth's

rotation, which is about four hundred and fifty meters

per second.

It is accelerating upward at an initial speed of about

twenty seven hundred meters per second.

These two independent movements define the motion of

the rocket.

Let's use the TI-Nspire to investigate this motion.

We will be using the parametric function

capabilities of the calculator.

Turn on the TI-Nspire.

Create a new document.

You may need to save a previous document.

Create a new graph window.

To activate the parametric equation capability press

MENU, and under graph types select PARAMETRIC.

Notice that unlike the function graph capability two

equations are shown, one for X and one for y.

With a parametric equation both X and Y are dependent on

another variable, in this case t.

In other words, the distance traveled along the x-axis

depends on how much time, t, has elapsed.

The same goes for the distance traveled along the Y axis.

The rocket is moving horizontally at a speed of

four hundred and fifty meters per second.

It's equation of motion is X of T equals 450t .

Vertically the rocket starts with an initial speed of

twenty seven hundred meters per second, but it slows down

with a constant acceleration.

The equation of motion in the vertical direction is Y of T

equals twenty seven hundred T minus 4.9 T squared.

Input these equations into the appropriate fields.

Input the equation for X of T and press the DOWN ARROW

to move to the Y of T field.

Then input the equation for Y of T and press the DOWN ARROW

to move to the field for setting the range for t.

We want the graph for the first ten minutes of the

flight, or six hundred seconds.

Change the value to the right of T to six hundred.

Press the CLEAR key to remove the previous value and replace

it with six hundred.

Then press ENTER.

To see all of the graph press MENU, and under WINDOW ZOOM

select the ZOOM FIT option.

Now you can see the whole graph.

And you can now see that this parametric graph is a parabola.

Notice that this graph passes the vertical line test so it

is a function.

But not all parabolas are functions.

This parametric graph, while still a parabola, does not

pass the vertical line test.

And this is why conic sections are known as

quadratic relations.

Returning to the path of the rocket, place a point on the

graph and follow its trajectory.

Press MENU and under POINTS AND LINES select POINT.

Use the nav pad to move the pointer so that it is on

the parabola.

press ENTER, then press ESCAPE.

Notice that a point is on the parabola with the

coordinates displayed.

Now make sure the pointer is above the point.

The pointer should change to an open hand.

Press and hold the CLICK key until the pointer changes to a

closed hand.

Use the RIGHT ARROW to move the point along the graph.

Notice how the coordinates change.

When the point reaches a Y coordinate of around three

hundred and seventy thousand meters, or three hundred and

seventy kilometers, it reaches the altitude of the

International Space Station, which is the first stop on the

trip to Mars.

Since this is the only portion of the parabola that we need

you can shorten its length.

Press tab to bring back the equation entry area and press

the UP and DOWN ARROWS as needed to get to the range of

t values.

Recall that you had set this to six hundred.

Press the CLEAR button to replace it with a value such

that the parabola ends at the point corresponding to the

point you added to the parabola.

Try to get your screen to look like this.

So, the next stop is the International Space Station.

As the Ares rocket leaves the Earth's atmosphere it has

discarded its rocket boosters, and what is revealed is the

Orion capsule, which is what will travel to Mars.

But first the Orion needs to dock at the International

Space Station.

In docking with the International Space Station

the Orion needs to change its motion to a circular orbit

around the Earth.

A circle is another conic section, and we can overlay a

circular orbit around a point on the parabola that

corresponds to docking with the International

Space Station.

We want to create a circle whose center is at the origin

and that intersects the point on the parabola that

represents where the Orion docks with the International

Space Station.

Before constructing the circle.

Change the window setting of the graph window.

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

Press and hold the CLICK key until the pointer changes to a

closed hand.

Then use the nav pad to move the origin to the middle of

the screen.

Press ESCAPE when you are done.

Now you want to rescale the graph window so that the

circle that you are about to draw will fit.

Use the nav pad to move to the horizontal axis.

Move the pointer so that is above one of the tick marks.

You will see an open hand.

As before, press and hold the CLICK key until it changes to

a closed hand.

Press the LEFT ARROW key and watch how the size of the

parabola shrinks as the graph window rescales.

Try to get your screen to look like this.

You are now ready to construct the circle.

Press MENU and under shapes select CIRCLE.

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

press ENTER.

This defines the center of the circle.

Now use a nav pad so that the pointer is on the point on

the parabola.

Press ENTER.

You should now see the circle.

Because of the way the graph was scaled your circular orbit

probably looks like an ellipse.

To make it look like a circle press MENU, and under WINDOW

ZOOM select ZOOM SQUARE.

The orbit should now look like a circle.

A circle is a conic section, and since all circles do not

pass the vertical line test the equation of a circle is

not a function.

The equation of this circle was found by pressing MENU,

and under ACTIONS select COORDINATES AND EQUATIONS.

Use the nav pad so that the pointer hovers over

the circle.

Press ENTER once, then use the DOWN ARROW to position the

label for the equation.

Press ENTER again.

The equation of this circle is X squared plus Y squared

equals three hundred and eighty seven thousand.

A circle can also be written as a pair of

parametric equations.

Press TAB to bring back the equation entry area.

Make sure the X two and Y two equations are shown.

Input the equations.

Press ENTER.

The equations for X and Y are trigonometric functions and

notice that the parametric circle overlaps the

first circle.

At any point in its orbit the coordinates of the spacecraft

are shown.

Because the parametric version uses sine and cosine, this

captures the idea that the orbit around Earth

is periodic.

In fact, the Orion and the International Space Station

complete one orbit around the Earth, or one period, every

ninety minutes.

Motion that causes you to retrace your steps creates a

relation, not a function.

As the Orion capsule orbits the Earth the crew is

preparing for the long trip to Mars.

The Orion departs from the International Space Station

carrying its crew and the module that will land on the

surface of Mars.

But what is the path that the Orion will take to Mars?

It may surprise you that a direct path is not the best

way to get there.

To see why this is so let's look at the path of Earth and

Mars around the sun.

The path of each planet in orbit around the sun is

an ellipse.

An ellipse is a conic section that has two foci.

In terms of the path of the planets around the sun, the

position of the sun is at one of the foci.

An ellipse is defined as the locus of points such that the

sum of the distances from the two foci to any point on the

ellipse is a constant.

Take a string and tie each end to two points that will become

the foci of the ellipse.

Extend the string and use it to draw an ellipse.

This is the path that each planet takes, and it is the

path that the Orion will take on its trek to Mars.

On a coordinate grid, and ellipse has a width of 2B and

a length of 2A.

The values of A and B are used in the equation of

the ellipse.

Once again, note that an ellipse is a quadratic

relation, not a function.

An ellipse can have different values for a and b.

The eccentricity is a measure of how wide an ellipse is.

The equation for eccentricity also uses the value of

A and B.

This window shows slider values for a and b.

As the values of A and B change so does the value of

the eccentricity.

When a and b are equal the ellipse is a circle and its

eccentricity is zero.

As the values of a and b change the eccentricity is

greater than zero.

It approaches one but doesn't reach the value of one.

In fact, one way to understand conic sections is to look at

their eccentricities.

This table summarizes the values of e for the different

conic sections.

All circles have an eccentricity of zero.

An ellipse has an eccentricity in the range from zero to one.

A parabola has an eccentricity of one, and a hyperbola has an

eccentricity greater than one.

Since Earth and Mars have elliptical orbits, each

ellipse has an eccentricity in the range of zero to one.

This table shows the eccentricity for each orbit

and the distance from the sun.

Notice that the distance from the sun is measured in units

of earth orbits, so Earth's distance is one, and Mars,

which is farther from the sun than Earth, has a distance of

one point five two four Earth orbits.

We can use the data in the table to construct a graph of

each planets orbit.

Let's use the equation for eccentricity.

From the data table we know the values of e and a for

each planet.

We can solve for b to find the orbital equation for

each planet.

Let's start with the eccentricity for Earth.

Replace e and a with the values from the table.

Solve for B.

We get a value of B equals zero point nine nine nine

eight six Earth orbits.

This means that the equation of the ellipse for Earth is x

squared over one squared plus Y squared over zero point

nine nine eight six squared equals one.

Now lets find the orbital equation for Mars.

Replace e and a with the values from the table.

Solve for B.

We get a value of b equal to one point five one seven four.

This means that the equation of the ellipse for Mars is X

squared over one point five two four squared plus Y

squared over one point five one seven four squared

equals one.

Each quadratic relation can be written as two

quadratic functions.

This table summarizes the two functions.

The plus or minus sign indicates that one function

graph is above the x-axis and the other is below the x-axis.

Let's use the TI-Nspire to graph these functions and

analyze them.

Turn on the TI-Nspire.

Create a new document.

You may need to save a previous document.

Create a new graph window.

Input the first function for the orbit of Earth.

Press the DOWN ARROW to graph it.

This is the top half of the ellipse.

To graph the bottom half simply graph the negative

version of F1 of X.

Press the DOWN ARROW.

Now input the first function for the orbit of Mars.

Press the DOWN ARROW to graph it.

To graph the other half of the ellipse input the negative

version of F3 of X.

press ENTER.

Your graph may look like two concentric circles.

To get a more accurate view change the window settings by

pressing MENU, and under WINDOW ZOOM select

WINDOW SETTINGS.

Change the range of X and Y values to negative three and

positive three.

Click OK to accept the changes.

Now your orbits should look like ellipses.

The outer ellipse is the orbit of Mars, and the inner ellipse

is the orbit of Earth.

To clean up the screen press MENU to activate the

hide-show option.

Use the nave pad to click on each of the equations to

hide them.

The path that the Orion will take in traveling from Earth

to Mars is another ellipse known as a transfer ellipse.

In fact, in shifting from the Earth orbit to the transfer

ellipse the Orion will momentarily travel along

the hyperbola.

This diagram shows the transfer ellipse relative to

the orbits of Mars and Earth.

In fact, since you know the values of a and b for each

ellipse you can derive the equation of the

transfer ellipse.

The equation will be of this form, where h and k are the

coordinates for the center of the ellipse.

We leave this as a final exploration.

Use the TI-Nspire to find the equation of the

transfer ellipse.

As the Orion makes its final approach to Mars we can look

back and see that this voyage has really been a tour of the

universe of conic sections.

[Music]