QuadraticFunctions--FullProgram.mp4

Algebra Nspirations

Quadratic Functions, Equations,
and Graphs

What does quadratic mean?

And how do we define a
quadratic function?

How can we predict the
geometry of its graph from the

algebra of its equation?

How can we solve a quadratic
equation using its graph?

Hello, my name is Monica
Neagoy and I will be your

guide as you explore the
quadratic world.

Functions, equations, and
graphs.

The TI-Nspire technology will
help us answer

these and other questions.

The term quadratic was coined
in the 17th century.

It's derived from "quadratus,"
the past participle of the

Latin verb, "quadrare," which
means to

square or to make square.

The connection with the
prefix, quadr- indicating the

number 4, as in quadrilateral,
is the fact that a square, or

quadratum in Latin, has four
sides.

I mention this because people
are often confused by the

apparent mismatch between the
prefix of 4 and

the power of 2.

Applications of quadratic
functions abound in the

physical world.

For one, they model the
trajectories of

projectiles in motion.

This was first discovered by
Galileo in the early 17th

century and later proved

mathematically by Isaac
Newton.

A projectile is anything that
is thrown, shot, ejected,

dropped, et cetera, in the
air.

So, a baseball's flight across
the field, a swimmer's dive

into the water, and a runner's
long jump can all be analyzed

with the help of quadratics.

The consecutive bounces of a
ball, or the water curve

jetting from a fountain are
additional examples of

quadratic application.

We'll begin with some
vocabulary, define a quadratic

function, and investigate the
connections between the

algebra of its equation and
the geometry of

its function graph.

We saw that quadratic
expressions and square terms

are related.

Here are two famous

expressions containing
squares.

The sum of consecutive odd
numbers always give a square,

as illustrated by this figure.

You can see, for instance,
that the sum of the first four

odd numbers, 1, 3, 5, and 7
gives 4 squared.

Generalizing this pattern, the
sum of n consecutive odd

numbers equals n squared.

Here we have a single squared
term.

This is another well-known
equation,

the Pythagorean theorem.

It contains 3 squared terms
and states that the square on

the hypotenuse of any right
triangle equals the sum of the

squares on the two shorter
sides.

Also called the legs of the
triangle.

This last example powerfully
illustrates why the phrase x

squared was chosen to denote x
to the second power.

And that's simply because x
squared is the area of a

square with side length x.

Just as x cubed, for x to the
third power, is the volume of

a cube with edge length x.

We're ready to state a
definition.

A quadratic function of one
real variable, x, is a

polynomial function of the
form f of x equals ax squared

plus bx plus c where a, b, and
c are constant and a is not

equal to zero.

This is called the general or
polynomial form.

Since the variable y is a
function of the variable x, we

write y equals f of x and use
y and f of x interchangeably.

Ax squared is called the
quadratic or squared term.

bx, the linear term.

And c, the constant term.

The expression ax squared plus
bx plus c is called a

polynomial of degree two.

Or a second degree polynomial
because the highest

exponent of x is 2.

It's important to think of a
quadratic function as a member

of a larger family of
functions

called polynomial functions.

The first in this family is a
linear function.

The second, a quadratic.

The third, a cubic.

The fourth, a quartic, and so
on.

In addition to knowing the
algebraic expression of a

quadratic, it's also important
to understand the effects of

the coefficient, a, b, and c,
on the geometry of its graph.

And thus, be able to predict
what the graph will look like.

Let's use the TI-Nspire to
better understand the meanings

of coefficients, a, b, and c,
and how they impact the shape

of the graph.

We'll begin with the leading
coefficient, a.

Turn on the TI-Nspire, press
the home key

to open a new document.

If a document is open, a
prompt asks you if you

wish to save it.

Click to choose yes, or press
tab, then click to choose no.

Select 2 to create a graph and
geometry page.

The blinking cursor is on the
function entry

line by f 1 of x.

Enter the simplest quadratic
function, x squared.

Note that, in this case, the
leading coefficient a is 1 and

the coefficients b and c are
both 0.

Press enter to graph.

Let's begin by moving the
graph's equation.

Hover over the equation with
the pointer.

It becomes an open hand.

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

Use the left and down arrows
to drag the equation to the

lower left area of the
monitor.

Press escape to exit grab and
drag mode.

Move the pointer near the top
of the screen to grab the

right branch of the graph.

The pointer changes to a
diagonal line segment over a

double arrow.

Click and hold until it turns
into a closed hand.

Use the arrows on the nav pad
to drag the right branch down

to open the graph.

Do so slowly to changes in the
value of the coefficient a.

Notice that if the graph opens
wider and wider, a decreases

and approaches 0.

Now, move the branch back up
and make

the graph very narrow.

Indeed, the value of a
increases.

Next, drag the branch down to
the right again.

But this time flipping the
cup-like

graph so it opens downward.

Move the graph slowly to
observe the

changing values of a.

Press escape to exit grab and
move movement

Pretty amazing how we can
change the shape of a graph

with this grab and move
technology.

We've learned a couple of
things.

For one, we now know the
general shape of a quadratic

function's graph.

It's cup-like and symmetric
about a

vertical axis of symmetry.

The name for this graph is a
parabola.

We also learned that for
positive values of the leading

coefficient a, the parabola
opens upward.

And have a lowest point called
the minimum.

For negative values of a, it
opens downward and has a

highest point called the
maximum.

The highest or lowest point of
a parabola

is called the vertex.

Finally, for small, absolute
values of a, we observe more

amplitude in the parabola.

And for large absolute values
of a, more compression.

You may not realize it, but
you're surrounded

by parabolic shapes.

The main cable of a suspension
bridge, subject to the uniform

load of the bridge deck.

Note that freely hanging
cables and the gateway arch in

Saint Louis are not parabolas,
but rather catenaries.

Other parabolic shapes include
umbrellas in the rain,

satellite dishes on roofs,
parabolic reflectors inside

car headlights and telescopes,
and even the soup spoons in

your kitchen.

Onto the next part of our
investigation, the effect of

coefficient c, or the constant
term, on

the shape of a parabola.

A quick exercise on the
TI-Nspire will help you

visualize this.

Press control and menu to
delete the graph.

Press tab to move to the entry
line.

Type x squared.

We recognize the basic
building block for all

quadratics.

Enter three more functions in
the same way.

x squared plus 3, x squared
plus 8, and x squared minus 7.

Notice that adding a positive
constant to x squared, like 3

or 8, shift the parabola
upward that many units.

Adding a negative constant
shifts it downward

accordingly.

Upward or downward shifts of a
graph are called vertical

translations.

In the general form of a
quadratic function, a change

in c produces a vertical
translation of the parabola.

Upward if c is positive,
downward if c is negative.

Knowing that x is 0 on the
y-axis, the fact that f of 0

equals c tells us,
furthermore, that c is the

y-coordinate of the
intersection point of the

parabola with the y-axis.

For this reason, c is called
the y-intercept.

Now, why don't you test your
ideas about the effect of b,

the coefficient of the linear
term, on the

shape of the parabola.

[MUSIC PLAYING]

I hope you're ready for our
second investigation.

It connects quadratic
functions to space travel.

So fasten your belts and let's
take off.

Space travel began in the late
1950s and it's still primarily

reserved for astronauts.

But there's an ongoing race
among aerospace companies to

develop the first commercial
carrier that will take

tourists into space.

In fact, several prototype
spaceships are expected to be

ready for test flights within
two years.

So in your lifetime, civilian
space travel may become an

everyday reality.

Understanding the mathematics
behind it will help you

appreciate it all the more.

Just think, one day you may
stare at the Earth the way

today you stare at the moon.

Let's take an imaginary trip
aboard a space shuttle to the

International Space Station,
or ISS, and investigate where

quadratic functions fit in.

Here is the problem, at
take-off, a space shuttle has

0 altitude and it's projected
upward by a

powerful initial velocity.

At about 120 kilometers above
the earth's surface, it cuts

off its main engine.

At that point, the shuttle's
vertical velocity is 2,110

meters per second, or 2.11
kilometers per second.

It still must rise to 340
kilometers above sea level,

the minimum orbiting altitude
of the ISS.

Here is the question, how long
will it take the shuttle to

reach the ISS located at 340
kilometers above the earth

after its main engine is
turned off?

Let's first examine the
algebraic expression of the

altitude and the function of
time.

The quadratic equation, giving
the vertical altitude or

height, h, of any projectile,
a space shuttle in our case,

as a function of time, t, is h
of t equals negative one half

gt squared plus b sub 0 t plus
h sub 0.

Where negative one half g is
the a coefficient, c sub 0 sub

d coefficient, h sub 0 the c
coefficient.

g stands for the acceleration
due to gravity.

Namely, 9.8 meters per second
squared.

Or 0.0098 kilometers per
second squared.

p sub 0 is the initial
velocity.

We call that the start time
for our problem is main engine

cut off time.

So v sub 0 is 2.11 kilometers
per second.

And a sub 0 is the initial
height.

Again, at main engine cut off
time.

So, 8 sub 0 equals 120
kilometers.

Putting all this together, we
obtain h of t equals negative

one half times 0.0098 t
squared, plus 2.11 t plus 120

or h of t equals negative
0.0049 t squared

plus 2.11 t plus 120.

You're now ready to set up the
equation you must solve in

order to answer the original
question, which was, how long

will it take the shuttle to
reach the ISS after its main

engine is turned off.

The question how long means
you're solving for t, the time

in seconds.

The target altitude is 340
kilometers.

So our equation is h of t
equals 340.

Or negative 0.0049 t squared
plus 2.11 t

plus 120 equals 340.

By solving this equation for
t, you will obtain the time

elapsed from main engine cut
off to the arrival at the ISS.

Instead of working this out
algebraically, let's solve it

graphically, using the
TI-Nspire.

Keep in mind that on the
handheld, t will be x and h of

t will be f of x.

Such mental gymnastics are
good for the mind.

Turn on the TI-Nspire.

Press the home key to open a
new document.

Save or delete the previous
document.

Select 2 to create a graph in
geometry page.

The blinking cursor is on the
function entry

line by f 1 of x.

Type in negative 0.0049 x
squared, plus 2.11 x plus 120

for our altitude function.

Press enter to graph.

Surprised there's no graph?

It's there, but out of the
viewing window.

Press menu 4 and 1 for the
window settings.

Set x min to negative 75 in
order to see the y-axis.

Press tab to move from one
entry to the next.

Set x max to 500.

That's more than eight
minutes.

Set y min to negative 50,
again to see the x-axis, and y

max to 400.

Well above our target altitude
of 340.

Then click OK.

Press escape to move the arrow
to the work area.

Now the parabola's in sight.

As expected, it opens
downwards because the leading

coefficient, a, is negative.

Since the graph plots height
over time, the y-intercept

represents the height y equals
120 kilometers at our starting

time, x equals zero.

So that's our initial height.

Now let's find the points
where y equals 340 using a

dynamic feature of the
TI-Nspire.

Use the nav pad to move the
arrow to a point on the graph.

About halfway between the
y-intercept and the vertex.

It becomes a finger pointing
hand.

Place a point on the graph.

Click and the pencil will plot
it.

Press escape to exit point on
mode.

An open hand is now hovering
over the point.

Using the nav pad, move the
open hand over the points

coordinate.

Click and hold until it
changes to a closed hand.

Use the nav pad to drag the
coordinate to the center of

the screen.

Press escape to exit grab and
drag mode.

Move the pointer back over to
that point on the graph until

it becomes an open hand.

Click and hold until it
changes to a closed hand.

You're now ready to slide the
point to the right along the

graph in search for the points
where y equals 340.

Press the right arrow until
the

y-coordinate approaches 340.

Notice this first occurs at x
equals about 177.

Keep pressing the right arrow
until you see an m for

maximum, the highest part of
the graph.

Keep sliding to the right
until y

approaches 340 once again.

This time, x is about 251.

So we solved our quadratic
equation graphically, making

sense of the meaning of the
parabola.

The quadratic expression on
the left is the altitude.

We needed an altitude of 340
kilometers.

So we traced the graph until
we found points

where y equals 340.

The first point yielded a
value of x close to 177.

That's our answer.

It takes the space shuttle
about 177 seconds, almost

three minutes, to reach the
International Space Station

after main engine cut off.

There were two solutions, as
for many quadratic equations.

But when solving a real world
quadratic problem, one often

discards one of the two
theoretical solution as

nonsensical.

That's the case for our second
point.

You also could solve the
problem algebraically by

subtracting 340 from both
sides of the equation and

obtaining the quadratic
equation negative 0.0049 t

squared plus 2.11 t minus 220
equals 0.

If you know the quadratic
formula, use it to compute the

two exact theoretical
solutions.

You would then discard the
greater one as we did.

Solving quadratic equations
algebraically is the focus of

another video in this same
series.

I hope you've enjoyed our
investigation.

In closing, let's recap the
meaning

of a quadratic function.

Algebraically, quadratic means
that the highest exponent of x

or the degree of the
polynomial is 2.

Geometrically, quadratic means
that the

function's graph is a
parabola.

Now that you know that
quadratic functions can be

found in all sorts of real
world phenomena and objects, I

hope you'll delve deeper into
the algebra and geometry of

these relationships with your
TI-Nspire.

[MUSIC PLAYING]