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.