FunctionsAndRelations--FullProgram.mp4

Algebra Nspirations

Relations and Functions

Environmentalists, meteorologists, economists.

People of all disciplines have always been interested in

studying relationships between changing

quantities called variables.

The number of polar bears or other endangered species at a

given time.

The variation in temperature with respect to height.

Or the profit made on a new movie,

depending on ticket sales.

To better understand relationships among two or

more variables, algebra is indispensable.

With that in mind, this program is designed to address

key concepts of 21st century school algebra,

relation and function.

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

explore the world of relation and function.

We'll connect these explorations to real world

application, and use the TI-Nspire to bring these

investigations alive.

We'll begin with a few examples of relations, then

we'll investigate functions, and

special kinds of relations.

In both cases, we'll explore definition, vocabulary,

notation, and representation.

For our first example, suppose your teacher, who often asks

her students to download data, video, and other large files

from the internet wanted to purchase USB flash drives for

her students.

She asks you to do the research and find the best

deal available.

So you search the web and find that Super-Cruz is having a

sale on flash drives, and offering these great prices.

This is the tabular representation of a relation

between two variables.

The variable capacity, measured in gigabytes.

And the variable, price, measured in dollars.

Tables can also be constructed horizontally.

Each capacity, value c, corresponds to a price value,

p, and thus, each row gives an ordered pair.

By definition, a relation is a set of ordered pairs.

So these five ordered pairs form a relation.

In set notation, the ordered pairs are separated by commas

and enclosed within braces.

Another form of representation is a mapping diagram, which

explicitly shows the relationship between the two

sets of variables.

The domain of a relation is the set of all first or

p-coordinates of the ordered pairs, cp, and a set of all

second or p-coordinates of the ordered pairs

constitutes the range.

The arrows show the correspondence between domain

and range values between capacity and price.

Since the price, p, of the flash drive depends on its

memory capacity, c, we've called p

the dependent variable.

Other synonyms include y-value, output, and ordinate.

c is the independent variable.

You'll also encounter x-value, input, and abscissa.

Next, let's use the TI-Nspire to look at a third way you can

represent a relation, graphically.

In this first example, we'll use the Calculator application

followed by the Graphs and Geometry application.

In the next, we'll use Lists and Spreadsheet followed by

Data and Statistics.

Turn on the TI-Nspire, press the Home key,

and select any document.

You may be prompted to save an open document.

After you decide, select 1 to create a calculator page.

To store the domain value as a list named c, press Control

followed by the right paren.

Then, key in 1, 2, 4, 8, and 16, inserting

a comma each time.

Press the right arrow to move outside the braces.

Then Control and Var to store these numbers.

Type the letter c, then press End.

This is now labelled as our c list.

To do the same for the range value, press Control and the

right paren, then type 5.99, 7.99, 12.99, 17.99, and 39.99.

Don't forget the commas.

When you're done, press the right arrow, and then Control

Var to store.

The letter p for price, and then enter.

This is now our p list.

Notice the 1 at the top left of the monitor.

It tells you we're on page one.

To graph this cp point, access the Graphs and Geometry

application by pressing Home.

Notice you're now on page two of your document.

Press Menu, and under Graph Type select Scatter Plots.

The domain, or x box, is highlighted.

Click to see your choices, select c, and press Enter.

Tab over to the range or y box.

Click again, but this time select p.

Press Enter.

To create an appropriate window for this graph, press

Menu, and under Window, select Zoom Data.

To see the point's coordinates, press Menu one

last time, and under Traits select Graph Traits.

Use the navigation pad, or nav pad, to move from one

point to the next.

This graph of our relation is called a scatter plot, a

scattering of points in the x-y plane, each representing

one ordered pair.

Another phrase used for the x-y plane is the Cartesian

plane, named after 17th century French mathematician

and philosopher, Rene Descartes.

Before him, relationships were defined by ordered pairs,

tables, or algebraic equations.

Descartes' innovative idea was to display these ordered pairs

of numbers as points in a two-dimensional

plane called a graph.

This flash of genius connected algebra and geometry like

never before, and analytic geometry was born.

This new way of graphing numbers in space forever

changed the face of modern mathematics.

And Descartes claimed that his idea of visualizing pairs of

numbers at points and plane, or graph, flashed before him

in a dream.

Revolutionary for that time, yet commonplace today,

especially given our graphing technology.

Let's now turn to our second x-y relation.

Suppose a group of students wanted to look at the

relationship between the numbers of hours spent

studying for a final exam and the final grade obtained, to

see if a general pattern emerged.

This time we'll use the list and spreadsheets and the data

and statistics application.

It's the more appropriate method to graph a scatter plot

when there are a lot of data.

In this relation we'll label h, the number of hours spent

studying, and g, the grade earned on the final exam.

Here we go.

Press the Home key for a new document.

Save or delete the open document and create a list and

spreadsheet page.

Enter these hours in the first column, and the

grade in the second.

Press Enter after each entry, and use the nav pad to move

around the spreadsheet.

Pause the video here to enter your data.

To graph the scatter plot, you'll need

to open a new page.

So press the Home key for a data and statistics page.

There's your graph.

To make some sense of this scatter plot, use the down

arrow to move the pointer to the x-axis.

A box appears with Click to Add Variable Inside.

Click, then select h for the x-axis variable.

Press Enter.

Next, use the left arrow to move the pointer to the y-axis

until the same box appears.

Click, then select g for the y-axis variable.

Then press Enter.

The scatter plot is now displayed with the hour values

along the bottom, and the grade values along the side.

We notice an upward trend from left to right, meaning more

study hours generally leads to higher grades, but there are a

couple of exceptions.

We'll learn more about analyzing statistical data and

drawing conclusion in another episode of this video series.

But I'd like to go back to our scatter plot for a moment, as

it provides a great segue from relations into functions.

Press Menu, and under Analyze or Actions, select plot value.

Enter 4 for v1, then press Enter.

The vertical line, x equals 4 passes through

two different points.

This means that two different students each studied the same

number of hours, 4, but earned two different grades.

Checking the ordered pair, we, indeed, find 472 and 491.

Perhaps, for some of you, this example brought to mind the

vertical line test.

Indeed, we often use the vertical line test to check

different relations of functions.

We found that an element in the domain has two assigned

values in the range.

That's fine for a relation, but a function's a special

kind of relation, satisfying a specific condition.

We're ready to define it.

A function is a relation that assigns exactly one value in

the range to each value in the domain.

In other words, each x value or input corresponds to

exactly one y value or output.

Can you figure out which of the following are functions

and which are not, and why?

So now you know that in a function, one input cannot be

assigned to more than one output.

But note that several x values can be assigned

to the same y value.

In this second diagram, what's important to see is that each

x is paired with exactly one y.

Next, some vocabulary and notation.

Suppose the function f assigned each domain value x

to the range value y equals 5x plus 1.

y equals 5x plus 1 is called the function rule or equation.

It defines the relationship between x and y.

In any function, the value of y depends on the value of x.

So we say y is a function of x, and we write

y equals f of x.

This is function notation.

So, in our example, y equals 5x, and f of x equals 5x are

equivalent.

Let's start by graphing four simple functions, and from

their graphical representation, intuitively

deduce the domain and range of each function.

Press the Home key to open a new document.

Decide if you wish to save the open one, and then press 2 for

a graphs and geometry page.

Under Page Layout, choose Select Layout and pick layout

2, the vertical split screen.

Press Control-Tab to move to the right.

Next, press Menu, and again, select 2.

We'll graph two functions on each of Cartesian planes.

The cursor is blinking by f1, so key in the letters I-N-T,

then the left paren, x, and finally the right paren.

For f2 type in negative x.

To zoom in, use the nav pad to move the pointer to the

first tic mark on the x-axis to the left of the origin.

When it becomes an open hand, press Control and click to

grab this point.

Use the left arrow to drag it all the way to the left edge

of the monitor.

Press Escape to exit grab and drag mode.

You can now see unit one on the x-axis.

The first function is called the integer part function.

Its graph is like a stair case, and often

called a step function.

It assigns the integer part of x to any real number.

The second is a linear function with

slope negative 1.

Negative slopes yield descending graphs,

from left to right.

Linear functions have a straight line graph, as you

may have guessed.

A vertical line test, from left to right, yields exactly

one intersection point on each graph for every value of x.

That confirms that we have two functions, and it also tells

us that in both cases, the domain and range

are all real numbers.

Indeed, int of x is defined for every x, since all real

numbers have an integer part, and the only one.

In particular, numbers between 0 and 1 have integer part 0.

And negative x is also defined for every real number, since

all numbers have an opposite.

For example, 1/2 has negative 1/2, and negative 5 has

negative negative 5, or positive 5.

For these two functions then, the domain, d, and range, r,

both equal the set of all real numbers.

Notice a thicker letter was a real number to distinguish it

from all other uses of the letter r, such as r for range.

Let's go back to the inspire and graph two more functions.

Press Control-Tab to move to the left.

Type in x squared for f, please.

For our fourth function, enter 1 divided by x for the

reciprocal function, 1 over x.

To zoom in this time, press Menu, and under Window, select

to zoom in.

A center box appears.

Press click twice, then Escape.

x square is a quadratic function, and the shape of all

quadratic function graphs is called a parabola.

This parabola lies in quadrants one and two, where

all y values are positive.

The two north-bound branches continue upwards to infinity.

A vertical line test yields exactly one intersection point

for every value of x.

1over x is called a rational function, and its graph is

called a rectangular hyperbola.

This hyperbola lies in quadrant one, where y is

positive, and quadrant three, where y is negative.

It has four infinite branches and nowhere does the graph

ever touch the x- or y-axis.

So, x and y are never 0.

A vertical line test yields exactly one intersection point

for every value of x, except 0.

Indeed, x squared is defined for all x, since all real

numbers have a square.

But the squares themselves, namely the y values, are all

positive, except 0 squared, which is 0.

So the domain is R, and the range is the set of all real

numbers that are greater than or equal to 0.

1 over x is defined for all real values of x,

except when x is 0.

A 0 in the denominator makes the fraction undefined.

So the domain of this function is all real numbers, except 0.

The same is true for the y value, as we saw that the

graph never approaches the x factors, which means y never

takes on the value of 0.

In separate episodes of this series, we look in more depth

at the different types of functions, including linear,

quadrtic, rational, exponential, and so on.

Another important concept is the notion of increasing and

decreasing functions.

Let's take a look.

f2 of x equals negative x is a decreasing function.

We can see that the graph descends from left to right.

This means that as x increases, y decreases.

f3 of x, on the other hand, is decreasing for negative values

of x, and increasing for positive values of x.

Finally, f4 of x is decreasing on two separate intervals, for

both negative and positive values of x.

I'll let you revisit f1 of x on your own.

I suggest you try graphing it by hand, on good paper, and

pay close attention to the endpoints of

each horizontal set.

We'll end this lesson with an exciting inspire exploration

that visually connects the circumference of a circle to a

function graph.

Press Control-N for a new document.

We're going to construct a circle.

Press Menu, and under shape, select circle.

Move the point toward the left of the monitor, and click to

place the circle's center.

Use the nav pad to create a circle, then click again.

Now, we measure the radius.

Press Menu, and under measurements, pick Length.

Move the cursor to the center, click, then drag it with the

down arrow until you reach a point on the circumference.

Click twice, and the radius length appears.

Next we measure the perimeter, or circumference.

With the nav pad, move to a point at the top of the circle

and click twice.

Now, the perimeter appears.

To name this variable, press the Var key and

select store Var.

Type the letter P for perimeter, then press Enter.

Do the same for the radius.

Hover over it, and likewise, press Var.

Choose 4 Var again, and this time type r for radius, and

then press Enter.

You've now labelled both variables.

We're ready to split the screen.

Under Page Layout, select Custom Split.

With the nav pad, drag the partition to the left, making partition to

the left screen about one-third the screen's width.

Then press Enter.

Press Control-Tab to move to the right. Menu to make a

selection. then 2 for a graph and geometry page.

You now have a plain geometry table on the left. and an

analytic geometry page on the right.

Since radii and perimeters, or circumferences in this case,

are positive, let's change the setting.

Press Menu. and under Window, select Window Settings.

Enter negative 2, 10, negative 4, then 50.

Click OK.

To place the point, press Menu, and under Points and

Lines, pick Point.

Click to plot a point.

Next, press Menu again, and under Actions, select

coordinates and equations.

Click twice to place the point's coordinates.

We now want to connect the x value with the radius and the

y value with the perimeter.

To do this, hover over the x-coordinate of the

point and press Var.

This time select Link To, and then the letter r.

Press Enter.

Do the same for the y-coordinate.

Press Var, Link To, the letter p, and then press Enter.

Finally, switch back over to the left screen by pressing

Control-Tab.

Move the pointer to a point on the circle.

When the word circle appears. click and

hold to grab the point.

And now for the amazing part.

Use the nav pad to drag the circle, making it very large

and then very small.

Notice that the point r-p travels along

an invisible graph.

What's the shape of this graph?

As the circle becomes very small. the point is closer to

the origin.

Why?

It looked like the point was tracing an ascending line.

starting at the origin.

A little algebra can help.

The perimeter of a circle is 2 pi r.

So the function rule is p equals 2 pi r.

That's approximately p equals 6.28r.

When r approaches 0. so does p.

When r grows very large. so does p.

r and p vary proportionally at a rate of 1 to 2 pi.

It's therefore. a linear function with slope 2 pi.

We've come to the end of our lesson.

I hope this dynamic approach to algebra has helped you

understand the meanings of relation, function, domain,

range, graph, and much more.

Oh wait, there's one more fun problem still to come.