Algebra Nspirations
Algebra Nspirations
Algebra Nspirations
Algebra Nspirations
Algebra Nspirations
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?