RationalFunctions--Segment01.mp4

Algebra Nspirations

Rational Functions and Expressions

Variables represent quantities that change, and algebra is

the study of relationships among these changing

quantities.

Some quantities, x and y, vary proportionally, like this time

elapsed when you're traveling and the distance covered by

any means of transportation, the measure of a circle's

radius and the measure of its perimeter, or the number of

hours you spend babysitting or doing some other job and the

amount of money you earn.

In these three cases, it's easy to see that if x

increases, so does y.

If x decreases, so does y.

Other quantities vary in opposite directions, such as

the length and the width of a rectangle whose area is kept

constant, the distance from a light source and its

intensity

think of an oncoming car headlight, for example

or the height of a fixed length ramp and the run time

of an object rolling down the ramp.

In these relationships, we observe a different situation.

If x increases, y decreases.

And if x decreases, y increases.

In the first set of examples, we say that y varies directly

as x, or y is directly proportional to x.

Algebraically, that means that their quotient equals some

constant, k.

The constant of proportionality, k, was the

average speed for the moving vehicle, 2pi for the circle,

and the hourly pay rate for the babysitting job.

In the case of the rectangle's dimensions, x and y, we say

that y varies inversely as x, or y is inversely

proportional to x.

Here their product is constant and equal to the content area.

For direct variation, the equivalent form y equals kx is

clearly a linear function and yields a straight line graph.

For inverse variation, the equivalent form of y equals k

over x is clearly not a linear function because the exponent

of x is not one, but rather negative 1.

We'll see that this is called a rational function.

The name, shape, and properties of this type of

graph are at the heart of this lesson's exploration.

While the last two situations regarding the light source and

ramp are also examples of inverse variations, they are

not necessarily rational functions.

Their function equations are of the form y equals k over x

square and y equals k over square root of x,

respectively.

By the end of this lesson, you'll know if either of these

is rational and why.

So to recap, here's an easy way to

remember what we reviewed.

Direct variation means that y is proportional to x.

And inverse variation means that y is proportional to the

reciprocal of x.

k is nonzero in both cases.

Now that we reviewed some important concepts, we'll

begin our lesson by dividing our first rational function, a

famous one known as Boyle's Law.

Then we'll define rational functions, drawing an analogy

between rational functions and numbers

and study their graphs.

We'll end up with a 2D geometry problem.

The time is the early 1660s, and the place is England.

The protagonists are Robert Boyle, a chemist and

physicist, and his assistant, Robert

Hooke, a natural scientist.

We are well into the scientific revolution, a

period known for the emergence of new laws and ideas in all

fields of science that eventually led to the

rejection of many long held doctrines rooted in folklore

or subjective belief.

Boyle was informed of a special relationship between

the absolute volume and pressure of a gas contained

within a closed system in which the

temperature is kept constant.

Hooke built an apparatus consisting of a U-shaped tube

to verify the conjecture.

Boyle trapped some air inside the tube by pouring mercury in

one end and sealing the other.

He measured the volume by the height of air in the tube and

the pressure by the height of mercury.

Boyle noticed that an increase in the amount of mercury, or

pressure exerted, caused a decrease in the volume of the

gas, or air.

He took successive measurements of each variable

in inches and recorded his data.

Here is a subset of Boyle's data.

We'll use the Nspire to plot these data points vt and find

the function that best fits the data.

To do this, the Nspire uses what we

call regression analysis.

Turn on the TI Nspire.

Press the Home key for a new document.

Save any previous documents if you wish, then select a list

and spreadsheet page.

Scroll to the top of Column A and type v for volume.

Press Enter, or the Down arrow.

Similarly, type p for pressure at the top of Column B.

We've chosen v as our independent variable and p as

our dependent variable.

Note that the opposite choice would yield the same function.

Pause the DVD now to enter the data from the chart.

Having noticed the inverse variation, we suspect the

product vp or pv remains constant.

Use the nav pad to move to the gray formula cell of Column C

and press the equal sign.

Press var, then enter to select p, insert the

multiplication symbol, then press var again to select v.

Press Enter one last time to populate Column C. The values

are indeed rather close, so we suspect a constant product pv

equals k or p equals k over v. Or, better yet, p equals kv to

the negative 1 power.

Keep in mind that all measurements have a margin of

error, especially the data obtained in Boyle's

experiment.

To perform a regression, you select the data.

Use the nav pad to move the cursor to the top of Column B.

Press the Up arrow once more to select Column B. It should

now be highlighted.

Then press and hold the Shift key while

pressing the Left arrow.

Now both column should be selected.

Next, press Menu and under Statistics select Stat

Calculations.

Since our independent variable is v and we anticipate the

negative 1 power of v, select Power

Repression and press Enter.

The x list, or volume values, are in Column A. And the y

list, or pressure values, are in Column B. So these default

values are fine.

Saving the regression equation in F1 is also fine.

So tap down to the 1st Result Column, which is the column

where the regression results will appear.

We occupied Column C with the product pv, so we need to

change C to D.

To do so, arrow left twice, press the Clear key, and type

the letter D in the place of C. Tab to OK and

press Enter or click.

Notice a few things.

One, the function equation is of the form a

times x to the v power.

Two, a is the constant we called k, the expected

constant product pv found in Column C.

It's value is 1,405.85.

And three, b is the exponent of our independent variable v.

And it's value is, in practical terms, negative 1.

To recap, we found, with the help of the Nspire statistical

regression analysis, the function y equals a times x to

b power, where a is 1,405.85 and b is extremely close to

negative 1.

This is an excellent approximation of the

anticipated inversely proportional relation p equals

k times v to the negative 1, which k id about 1,406.

This is how Robert Boyle proved his conjecture, which

ultimately was coined and is still known as Boyle's Law.

Let's go back to the Nspire and ploy Boyle's data points

and graph the regression function, which

we stored in F1.

To insert a graph and geometry page into your document, press

Control i and select 2.

Notice the 1.2 at top left, indicating page 2.

Press Menu, and under Graph Type select Scatter Plot.

The x variable is highlighted.

Select v for volume.

Tab over to y.

And this time select p.

To adjust the window to our data, press Menu.

And under Windows, select View Data.

There's our scatter plot in full screen.

To graph the regression equation over these points,

press Menu again.

And under Graph Type choose Function by pressing Enter.

F2 is displayed.

To access F1, where our regression equation is stored,

press the Up arrow once , and F1 is displayed.

Press Enter to graph F1.

Hover over F1 until the pointer becomes an open hand.

Press Control click to graph the function equation, then

use the nav pad to drag it to the top right of the monitor.

Press Escape to exit Grab and Drag.

You can now see the five original points, vp, along

with F1's continuous curve.

We see from the shape of this graph that as v increases

along the x-axis from left to right, p decreases along the

y-axis moving downward.

Indeed, p is inversely proportional to v and their

product remains constant and equal to 1,405.85.

To finish, press Menu.

And under Window, select Zoom Out.

Use the nav pad to move the center box to the center of

your monitor, pressing Enter twice and then Escape.

You can now see more of the graph of the function p equals

1,405.85 over v. Clearly, only a piece of the graph in

quadrant one is the mathematical model for Boyle's

real world experiment.

Back in the 17th century, Robert Boyle did all his

calculations by hand without the help of technology, and

proved that the product of pressure and

volume remains constant.

In 1660, he published his results in a book whose

subtitle is Touching the Spring of

the Air and its Effects.

Let's finish this exploration with a clear statement of

Boyle's Law.

The volume v occupied by a given mass of gas, is

inversely proportional to the pressure p

to which it is subjected.

This assumes a closed system in which the temperature is

maintained constant.

Algebraically, this means that the product pv equals some

constant k related to the particular gas or p equals k

over v.

You've also reviewed these equivalent forms of k over v.

Being flexible in our use of different letters to represent

variables and constants, we know that Boyle's Law

p equals k over v is a particular instance of the

general form y equals a over x.

This is the simplest type of rational

function you will encounter.

So what are rational functions?

And what are their graph characteristics?

We'll find out in our second exploration.

If you're viewing this video on rational functions, you're

probably already acquainted with the family

of polynomial functions.

The first member of this family is the linear function,

in which the highest exponent of x is 1.

We say the degree of this polynomial is 1.

The second member is the quadratic function, or second

degree polynomial.

The third is a cubic function of degree 3.

The fourth, a quartic of degree 4, and so on.

Three things to remember.

The numerical coefficients are real numbers.

The exponents are non-negative integers.

And the highest exponent is the degree of the polynomial.

The addition, subtraction, and multiplication of any two

polynomials yields another polynomial.

But division does not, unless the divisor is a factor of the

dividend, and this divided exactly.

Like, for example, x squared minus 4 divided by x plus 4.

Since x squared minus 4 has two factors, one of which is

the divisor, we have an exact division and the quotient is

another polynomial function.

But the division x squared minus 4 by 2x plus 3, for

example, does not yield a new polynomial.

This is a rational function.

In general, f of x is a rational function if it is the

quotient or ratio of any two polynomial functions, N of x

and D of x, provided these two have no

non-constant common factor.

Here's a question for you.

Are polynomial functions a subset of

the rational function?

Well, to answer that, let's do a quick

review of rational numbers.

By definition, Q is the set of all numbers that can be

written as the quotient or ratio of two integers, p and

q, provided the denominator q is non-zero.

If you know the symbol Z for the set of integers, the

symbolic definition of set Q can be written like this.

Here is a question about numbers analogous to our

question about functions.

Are integers rational numbers?

Or, similarly, is Z a subset of Q?

The answer is yes, because any integer n can be written as

the quotient n over 1, which satisfies the definition of

the rational.

Note that n also has many other equivalent fractional

expressions of integers.

So the answer to our original question is also yes, because

any polynomial function, p of x, can be written as the

quotient p of x over 1.

1 qualifies as a polynomial function since 1 equals 1

times x to the 0 power.

It's called a constant polynomial.

To recap, the set of polynomial functions is

therefore a special subset of the rational function, just as

the set of integers is a special subset of

the rational numbers.

Before examining the graphs of a rational

functions, try this.