RationalFunctions--FullProgram.mp4

Algebra Nspirations

Rational Functions and Expressinos

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.

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The simplest non-trivial rational function is f of x

equals 1 over x.

We saw that 1 is a polynomial.

We will first explore what this graph looks like, and

then see how it changes as we vary the value of the

numerator, making it larger or smaller.

The general form for this simplest types of rational

function is a over x, where a is some constant.

Here we go.

Press Home for a new document.

Save the previous one if you wish, then select 2 for a

Graph and Geometry page.

To define f1 of x, press Control division key for a

fraction placeholder.

Then type a in the numerator, x in the denominator.

Press Enter to graph.

No graph appears because we haven't defined a, our

parameter, meaning a constant that will take on different

values for each function.

To do so, we insert a slider.

Press Menu, and under Actions select Insert Slider.

Type a to replace the default variable v1.

Press Escape or Tab and the graph will appear.

Notice the graph lies in quadrants one and three.

The default range for the slider is 0 to 10.

In order to observe all values from negative 10 to positive

10, let's modify the slider settings.

With the pointer inside the slider box, press Control Menu

to access the Slider Context Menu.

Select Settings.

Tab down to Minimum and enter negative 10.

All other default values are fine, so Tab

down to OK and click.

Next, use the nav pad to move the pointer over the slide

bar, also called the thumb of the slider.

Press Control click to grab the thumb.

Use the Left and Right arrows to slide the thumb slowly

along the slider track, from one end to the other.

Take note of the changing values of the a and how they

affect the shape of the rational function graph.

Examine what happens when a changes from positive values

to negative values or vice versa, and when a equals 0.

When you're done, stop the thumb at a equals 10.

I trust you noticed several properties of

this family of graphs.

There are some striking differences between these and

the graphs of all polynomial functions with a

denominator of 1.

First, we noticed a discontinuity or a break in

the curve at x equals 0.

That makes sense, since the fraction is undefined when the

denominator is 0.

Therefore, f of x is defined for all real numbers x, except

for x equals 0, where f of x is undefined.

We call the y-axis the vertical asymptote of f of x

because its graph approaches this line as x gets closer to

0 from either side.

We call the x-axis the horizontal asymptote of f of x

because its graph approaches this line as x gets extremely

small or extremely large.

We say, when x approaches negative or positive infinity.

If the parameter a is positive, the graph lies in

quadrants one and three.

If a is negative, in quadrants two and four.

When a equals 0, f of x becomes the constant function

f of x equals 0, so the graph is the x-axis.

That's why you saw nothing when the slider was on 0.

The name of this graph is a hyperbola.

Moreover, when the asymptotes are perpendicular, it's called

a rectangular hyperbola.

Lastly, the intercept.

f of equals y equals 0 yields the x-intercept.

The zeros of a rational function are the zeros of its

numerator, since a fraction is 0 when it's numerator is 0.

For all non-zero a values, f has no x-intercept.

x equals 0 yields the y-intercept.

But f has no y-intercept either since x can't be 0.

While the results we just stated apply to the rational

functions of the form a over x, we can use the same logic

for all rational functions.

Let's practice.

f of x is not defined at 2 or negative 2, since these values

make the denominator 0.

So f has two vertical asymptotes, x equals 2 and x

equals negative 2.

This is trickier.

If x approaches plus infinity, the denominator gets so much

larger than the numerator that f approaches 0.

So again, the horizontal asymptote is y

equals 0, the x-axis.

Check negative infinity on your own.

f of negative 3/2 equals 0, since negative 3/2 makes the

numerator 0.

So the graph x-intercept is negative 3/2, 0.

f of zero equals negative 3/4.

So the graph's y-intercept is 0, negative 3/4.

Let's graph this function to visualize these features.

Press Control Tab.

Go back to page one.

This is the graph of y equals 10 over x.

Press Menu.

And under Window, select A Square.

Press Menu again, and under Shape select Rectangle.

To construct one, use the nav pad to move the pencil to the

origin and click.

Arrow up and click on another point on the y-axis.

Then arrow right and click somewhere near the graph in

quadrant one.

Press Escape.

Now got back to Menu again.

And me Measurement, select Area.

With the pointer on the rectangle, click twice in and

its area will appear.

Press Escape.

Press Control click to grab the points.

Drag the point along the graph until you get an area of 10

exactly or very, very close to 10.

Press Escape to exit Grab and Drag.

Hover over the area of value and click to highlight it.

Press Menu, and under Actions select Attributes.

You now see an arrow.

Press Enter and lock in this area value.

Press Escape a final time.

Lastly, like before, grab the top right point of the

rectangle and use the nav pad to drag it along the graph to

which it is now locked.

We now can see many different rectangles, all of which have

a fixed area of 10 square units.

I really appreciate when technology enables us to see

such connections between geometric concepts like the

fixed area of a rectangle on one hand and their algebraic

expressions

in this case the equation of a rational function

on the other hand.

I hope you do too.

I also hope you have a better idea about rational numbers

and functions.

Now it's your turn to test your skills.

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