Algebra Nspirations

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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