Experiments with Linear Functions

Karyn Carson

Now that
weÕve covered linear functions, letÕs examine what we can do with them.

Take two
linear functions, f(x) and g(x) and graph them.

For
example: f(x) = (x+1) and g(x) = (x-1).

If you
add the two expressions, you get a new function, h(x) = 2x.
What kind of graph is this?

What
happens if you multiply the expressions, f(x) and g(x)? What will the graph look like?

Again,
graph the linear functions of f(x) = x+1 and g(x)
= x-1 and then graph their product.

The
product of the expressions is h(x)
= x^{2 }– 1.
How is this graph different from the original graphs? How is the function different from the
original functions?

This
graph is called a parabola. The
equation is different because the x is raised to the second power.

What do
you think will happen if we divide the original linear functions? LetÕs seeÉ

If you
divide the two linear expressions (this is as simple as it gets):

__x+1__

x
-1

you
notice something new. What
happened to the graph? Do you know
what shape it has now become?

This is
called a hyperbola.

Now,
letÕs see what happens if our ÔxÕ in the f(x) function is set equal to our g(x)
function (this is called a composite function):

[(x-1)
+1] = h(x)

x = h(x)

What
would this graph look like?

This,
again, is a linear function. What
is itÕs relationship to f(x) and g(x)?

It is
parallel.

LetÕs
continue our investigation. Do all
of these observations only occur with the functions I have explored with
you? IÕd like for you to
investigate these same relationships with your own linear functions. Make up two linear functions and
perform all the operations we performed today (add, multiply, divide and
combine). When we come back
together, letÕs discuss your findings.

__ __

Example
#2:

f(x) = 2x + 3 and g(x) = x – 8

SUM: (2x + 3) + (x – 8)

3x – 5

h(x) = 3x – 5 line

PRODUCT: (2x + 3) (x
– 8)

2x^{2} – 13x –
24

h(x) = 2x^{2} – 13x – 24 Parabola

QUOTIENT: 2x + 3/x-8

h(x) = __2x + 3__

x – 8 Hyperbola

COMPOSITE: 2(x – 8) + 3

2x – 13

h(x) = 2x – 13 line,
parallel to f(x)

Will all
quotients of linear functions result in a hyperbola?

What if
the denominator is a factor of the numerator?

Use f(x) = 6x – 3 and g(x) = 2x – 1.

__6x - 3__

2x
– 1 = 3/1,
so y = 3.

Will all
composite functions be parallel to f(x)?

No:

If f(x) =
-3x + 5 and g(x) = -3x – 5, then f(g(x)) will have a positive slope, as
opposed to the original functions.
Regardless, the composite function will be linear.

In
conclusion:

The sum
of linear expressions will result in a line, because the degree of the
resulting function is one.

The
product of linear expressions will result in a parabola, because the degree of
the resulting function is two.

Based on
my examples, the quotient of linear expressions will result in a hyperbola,
unless the denominator is a factor of the numerator. The reason the hyperbolas result from some of the quotients
is because the value of x has limitations in that the denominator cannot equal
zero.

Composite
functions will result in a line, but not necessarily parallel to the original
function. ItÕs a line because the
degree of the function will always be one.