Hello. I'm Professor Von Schmohawk and welcome to Why U.
In the last several lectures we introduced polynomial functions
and said that their graphs can have many different interesting shapes.
We showed that polynomial functions
are built from a sum of one or more terms called "monomials".
Monomials with a single real variable have the form "a times x to the nth power"
where "a" is a positive or negative constant
and "x" is a real variable with a non-negative integer exponent.
The terms of polynomials written in "standard form"
are arranged so that the exponents decrease going from left to right.
Therefore, the term with the largest exponent is always the "leading term".
Likewise, the coefficient of the leading term is called the "leading coefficient".
As we embark upon building polynomials
we will see that the leading term determines whether the function's graph
will ultimately grow in the positive or negative direction
as x takes on very large positive or negative values.
This is referred to as the "end behavior" of the graph.
As we saw in previous lectures, when writing monomials
it is not necessary to write "x to the zero power"
since in Algebra, any variable with an exponent of "zero"
by convention is considered to have a value of "one".
Likewise, "x to the first power"
is typically written without an exponent
since variables with no exponent have an implied exponent of "one".
If we think of each monomial as a separate function
then a polynomial function can be thought of as a sum of these monomial functions.
But how can adding various powers of x
produce graphs with such a large variety of shapes?
As we saw in the previous lecture the graphs of powers of x are relatively simple.
We also saw how a monomial's coefficient affects the shape of its graph
by vertically stretching it
or compressing it
or in the case of negative coefficients reflecting it across the x-axis.
So let's add some monomial functions and see what kind of graphs are produced.
As an example, let's add "x-squared"
and "x to the fourth power".
This produces a polynomial with two terms otherwise known as a "binomial".
When we add functions, the value of the resulting function
is the sum of the individual functions at each value of x.
For example, when x equals "one"
both functions have a value of "one".
Therefore, the value of the function "x to the fourth power plus x-squared"
is "one plus one"
or "two".
Doing this for every value of x
produces the graph of the polynomial function "x to the fourth power plus x-squared".
This function doesn't look much different from the graphs of the two functions we added.
But let's see what happens if we change the coefficient of "x to the fourth power"
to "negative one".
When we add these two monomials
we get the polynomial function "negative x to the fourth power plus x-squared"
producing a more interesting graph.
Even though both monomial functions intersect the x-axis at a single point
the polynomial function intersects the x-axis at three points.
This is because there are three different x values
where the sum of the two monomial functions is zero.
For example, when x is zero both functions have a value of zero.
Therefore their sum also has a value of zero.
Likewise, for the two monomial functions
when x is "one"
or negative "one"
the sum of the two functions is also "zero"
producing two additional zeros for the polynomial function.
Taking a closer look at the graph of this polynomial function
we see that the function has positive values for x greater than "zero" and less than "one"
and for x less than "zero" and greater than "negative one".
This is because for x values in this range, the magnitude of "x-squared"
is greater than the magnitude of "negative x to the fourth power".
For example, when x is "one-half"
"negative x to the fourth power" is "negative one-sixteenth"
and "x-squared" is "one-fourth".
The function's value is thus "negative one-sixteenth plus one-fourth"
or "positive three-sixteenths".
On the other hand, for x values greater than "one"
or less than "negative one"
the value of the polynomial function becomes negative.
This is because for that range of x values the magnitude of "x -squared"
is smaller than the magnitude of "negative x to the fourth power".
For example, when x is "two"
"negative x to the fourth power" is "negative sixteen"
and "x-squared" is "four".
The function's value is thus "negative twelve".
So by adding these two monomial functions
we produced a graph with a more complicated shape.
As x gets further from the origin, the graph initially grows in the positive direction
but then reverses direction, becoming negative.
From then on, "negative x to the fourth power" increases much faster than "x-squared"
causing the graph's end behavior to grow without limit in the negative direction.
As we mentioned earlier
we can always predict the end behavior of a polynomial function's graph
by looking at the end behavior of the leading term.
The leading term in this case is "negative x to the fourth power".
Since the end behavior of the leading term becomes infinitely negative
for both positive and negative x
the end behavior of the polynomial does so as well.
So let's see if it's possible by adding a third monomial term with a larger exponent
to make the graph change directions a second time, once again curving upward.
Let's try adding "x to the sixth power" to our polynomial
and see how this effects the function's graph.
Adding this term to the polynomial
produces a graph that looks quite similar to the graph of "x to the sixth power".
Perhaps this is because "x to the sixth power" grows so fast
that it overpowers the other terms in the polynomial.
However, we can try mitigating this by choosing a coefficient for this additional term
with a value smaller than the implied coefficient of "one".
So let's see what happens when we reduce the value of this coefficient.
With a coefficient of "two-tenths"
the effect of this new term is reduced enough
to allow each term to control the direction of the graph
over some range of x values.
As x gets further from the origin
the graph first switches from a positive to a negative direction
and then reverses a second time, heading back in the positive direction.
No matter how small we make the coefficient
since the leading term determines the graph's end behavior
as x takes on large positive or negative values
the graph will continue to grow infinitely in the positive direction
since the leading term "x to the sixth power" does so as well.
So far, by combining the two monomial functions
"x-squared"
and "negative one times x to the fourth power"
we were able to make the graph reverse directions
first growing in the positive direction
and then in the negative direction.
We then added a third term "two-tenths times x to the sixth power"
producing a graph that reverses direction twice
first heading in a positive direction
then reversing direction, becoming negative
and then reversing direction again, with its end behavior becoming infinitely positive.
So let's see if by adding more terms
we can produce a graph with even more changes of direction.
We will start with the monomial "x-squared"
and add "negative x to the fourth power"
with a coefficient of "negative zero point one two".
Next let's add "x to the sixth power" with a coefficient of "point zero zero four".
Let's then add "negative x to the eighth power" with a smaller coefficient
and finally "x to the tenth power" with an even smaller coefficient.
But instead of increasing the number of changes in direction in our graph
this new term eliminated the previous change in direction.
This is because, as we get further from the origin
x to the tenth power increases so much faster than the other powers of x in the polynomial
that it overpowers the changes in direction caused by some of the other terms.
However, by a process of trial and error, the other coefficients can be readjusted
so that each term contributes a positive or negative change in direction
producing a graph with even more peaks and valleys.
We have seen how a polynomial function's graph
can be affected by its individual monomial building blocks.
Our next challenge, given a particular polynomial function
is to produce that function's graph.
There are computer programs and electronic calculators
that are capable of creating accurate graphs of polynomial functions.
However, when those devices are not readily available
it is often desirable to sketch a quick representation of a function's graph
to get an idea of its general behavior.
In the next lecture, we will see how this can be accomplished.