Hello. I'm Professor Von Schmohawk and welcome to Why U.
So far, we have discussed polynomial functions
and showed that their graphs can have interesting and varied shapes.
As in previous lectures, the polynomial functions we will be discussing
are functions of a single real variable "x".
But given a particular polynomial function, how can we produce that functions graph?
There are various graphing utilities
that are capable of creating accurate graphs of polynomial functions.
However, it is often desirable to sketch a quick representation of a function's graph
to get a general idea of its behavior.
In this lecture, we will see how this can be accomplished.
The graphs of all polynomial functions have certain similar characteristics.
For instance, the graphs of polynomials are "continuous".
This means that their graphs have no breaks or jumps
so the line of the graph is uninterrupted.
A simple way of thinking of a continuous graph
is that it can be drawn in a single stroke without lifting your pen off the paper.
Functions that are not continuous are said to be "discontinuous".
There are a number of ways that a function can be discontinuous.
Take for example the function "one over x".
This function is discontinuous at an x value of "zero".
At that point there is break in the graph called a "discontinuity".
When the values of x are small the function "one over x" is large.
The function grows infinitely positive as x approaches zero from the right
and infinitely negative as x approaches zero from the left.
Notice that when x is zero
the value of "one divided by zero" is undefined.
Therefore the function's domain includes all values of x except zero.
Another way that a function can be discontinuous is for a finite jump to occur in its graph.
For example, consider the function "x divided by the absolute value of x".
When x is positive, the value of x and its absolute value are the same
so "x divided by its absolute value" is "one".
On the other hand, when x is negative
x and its absolute value have the same magnitude but opposite signs.
So in that case "x divided by its absolute value" is "negative one".
Notice that when x is zero
x and its absolute value are both zero.
Since "zero divided by zero" is undefined
the function's domain includes all values of x except zero.
An additional type of discontinuity
is a single point missing from an otherwise continuous interval of a graph.
An example of this is illustrated by the function "x-squared divided by x".
The value of the function "x-squared divided by x"
is the same as the value of the function "f of x equals x"
for all values of x except when x is zero.
In that case, evaluating "x-squared divided by x"
gives us "zero divided by zero" which is undefined.
Therefore, this function's domain includes all values of x except zero
and its graph does not include that single point at the origin.
There are many other variations on discontinuities
but the important point is that polynomial functions are continuous
and so their graphs have no gaps, jumps or other discontinuities.
One additional feature shared by all polynomial functions
is that their graphs are "smooth".
So what exactly is meant by the term "smooth"?
A graph that is smooth has no sharp corners.
Its slope is continuous at every point, never instantaneously changing to a new value.
An example of a function that is not smooth is the "absolute value of x".
For positive values of x, its graph has a slope of "one"
and when x is negative its graph has a slope of "negative one".
At the origin, the graph's slope instantaneously changes value
jumping from "negative one" to "one" as the value of x goes from negative to positive.
Therefore, this function and its graph are not smooth.
However, since the function is defined at every point including when x is "zero"
the line of the graph is uninterrupted and therefore the function is continuous.
Concepts such as continuity and smoothness
will become more important later when studying calculus
but knowing that all polynomial functions are continuous and smooth
gives us some guidance when sketching a polynomial function's graph.
So how do we go about sketching the graph of a polynomial function?
The graph of a function is a visual representation
of the function's output values over some range of input values.
The domain of polynomial functions includes the set of all real numbers
extending infinitely in both the negative and positive directions.
But when creating a graph, we would like to select a subset of this domain
that displays the most interesting or important characteristics of the function.
For instance, we could choose to graph the function "x-cubed, plus x-squared, minus two x"
for input values between "negative fifty" and "positive fifty".
But if instead, we select a more limited set of input values
we get a more informative graph.
Although this graph shows only a small portion of the function's domain
it gives us a better idea about the function's general behavior.
This example also illustrates that the graph's "x intercepts"
which are the points where the function's graph intersects the x-axis
can often indicate a region of the function's domain
that will produce the most interesting graph.
In this example, at each of its three intercepts the graph crosses the x-axis
and therefore has opposite signs on either side of the intercept.
For example, at the intercept "negative two, zero"
the graph is negative on one side and positive on the other.
Likewise at the intercept at the origin
the graph is positive on one side and negative on the other.
And at the intercept "one, zero"
the graph is once again negative on one side and positive on the other.
However, at an x-intercept it is also possible for a graph to intersect the x-axis
and then bounce back without crossing it.
In that case, the sign of the function's value will remain the same on both sides of the intercept.
So the x-intercepts are important points to identify when sketching the graph of a polynomial function.
These x values are called the "zeros" of the function
since they're the values of x where the function's output value is zero.
The zeros of a function are also the solutions to the equation
produced by setting that function equal to "zero".
For example, "x equals one" and "x equals negative two"
are the solutions to the equation "x-cubed minus three x, plus two, equals zero".
These solutions are also called "roots" of the function
and have been an area of great interest in mathematics for many centuries.
So how do we go about finding these roots?
When we first studied linear functions, which are first-degree polynomial functions
we saw that the solution to the corresponding linear equation
produced by setting that function equal to zero
could always be found by adding, subtracting, multiplying or dividing terms
until the variable x is alone on one side of the equation.
Likewise, when we studied quadratic functions, which are second-degree polynomial functions
we saw that the solutions to the corresponding quadratic equation
could be found by factoring the polynomial into a pair of linear terms.
Once we have factored the polynomial, the solutions to the quadratic equation
can be determined by using the zero product property
setting each term equal to zero
and then solving the two resulting linear equations.
As we also saw, in cases where factoring the quadratic is difficult
the solutions to the corresponding quadratic equation
can always be found using the quadratic formula.
This famous formula works in every case
producing the roots of any second-degree single-variable polynomial equation.
So we know that the roots of first and second-degree polynomial functions can always be found.
But what about the roots of higher-degree polynomials?
In fact, there are algebraic methods
for finding the roots of any third-degree "cubic" polynomial function
or fourth-degree "quartic" polynomial function.
However, the methods that work in every case
can be extremely complicated and often impractical to use.
In addition, no general methodology exists
for calculating the roots of polynomials of degrees higher than four
although these roots can be approximated using computational techniques.
But all is not lost.
Polynomial functions of degree higher than two
can often be easily factored into products of first-degree polynomials
whose roots can then be found.
The "zero product property" tells us
that those roots will also be the roots of the higher-degree polynomial function.
For example, let's say we want to find the roots
of the third-degree polynomial function "x-cubed plus x-squared minus two x".
Since x is a common factor in each of the three terms
we can factor this third-degree polynomial into a product of x times a second-degree polynomial.
That second-degree polynomial
can then be factored into the product of two first-degree polynomials.
The roots can then be determined
setting each term equal to zero and solving the three resulting linear equations.
These x values are the roots of the polynomial function
and the x-intercepts of that function's graph.
Once the x-intercepts of the function's graph have been determined
we know that the graph must intersect the x-axis at each of these points.
Since these x values are the function's only roots
we also know that the its graph cannot intersect the x-axis anywhere else.
Therefore, between each consecutive pair of points
the graph must either lie entirely above
or entirely below the x-axis.
However, there are many shapes that a graph could take
while intersecting the x-axis at only these three points.
In the next lecture, we will see how to choose between these various possibilities
and sketch the approximate shape of the polynomial function's graph
once we know it's roots and thus it's x -intercepts.