5.1 Graphs of Polynomials

Introduction

Polynomials have intrigued mathematicians for centuries. Ancient Egyptian, Babylonian, and Greek mathematicians all studied simple polynomials.

The Chinese are known to have worked with cubics in the early 7th century, at the same time mathematicians in India were busy creating the quadratic formula.

Two centuries later the Persian mathematician al-Khwārizmī introduced the world to “al-jabr”, or “algebra”, as a process for solving polynomial equations.

During the Renaissance polynomials spread throughout western Europe and contests were held to see who could solve the hardest polynomial problem.

As we explore the properties and behaviors of polynomial functions in this chapter you might begin to see why they have been able to hold the attention of great minds throughout history.

Standard Form of a Polynomial

One attractive feature of polynomial functions is the simplicity of their equations. In Chapter 4 you learned that polynomials are sums of power functions with non-negative integer powers.

For example, the polynomial $p(x)=5x^{3}+7x^{2}-4x+8$ is a sum of the four power functions $5x^{3}$,   $7x^{2}$,   $-4x$   and   $8$.

Each power function is called a term of the polynomial. The highest power is called the degree of the polynomial, and the coefficient of that term is called the leading coefficient. The term without an $x$ is referred to as the constant term.

It is standard to write a polynomial so that the powers are in descending order.

End Behavior of Polynomials

It's important to be able to identify the degree and leading coefficient of a polynomial because they influence the overall shape of its graph.

Even vs Odd Degrees

Since degrees of polynomials are always whole numbers, the degree must either be an even number or an odd number. The ends of polynomials with even degrees behave differently than those with odd degrees.

To investigate this, random polynomials with different degrees are shown in the figure below. As you switch from one to another, try to decide how even degree polynomials are different from odd degree polynomials. In particular, pay attention to the end behavior of the graphs.

When a polynomial has an even degree, both ends of its graph will point in the same direction and it will look somewhat similar to a parabola. When the degree is odd, however, one end will go up while the other goes down, just like a line.

Positive vs Negative Leading Coefficients

The leading coefficient of a polynomial can change the direction of the graph, depending on if it is positive or negative. In this figure you can change the sign of the leading coefficient for several polynomials.

If the degree is even then the graph mimics the behavior of a parabola, with both ends pointing up when the leading coefficient is positive and down when it is negative.

If the degree is odd then the graph behaves like a line, pointing up to the right when the leading coefficient is positive and pointing down when the leading coefficient is negative. This mirrors the behavior of a line with positive or negative slope.

Since the end behavior of a polynomial depends only on the degree and the leading coefficient, in the long run its graph will look like the graph of its leading term.

In this figure you can zoom out on the graphs of $\color{red}{p(x)=\frac{1}{2}x^4-2x^{2}+x+1}$ and $\color{black}{f(x)=\frac{1}{2}x^4}$.

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We can see that the graph of $\color{red}{p(x)=\frac{1}{2}x^4-2x^{2}+x+1}$ is nearly identical to the graph of the power function $\color{black}{f(x)=\frac{1}{2}x^4}$ as we zoom out, even though they are very different for small values of $x$.

In general, any polynomial will eventually behave like the power function $ax^n$, where $a$ is the leading coefficient and $n$ is the degree of the polynomial.

Determine Degree & Leading Coefficient from a Graph

Knowing how the end behavior is connected to the degree and leading coefficient allows us draw conclusions about the equation of a polynomial simply by observing its graph. Specifically, we know that

• If both ends point in the same direction, then the degree is even.
• If one end is up while the other is down, then the degree is odd.
• If the right end points up, then the leading coefficient is positive.
• If the right end points down, then the leading coefficient is negative.

Turning Points

Another thing that makes polynomials are useful is their ability to change direction. The point where a polynomial changes from increasing to decreasing, or vice versa, is known as a turning point. Turning points are always local maximums or local minimums.

The number of turning points is related to the degree of the polynomial. Look again at the graphs above. Can you spot a connection between the degree and the number of turns?

From the graphs it would seem that the number of turns is always one less than the degree, since the 2nd degree polynomial had 1 turn, the 3rd degree had 2 turns, and the 4th degree had 3 turning points.

Unfortunately, that's not always true. Take a look a the three graphs below.

All three are 5th degree polynomials but each graph has a different number of turns. It seems that a 5th degree polynomial can have 4 turns, but it could also have less than 4.

A good way to describe this is to say that the maximum number of turning points is always one less than the degree.

Find the Minimum Degree from a Graph

Since we can find the maximum number of turns by looking at the equation, we should also be able to do the reverse: find the minimum degree by looking at the graph. This is done by counting the number of turns and adding 1.

As an example, consider the following polynomial.

Since the graph has 3 turning points, the degree of the polynomial must be at least 4. The degree could be higher, but it must be at least 4.

We actually know a little more than that. Since both ends point in the same direction, the degree must be even. So the actual degree could be any even degree of 4 or higher. It cannot, for instance, be a 5th degree polynomial.

Constant Term

The behaviors we have investigated so far were connected to the degree and the leading coefficient of the polynomial. Next we will look at how the constant term affects a polynomial's graph.

In the interactive figure below you can adjust the value of the constant term in a 3rd degree polynomial. Look for a connection between the constant term and the graph. Make a note of your observations before continuing.

You should have found that the constant term of a polynomial is the value of y-intercept of the graph. The reason for this is simple.

Recall that the y-intercept of any function occurs when $x=0$. Replacing $x$ with $0$ will eliminate every term containing a power of $x$, leaving just the constant term.

Take $p(x)=-4x^7+6x^4-18x^{2}+9$ as an example. If we evaluate $p(0)$ we find

[p(0)=-4(0)^7+6(0)^4-18(0)^2+9=9]

which verifies that the y-intercept is $y=9$, exactly the same value as the constant term.

Real Zeros of Polynomials

While it is easy to see the y-intercept from the standard equation of a polynomial, finding the x-intercepts is much more challenging.

However, if the polynomial has been factored Factoring is the process of breaking a polynomial into simpler pieces which, when multiplied together, are the same as the original polynomial. For instance, $(x+2)(x+3)$ is the factored form of $x^{2}+5x+6$. , then finding the x-intercepts is simple.

The Factor Theorem

In the interactive figure below, you can change the location of the x-intercepts. As you do so, pay attention to how the factored equation changes.

You probably noticed that there is a factor for each x-intercept. The reason for this is that x-intercepts are real number solutions to $p(x)=0$. And if $(x-c)$ is a factor, then $x=c$ will cause the entire function to equal zero. In fact, x-intercepts are often called real zeros for this very reason.

In general, $p(c)=0$ if and only if $(x-c)$ is a factor of the polynomial. In other words, knowing that $(x-c)$ is a factor tells us that $x=c$ is a zero. The reverse is also true. If $x=c$ is a zero then $(x-c)$ is a factor. This is known as the factor theorem.

With the factor theorem we can quickly find the x-intercepts of any factored polynomial. For instance, if $p(x)=(x-3)(x+6)(x+15)$, then the x-intercepts are $x=3$, $x=-6$, and $x=-15$.

Multiple Zeros

Consider a polynomial such as $f(x)=(x-2)(x-2)$. The factor theorem tells us that this polynomial has real zeros at $x=2$ and $x=2$. But what does it mean for a polynomial to have two zeros with the same value? Can it have more than one x-intercept at $x=2$?

The figure below will help answer this question. You are fee to move the real zeros (blue points) anywhere you want. Look for a change in the shape of the graph and a corresponding change in the equation when multiple zeros are put in the same spot.

Putting multiple zeros in the same spot creates a factor of the form $(x-c)^m$ where the power $m$ corresponds to the number of zeros. If the multiplicity of the zero is odd, then the graph crosses the x-axis. If the multiplicity is even, then the graph does not cross the x-axis but instead just touches it and then bounces off.

Determine Polynomial Functions from a Graph

We have now arrived at a point where we can create an equation for a polynomial simply by analyzing its graph. To see how this might work, let's start with this graph.

Since the graph has just one turn and both ends point upward, we expect the equation to be at least a 2nd degree polynomial with a positive leading coefficient. It should also have a constant term of $-2$, since the y-intercept is at $(0, -2)$.

In addition, the zeros at $x=-2$ and $x=1$ tell us that $(x+2)$ and $(x-1)$ must be factors. This suggests that the equation might be $p(x)=(x+2)(x-1)=x^{2}+x-2$.

This equation meets all of the expectations we had above; it is a 2nd degree polynomial with a positive leading coefficient and the y-intercept is at $-2$. Graphing this equation on a calculator shows that it does have the right shape.

Let's try a few more examples, starting with the graph below.

We can see 2 turning points which suggests that the degree is at least 3. We also know that the leading coefficient is negative and the y-intercept is $(0, 2)$.

The graph has real zeros at $x=-1$, $x=-2$ and $x=3$, which means $(x+1)$, $(x+2)$ and $(x-3)$ must be factors, so one possible equation is

[p(x) = (x + 1) (x + 2) (x - 3)]

To see if this is a good fit we'll expand the equation and check its leading coefficient and y-intercept.

[ \begin{align} p(x) &= (x + 1) (x + 2) (x - 3) && \newline &= (x+1)(x^{2}-3x+2x-6) && \text{multiply} (x+2)(x-3) \newline &= (x+1)(x^{2}-x-6) && \text{combine} -3x+2x \newline &= (x^{3}+3x^{2}+2x-3x^{2}-9x-6) && \text{distribute} (x+1) \newline &= x^{3}-7x-6 && \text{simplify} \end{align} ]

The leading coefficient is positive and, since the constant term is $-6$, this function has a y-intercept at $(0, -6)$. Since neither of these matches our graph, we'll need to find a way to change both without altering the real zeros.

We can change both the leading coefficient and the y-intercept if we multiply it by a constant $a$, making the equation $p(x)=a(x^{3}-7x-6)$. Remember, multiplying a function by a constant vertically stretches and/or flips the graph, it doesn't change the x-intercepts or zeros.

Since we want to have a y-intercept of $2$, the trick will be to find $a$ when $\color{red}{p(0)=2}$.

[ \begin{align} p(x) &= a(x^{3}-7x-6) && \newline p(0) &= a((0)^3-7(0)-6) && \text{replace each } x \text{ with } 0 \newline p(0) &= -6a && \text{simplify} \newline \color{red}{2} &= -6a && \text{since } \color{red}{p(0)=2} \newline -\frac{1}{3} &= a && \text{divide by } -6 \end{align} ]

This gives us $p(x)=-\frac{1}{3}(x^{3}-7x-6)=-\frac{1}{3}x^{3}+\frac{7}{3}x+2$, which has all the properties (zeros, degree, leading coefficient, and y-intercept) of the graph we were trying to match. Plotting this new equation on a calculator

shows that it is, indeed, a good fit.

As our final example consider this graph.

This graph appears to have a single zero at $x=0$, a multiple zero with even multiplicity at $x=-1$, and a multiple zero with odd multiplicity at $x=2$.

When faced with graphs that have multiple zeros we will choose the lowest power possible unless directed otherwise. Thus we start with the equation

[ p(x)=a(x)(x+1)^2(x-2)^3 ]

To find $a$ we need another point on the graph. Note that the point $(1, 2)$ is on the graph, so $p(1)=2$ which will help us solve for $a$.

[ \begin{align} p(x) &= a(x)(x+1)^2(x-2)^3 \newline p(1) &= a(1)(1+1)^2(1-2)^3 && \text{Replace each } x \text{ with } 1 \newline 2 &= -4a && \text{since } p(1)=2 \newline -\frac{1}{2} &= a && \text{divide by }-4 \end{align} ]

Using $a=-\frac{1}{2}$ we arrive at the equation $p(x)=-\frac{1}{2}(x+1)^2(x)(x-2)^3$. Checking this on a graphing calculator

shows that we have found a good match.

Looking Ahead

We should point out that this process of finding the factored equation of a polynomial from a graph only works if the number of real zeros equals the degree.

As a case in point, look at the graph below.

There is only one real zero, $x=-2$, implying that the equation only has one factor $(x+2)$ and is the first degree polynomial $p(x)=x+2$.

But we can clearly see 2 turning points! Since a first degree polynomial cannot have 2 turning points, there must be more to this polynomial than meets the eye.

What we can't see are the zeros that are imaginary numbers. While real zeros can be picked out as the x-intercepts, these imaginary zeros cannot be seen on a regular graph. In the next section we will discuss ways to visualize and find imaginary zeros.