In Chapter IntroductiontoFunctions, we studied functions of the form \(f(x) = b\) (constant functions), \(f(x) = mx+b\), \(m \neq 0\) (linear functions), and \(f(x) = ax^2+bx+c\), \(a \neq 0\) (quadratic functions). In each case, we learned how to construct graphs, find zeros, describe behavior, and use the functions in each family to model real-world phenomena. One might wonder about functions of the form \(f(x) = ax^3+bx^2+cx+d\), \(a \neq 0\), or functions containing even higher powers of \(x\). These are the polynomial functions and are the subject of study in this chapter. As you may recall, polynomials are the result of adding monomials, so we begin our study of polynomial functions with monomial functions.

1 Monomial Functions

Monomial functions, by definition, contain the constant functions along with a two parameter family of functions, \(f(x) = ax^n\). We use \(x\) as the default independent variable here with \(a\) and \(n\) as parameters. From Section ??, we recall that the set \(\mathbb {N} = \{ 1, 2, 3, \ldots \}\) is the set of natural numbers, so examples of monomial functions include \(f(x) = 2x = 2x^{1}\), \(g(t) = -0.1 t^2\), and \(H(s) = \sqrt {2} \, s^{117}\). Note that the function \(f(x) = x^0\) is not a monomial function. Even though \(x^0 = 1\) for all nonzero values of \(x\), \(0^{0}\) is undefined, and hence \(f(x) = x^0\) does not have a domain of \((-\infty , \infty )\).

We begin our study of the graphs of polynomial functions by studying graphs of monomial functions. Starting with \(f(x) = x^n\) where \(n\) is even, we investigate the cases \(n = 2\), \(4\) and \(6\) using Desmos.

Adjusting the slider, we can observe that as \(n\) increases, the graphs ‘flatten’ for \(-1<x<1\) and ‘narrow’ for \(x<-1\) or \(x>1\). Numerically, we see that if \(|x|<1\), \(x^n\) becomes much smaller as \(n\) increases whereas if \(|x|>1\), \(x^n\) becomes much larger as \(n\) increases.

\(\begin{array}{|r||c|c|c|} \hline x & x^2 & x^4 & x^6 \\ \hline -2 & 4& 16& 64 \\ \hline -1 & 1 & 1& 1\\ \hline -0.5 & 0.25 & 0.0625& 0.015625 \\ \hline 0 & 0 & 0 & 0 \\ \hline 0.5 & 0.25 & 0.0625 & 0.015625 \\ \hline 1& 1 & 1& 1 \\ \hline 2 & 4 & 16 & 64 \\ \hline \end{array}\)

From the graphs, it appears as if the range of each of these functions is \([0, \infty )\). When \(n\) is even, \(x^n \geq 0\) for all \(x\) so the range of \(f(x) = x^{n}\) is contained in \([0, \infty )\). To show that the range of \(f\) is all of \([0, \infty )\), we note that the equation \(x^n = c\) for \(c \geq 0\) has (at least) one solution for every even integer \(n\), namely \(x = \sqrt [n]{c}\). (See Section ?? for a review of this notation.) Hence, \(f(\sqrt [n]{c}) = (\sqrt [n]{c})^n = c\) which shows that every non-negative real number is in the range of \(f\).

Another item worthy of note is the symmetry about the line \(x =0\) a.k.a the \(y\)-axis. (See Definition ?? for a review of this concept.) With \(n\) being even, \(f(-x) = (-x)^n = x^n = f(x)\). At the level of points, we have that for all \(x\), \((-x, f(-x)) = (-x,f(x))\). Hence for every point \((x, f(x))\) on the graph of \(f\), the point symmetric about the \(y\)-axis, \((-x, f(x))\) is on the graph, too. We give this sort of symmetry a name honoring its roots here with even-powered monomial functions:

We now turn our attention to functions of the form \(f(x) = x^{n}\) for odd powers of \(n\) (\(n \geq 3\)) and investigate three examples using Desmos.

As the slider increases \(n\) from \(3\) to \(5\) to \(7\), we see similar behavior to even powered monomials with the major difference being that when a negative number is raised to an odd natural number power the result is still negative. Numerically we see that for \(|x| > 1\) the values of \(|x^n|\) increase as \(n\) increases and the values of \(|x^n|\) get closer to \(0\) as \(n\) increases. This translates graphically into a flattening behavior on the interval \((-1, 1)\) and a narrowing elsewhere.

\(\begin{array}{|r||c|c|c|} \hline x & x^3 & x^5 & x^7 \\ \hline -2 & -8& -32& -128 \\ \hline -1 & -1 & -1& -1\\ \hline -0.5 & 0.125 & -0.03125& -0.0078125 \\ \hline 0 & 0 & 0 & 0 \\ \hline 0.5 & 0.125 & 0.03125 & 0.0078125 \\ \hline 1& 1 & 1& 1 \\ \hline 2 & 8 & 32 & 128 \\ \hline \end{array}\)

The range of these functions appear to be all real numbers, \((-\infty , \infty )\) which is algebraically sound as the equation \(x^n = c\) has a solution for every real number, namely \(x = \sqrt [n]{c}\). Hence, for every real number \(c\), choose \(x = \sqrt [n]{c}\) so that \(f(x) = f(\sqrt [n]{c}) = (\sqrt [n]{c})^n =c\). This shows that every real number is in the range of \(f\).

Here, since \(n\) is odd, \(f(-x) = (-x)^n = -x^n = -f(x)\). This means that whenever \((x, f(x))\) is on the graph, so is the point symmetric about the origin, \((-x, -f(x))\). (Again, see Definition ??.) We generalize this property below. Not surprisingly, we name it in honor of its odd powered heritage:

The most important thing to take from the discussion above is the basic shape and common points on the graphs of \(y = x^n\) for each of the families when \(n\) even and \(n\) is odd. While symmetry is nice and should be noted when present, even and odd symmetry are comparatively rare. The point of Definitions 2 and 3 is to give us the vocabulary to point out the symmetry when appropriate.

Moving on, we take a cue from Theorem ?? and prove the following.

Proof. Our goal is to start with the graph of \(f(x) = x^n\) and build it up to the graph of \(F(x) = a(x-h)^n+k\). We begin by examining \(F_{1}(x) = (x-h)^n\). The graph of \(f(x) = x^n\) can be described as the set of points \(\{ (c, c^n) \, | \, c \in \mathbb {R} \}\). Likewise, the graph of \(F_{1}\) can be described as the set of points \(\{(x, (x-h)^n) \, | \, x \in \mathbb {R} \}\). If we relabel \(c =x-h\) so that \(x = c+h\), then as \(x\) varies through all real numbers so does \(c\). Hence, we can describe the graph of \(F_{1}\) as \(\{ (c+h, c^n) \, | \, c \in \mathbb {R} \}\). This means that we can obtain the graph of \(F_{1}\) from the graph of \(f\) by adding \(h\) to each of the \(x\)-coordinates of the points on the graph of \(f\) and that establishes the first step of the theorem.

Next, we consider the graph of \(F_{2}(x) = a(x-h)^n\) as compared to the graph of \(F_{1}(x) = (x-h)^n\). The graph of \(F_{1}\) is the set of points \(\{ (x, (x-h)^n \, | \, x \in \mathbb {R} \}\) while the graph of \(F_{2}\) is the set of points \(\{ (x, a(x-h)^n) \, | \, x \in \mathbb {R} \}\). The only difference between the points \((x, (x-h)^n)\) and \((x, a(x-h)^n)\) is that the \(y\)-coordinate in the latter is \(a\) times the \(y\)-coordinate of the former.

In other words, to produce the graph of \(F_{2}\) from the graph of \(F_{1}\), we take the \(y\)-coordinate of each point on the graph of \(F_{1}\) and multiply it by \(a\) to get the corresponding point on the graph of \(F_{2}\). If \(a>0\), all we are doing is scaling the \(y\)-axis by \(a\). If \(a<0\), then, in addition to scaling the \(y\)-axis, we are also reflecting each point across the \(x\)-axis. In either case, we have established the second step of the theorem.

Last, we compare the graph of \(F(x) = a(x-h)^n + k\) to that of \(F_{2}(x) = a(x-h)^n\). Once again, we view the graphs as sets of points in the plane. The graph of \(F_{2}\) is \(\{ (x, a(x-h)^n) \, | \, x \in \mathbb {R} \}\) and the graph of \(F\) is\(\{ (x, a(x-h)^n+k) \, | \, x \in \mathbb {R} \}\). Looking at the corresponding points, \((x, a(x-h)^n)\) and \((x, a(x-h)^n+k)\), we see that we can obtain all of the points on the graph of \(F\) by adding \(k\) to each of the \(y\)-coordinates to points on the graph of \(F_{2}\). This is equivalent to shifting every point vertically by \(k\) units which establishes the third and final step in the theorem. □

This argument should sound familiar. The proof we presented above is more-or-less the same argument we presented after the proof of Theorem ?? in Section ?? but with ‘\(| \cdot |\)’ replaced by ‘\((\cdot )^n\).’ Also note that using \(n =2\) in Theorem 1 establishes Theorem ?? in Section ??.

We now use Theorem 1 to graph two different “transformed” monomial functions. To provide the reader an opportunity to compare and contrast the graphical behaviors exhibited in the case when \(n\) is even versus when \(n\) is odd, we graph one of each case.

Example 1 demonstrates two big ideas in mathematics: first, resolving a complex problem into smaller, simpler steps, and, second, the value of changing form.

Next we wish to focus on the so-called end behavior presented in each case. The end behavior of a function is a way to describe what is happening to the outputs from a function as the inputs approach the ‘ends’ of the domain. Since domain of monomial functions is \((-\infty , \infty )\), we are looking to see what these functions do as their inputs ‘approach’ \(\infty \) and \(-\infty \). The best we can do is sample inputs and outputs and infer general behavior from these observations. The good news is we’ve wrestled with this concept before. Indeed, every time we add ‘arrows’ to the graph of a function, we’ve indicated its end behavior. Let’s revisit the graph of \(f(x) = x^2\) using the table below.

\(\begin{array}{|r||c|} \hline x & f(x) = x^2 \\ \hline -1000 & 1000000 \\ \hline -100 & 10000 \\ \hline -10 & 100 \\ \hline 0 & 0 \\ \hline 10 & 100 \\ \hline 100& 10000 \\ \hline 1000 & 1000000 \\ \hline \end{array}\)

As \(x\) takes on negative values that are larger in absolute value, we see \(f(x)\) takes on larger and larger positive values, seemingly without bound. It should be stressed that since ‘\(-\infty \)’ and ‘\(\infty \)’ aren’t real numbers, we can’t write ‘\(f(-\infty ) = \infty \),’ so in order to communicate this behavior, we write as \(x \rightarrow -\infty \), \(f(x) \rightarrow \infty \), or, more succinctly, \(\ds { \lim _{x \rightarrow - \infty } f(x) = \infty }\). Note that this latter notation is borrowed from Calculus and is read ‘the limit as \(x\) approaches \(- \infty \) of \(f(x)\) is \(\infty \)’.

Graphing \(f(x) = x^2\) using GeoGebra, we can use the embedded slider to ‘zoom out’ to see what these data mean visually:

Graphically, the farther to the left we select inputs on the \(x\)-axis, the farther up the \(y\)-axis the output (function) values are. We indicate this by attaching an ‘arrow’ on the graph in Quadrant II indicating the graph continues to head upward to the left.

Similarly, observing the behavior of \(f\) as \(x \rightarrow \infty \), we get that \(\ds { \lim _{x \rightarrow \infty } f(x) = \infty }\) since as the \(x\) values increase without bound, so do the \(f(x)\) values. Graphically we indicate this by an arrow on the graph in Quadrant I heading upwards to the right. This behavior holds for all functions \(f(x) = x^n\) where \(n \geq 2\) is even.

Repeating this investigation for for \(f(x) = x^3\), we find as \(\ds { \lim _{x \rightarrow - \infty } f(x) = -\infty }\) and \(\ds { \lim _{x \rightarrow \infty } f(x) = \infty }\). This trend holds for all functions \(f(x) = x^n\) where \(n\) is odd.

\(\begin{array}{|r||c|} \hline x & f(x) = x^3 \\ \hline -1000 & -1000000000 \\ \hline -100 & -1000000 \\ \hline -10 & -1000 \\ \hline 0 & 0 \\ \hline 10 & 1000 \\ \hline 100& 1000000 \\ \hline 1000 & 1000000000 \\ \hline \end{array}\)

Theorem 2 summarizes the end behavior of monomial functions. The results are a consequence of Theorem 1 in that the end behavior of a function of the form \(y = ax^n\) only differs from that of \(y = x^n\) if there is a reflection, that is, if \(a<0\).

2 Polynomial Functions

We are now in the position to discuss polynomial functions. Simply stated, polynomial functions are sums of monomial functions. The challenge becomes how to describe one of these beasts in general. Up until now, we have used distinct letters to indicate different parameters in our definitions of function families. In other words, we define constant functions as \(f(x) = b\), linear functions as \(f(x) = mx+b\), and quadratic functions as \(f(x) = ax^2+bx+c\). We even hinted at a function of the form \(f(x) = ax^3+bx^2+cx+d\). What happens if we wanted to describe a generic polynomial that required, say, 117 different parameters? Our work around is to use subscripted parameters, \(a_k\), that denote the coefficient of \(x^{k}\). For example, instead of writing a quadratic as \(f(x) = ax^2+bx+c\), we describe it as \(f(x) = a_2 x^2 + a_1 x + a_0\), where \(a_2\), \(a_1\), and \(a_0\) are real numbers and \(a_2 \neq 0\). As an added example, consider \(f(x) = 4x^5 - 3x^2 + 2x - 5\). We can re-write the formula for \(f\) as \(f(x)= 4x^5 + 0 x^{4} + 0 x^{3} + (-3)x^2 + 2 x + (-5).\) and identify \(a_5 = 4\), \(a_4 = 0\), \(a_3 = 0\), \(a_2 = -3\), \(a_1 = 2\) and \(a_0 = -5\). This is the notation we use in the following definition.

As usual, \(x\) is used in Definition 4 as the independent variable with the \(a_k\) each being a parameter. Even though we specify \(n \in \mathbb {N}\) so \(n \geq 1\), the value of the \(a_k\) are unrestricted. Hence, any constant function \(f(x) = b\) can be written as \(f(x) = 0 x + a_0\), and so they are polynomials. Polynomials have an associated vocabulary, and hence, so do polynomial functions.

Again, constant functions are split off in their own separate case Definition 5 because of the ambiguity of \(0^0\). (See the remarks following Definition 1.) A consequence of Definition 5 is that we can now think of nonzero constant functions as ‘zeroth’ degree polynomial functions, linear functions as ‘first’ degree polynomial functions, and quadratic functions as ‘second’ degree polynomial functions.

We now turn our attention to graphs of polynomial functions. Since polynomial functions are sums of monomial functions, it stands to reason that some of of the properties of those graphs carry over to more general polynomials. We first discuss end behavior. Consider \(f(x) = x^3-75x+250\). Using Desmos, we graph \(y = f(x)\) along with the graph of its leading term, \(y = x^3\). Near the origin, the two graphs look quite different; however, as we zoom out, the two graphs begin look similar:

This observation is borne out numerically as well. Based on the table below, as \(x \rightarrow \infty \) or \(x \rightarrow -\infty \), it certainly appears as if \(f(x) \approx g(x)\):

\[ \begin{array}{|r||c|c|c|c|c|c|} \hline x & f(x) = x^3 -75x+250 & x^3 & -75 x & 250 & \frac {75}{x^2} & \frac {250}{x^3} \\ \hline -1000 & \approx -1 \times 10^9 & -1 \times 10^9 &75000 & 250 & 7.5 \times 10^{-5} & -2.5 \times 10^{-7} \\ \hline -100 & \approx -9.9 \times 10^5 & -1 \times 10^6 & 7500 & 250 & 0.0075 & -2.5 \times 10^{-4} \\ \hline -10 & 0 & -1000 & 750 & 250 & 0.75 & -0.25\\ \hline 10 & 500 & 1000 & -750 & 250 & 0.75 & 0.25 \\ \hline 100 &\approx 9.9 \times 10^5 & 1 \times 10^6 & -7500 & 250 & 0.0075 & 2.5 \times 10^{-4} \\ \hline 1000 & \approx 1 \times 10^9 & 1 \times 10^9 & -75000 & 250 & 7.5 \times 10^{-5} & 2.5 \times 10^{-7} \\ \hline \end{array} \]

One way to think about what is happening numerically is that as \(x \rightarrow \infty \) or \(x \rightarrow -\infty \), the leading term \(x^3\) dominates the lower order terms \(-75x\) and \(250\). In other words, \(x^3\) grows so much faster than \(-75 x\) and \(250\) that these ‘lower order terms’ don’t contribute anything of significance to the \(x^3\) so \(f(x) \approx x^3\). To see this, we rewrite \(f(x)\) as

\[f(x) = x^3 - 75x + 250 = x^3 \left (1 - \frac {75}{x^2} + \frac {250}{x^3} \right ).\]

As \(x \rightarrow \infty \) or \(x \rightarrow -\infty \), both \(\frac {75}{x^2}\) and \(\frac {250}{x^3}\) have constant numerators but denominators that are becoming unbounded. As such, both \(\frac {75}{x^2}\) and \(\frac {250}{x^3} \rightarrow 0\). Therefore, as \(x \rightarrow \infty \) or \(x \rightarrow -\infty \),

\[ f(x) = x^3 - 75x+250 = x^3 \left (1 - \frac {75}{x^2} + \frac {250}{x^3} \right ) \approx x^3 (1 + 0 + 0) = x^3. \]

Next, consider \(g(x) = -0.01x^4 + 5x^2\). Following the logic of the above example, we would expect the end behavior of \(y=g(x)\) to mimic that of \(y = -0.01 x^4\). When we use Desmos to graph \(y = g(x)\) and \(y = -0.01x^4\), a view near the origin seems to suggest the exact opposite. However, zooming out reveals that the two graphs do share the same end behavior.

Algebraically, for \(x \rightarrow \infty \) or \(x \rightarrow -\infty \), even with the small coefficient of \(-0.01\), \(-0.01x^4\) dominates the \(5x^2\) term so \(g(x) \approx -0.01 x^4\). More precisely,

\[g(x) = -0.01x^4+5x^2 = x^4 \left (-0.01 + \frac {5}{x^2} \right ) \approx x^4(-0.01 + 0)= -0.01x^4. \]

The results of these last two examples generalize below in Theorem 3.

We argue Theorem 3 using an argument similar to ones used above. As \(x \rightarrow \infty \) or \(x \rightarrow -\infty \),

\[ f(x) = x^{n} \left ( a_{n} +\dfrac {a_{n-1}}{x}+ \ldots + \dfrac {a_2}{x^{n-2}} + \dfrac {a_1}{x^{n-1}}+\dfrac {a_0}{x^{n}}\right ) \approx x^n( a_{n} + 0 +\ldots 0) = a_{n} x^n \]

If this argument looks a little fuzzy, it should. As with all things involving infinity, the precision of Calculus is required here For now, we’ll rely on number sense and algebraic intuition.

Now that we know how to determine the end behavior of polynomial functions, it’s time to investigate what happens ‘in between’ the ends. First and foremost, polynomial functions are continuous. Recall from Section ?? that, informally, graphs of continuous functions have no ‘breaks’ or ‘holes’ in them. Since monomial functions are continuous (as far as we can tell) and polynomials are sums of monomial functions, it turns out that polynomial functions are continuous as well.

Moreover, the graphs of monomial functions, hence polynomial functions, are smooth. Once again, ‘smoothness’ is a concept defined precisely in Calculus, but for us, functions have no ‘corners’ or ‘sharp turns’. Below we find the graph of a function which is neither smooth nor continuous, and to its right we have a graph of a polynomial, for comparison.

The function whose graph appears on the left fails to be continuous where it has a ‘break’ or ‘hole’ in the graph; everywhere else, the function is continuous. The function is continuous at the ‘corner’ and the ‘cusp’, but we consider these ‘sharp turns’, so these are places where the function fails to be smooth. Apart from these four places, the function is smooth and continuous. Polynomial functions are smooth and continuous everywhere, as exhibited in the graph on the right.

The notion of smoothness is what tells us graphically that, for example, \(f(x) = |x|\), whose graph is the characteristic ‘\(\vee \)’ shape, cannot be a polynomial function, even though it is a piecewise-defined function comprised of polynomial functions. Knowing polynomial functions are continuous and smooth gives us an idea of how to ‘connect the dots’ when sketching the graph from points that we’re able to find analytically such as intercepts.

Pathologies not found on graphs of polynomial functions.
The graph of a polynomial function.

Speaking of intercepts, we next focus our attention on the behavior of the graphs of polynomial functions near their zeros. Recall a zero \(c\) of a function \(f\) is a solution to \(f(x) = 0\). Geometrically, the zeros of a function are the \(x\)-coordinates of the \(x\)-intercepts of the graph of \(y = f(x)\).

Consider the polynomial function \(f(x) = x^3 (x-2)^2 (x+1)\). To find the zeros of \(f\), we set \(f(x) = x^3 (x-2)^2 (x+1) = 0\). Since the expression \(f(x)\) is already factored, we set each factor equal to zero.

Solving \(x^3 = 0\) gives \(x = 0\), \((x-2)^2 = 0\) gives \(x = 2\), and \(x+1 = 0\) gives \(x = -1\). Hence, our zeros are \(x = -1\), \(x = 0\), and \(x = 2\).

Using Desmos, we graph \(y = f(x)\). By selecting each \(x\)-intercept, we can note the behavior of the graph near each of these points.

We first note that the graph crosses through the \(x\)-axis at \((-1,0)\) and \((0,0)\), but the graph touches and rebounds at \((2,0)\). Moreover, at \((-1,0)\), the graph crosses through the axis is a fairly ‘linear’ fashion whereas there is a substantial amount of ‘flattening’ going on near \((0,0)\). Or aim is to explain these observations and generalize them.

First, let’s look at what’s happening with the formula \(f(x) = x^3 (x-2)^2 (x+1)\) when \(x \approx -1\). We know the \(x\)-intercept at \((-1,0)\) is due to the presence of the \((x+1)\) factor in the expression for \(f(x)\). So, in this sense, the factor \((x+1)\) is determining a major piece of the behavior of the graph near \(x = -1\). For that reason, we focus instead on the other two factors to see what contribution they make.

We find when \(x \approx -1\), \(x^3 \approx (-1)^3 = -1\) and \((x-2)^2 \approx (-1-2)^2 = 9\). Hence, \(f(x) = x^3 (x-3)^2 (x+1) \approx (-1)^3 (-1-2)^2 (x+1) = -9(x+1)\).

Near \(x = 0\). The \(x\)-intercept \((0,0)\) is due to the \(x^3\) term. For \(x \approx 0\), \((x-2)^2 \approx (0-2)^2 = 4\) and \((x+1) \approx (0+1) = 1\), so \(f(x) = x^3 (x-3)^2 (x+1) \approx x^3 (-2)^2(1) = 4x^3\).

Last, but not least, we analyze \(f\) near \(x = 2\). Here, the intercept \((2,0)\) is due to the \((x-2)^2\) factor, so we look at the \(x^3\) and \((x+1)\) factors. If \(x \approx 2\), \(x^3 \approx (2)^3 = 8\) and \((x+1) \approx (2+1) = 3\). Hence, \(f(x) = x^3 (x-3)^2 (x+1) \approx (2)^3 (x-2)^2 (2+1) = 24(x-2)^2\).

Using Desmos, we graph \(f\) and can select to display the graphs of \(y=-9(x+1)\), \(y = 4x^3\), and \(y=24(x-2)^2\).

Near \((-1,0)\), the graph of \(f\) closely resembles the line \(y = -9(x+1)\). Near \((0,0)\), the graph of \(f\) closely resembles the graph of the cubic monomial \(y = 4x^3\). Finally, near \((2,0)\), the graph resembles the parabola \(y=24(x-2)^2\).

We generalize our observations in Theorem 4 below. Like many things we’ve seen in this text, a more precise statement and proof can be found in a course on Calculus.

Let’s see how Theorem 4 applies to our findings regarding \(f(x) = x^3 (x-2)^2 (x+1)\). For \(c = -1\), \((x-c) = (x-(-1)) = (x+1)\). We rewrite \(f(x) = x^3 (x-2)^2 (x+1) = (x-(-1))^1 \left [x^3(x-2)^2\right ]\) and identify \(m=1\) and \(q(x) = x^3 (x-2)^2\). We find \(q(c) = q(-1) = (-1)^3(-1-2)^2 = -9\) so Theorem 4 says that near \((-1,0)\), the graph of \(y=f(x)\) resembles \(y = q(-1)(x-(-1))^1 = -9(x+1)\).

For \(c=0\), \((x-c) = (x-0) = x\) and we can rewrite \(f(x) = x^3 (x-2)^2 (x+1) = (x-0)^3 \left [(x-2)^2 (x+1)\right ]\). We identify \(m=3\) and \(q(x) = (x-2)^2(x+1)\). In this case \(q(c) = q(0) = (0-2)^2(0+1) = 4\), so Theorem 4 guarantees the graph of \(y = f(x)\) near \(x=0\) resembles \(y = q(0)(x-0)^3 = 4x^3\).

Lastly, for \(c = 2\), we see \(f(x) = (x-2)^2 \left [x^3 (x+1)\right ]\) and identify \(m = 2\) and \(q(x) = x^3(x+1)\). We find \(q(2) = 2^3 (2+1)= 24\), so per Theorem 4, the graph of \(y = f(x)\) resembles \(y = 24(x-2)^2\) near \(x=2\).

As we already mentioned, the formal statement and proof of Theorem 4 require Calculus. For now, we can understand the theorem as follows.

If we factor a polynomial function as \(f(x) = (x-c)^m q(x)\) where \(m \geq 1\), then \(x=c\) is a zero of \(f\), since \(f(c) = (c-c)^m q(c) = 0 \cdot q(c) = 0\). The stipulation that \(q(c) \neq 0\) means that we have essentially factored the expression \(f(x) = (x-c)^m q(x) = (\text {going to $0$}) \cdot (\text {not going to $0$})\).

Thinking back to Theorem 1, the graph \(y = q(c) (x-c)^m\) has an \(x\)-intercept at \((c,0)\), a basic overall shape determined by the exponent \(m\), and end behavior determined by the sign of \(q(c)\).

The fact that if \(x=c\) is a zero then we are guaranteed we can factor \(f(x) = (x-c)^m q(x)\) were \(q(c) \neq 0\) and, moreover, such a factorization is unique (so that there’s only one value of \(m\) possible for each zero) is a consequence of two theorems, Theorem ?? and The Factor Theorem, Theorem ?? which we’ll review in Section ??. For now, we assume such a factorization is unique in order to define the following.

So, for \(f(x) = x^3 (x-2)^2 (x+1) = (x-0)^3(x-2)^2(x-(-1))^1\), \(x=0\) is a zero of multiplicity \(3\), \(x=2\) is a zero of multiplicity \(2\), and \(x =-1\) is a zero of multiplicity \(1\). Theorems 3 and 4 give us the following:

Our next example showcases how all of the above theory can assist in sketching relatively good graphs of polynomial functions without the assistance of technology.

A couple of remarks about Example 3 are in order. First, notice that the factor \((x^2+1)\) was more of a spectator in our discussion of the zeros of \(p\). Indeed, if we set \(x^2+1 = 0\), we have \(x^2=-1\) which provides no real solutions. That being said, the factor \(x^2+1\) does affect the shape of the graph.

Next, when connecting up the graph from \((-1,0)\) to \((0,-1)\) to \(\left (\frac {1}{2}, 0 \right )\), there really is no way for us to know how low the graph goes, or where the lowest point is between \(x = -1\) and \(x = \frac {1}{2}\) unless we plot more points. Likewise, we have no idea how high the graph gets between \(x = \frac {1}{2}\) and \(x = 1\). While there are ways to determine these points analytically, more often than not, finding them requires concepts from Calculus which we’ll investigate later. Since these points do play an important role in many applications, we’ll need to discuss them in this course and, when required, we’ll use technology to find them. For that reason, we have the following definition:

Once again, the terminology used in Definition 7 blurs the line between the function \(f\) and its outputs, \(f(x)\). Also, some textbooks use the terms ‘relative’ minimum and ‘relative’ maximum instead of the adjective ‘local.’ Lastly, note the definition of local extrema requires an open interval exist in the domain containing \(a\) in order for \((a, f(a))\) to be a candidate for a local maximum or local minimum. We’ll have more to say about this in later chapters. If our open interval happens to be \((-\infty , \infty )\), then our local extrema are the extrema of \(f\).

Let’s take a moment to analyze the graph of \(p(x) = (2x-1)(x+1)(1-x^4)\) furnished by Desmos.

We first consider the point \((-1,0)\). Even though there are points on the graph of \(y = p(x)\) that are higher than \((-1,0)\), locally, \((-1,0)\) is the top of a hill. To satisfy Definition 7, we need to provide an open interval on which \(p(-1) = 0\) is the largest, or maximum function value. Note the definition requires us to provide just one open interval. One that works is the interval \((-1.5, -0.5)\). We could use any smaller interval or go as large as \(\left (-\infty , \frac {1}{2} \right )\).

Next we encounter a ‘low’ point at approximately \((-0.2353, -1.1211)\). More specifically, for all \(x\) in the interval, say, \((-0.5, 0)\), \(p(x) \geq 1.1211\), Hence, we have a local minimum at \((-0.2353, -1.1211)\). Lastly, at \((0.811, 0.639)\), we are back to a high point. In fact, \(0.639\) isn’t just a local maximum value, based on the graph, it is the maximum of \(p\). Here, we may choose the open interval \((-\infty , \infty )\) as the open interval required by Definition 7, since for all \(x\), \(p(x) \leq 0.639\). It is important to note that there is no minimum value of \(p\) despite there being a local minimum value.

We close this section with a classic application of a third degree polynomial function.

Notice that there is a very slight, but important, difference between the function \(V(x) = x(10-2x)(12-2x)\), \(0 < x < 5\) from Example 4 and the function \(p(x) = x(10-2x)(12-2x)\): their domains. The domain of \(V\) is restricted to the interval \((0,5)\) while the domain of \(p\) is \((-\infty , \infty )\). Indeed, the function \(V\) has a maximum of (approximately) \(96.771\) at (approximately) \(x = 1.811\) whereas for the function \(p\), \(96.771\) is a local maximum value only. We leave it to the reader to verify that \(V\) has neither a minimum nor a local minimum.