5.3: Power Functions and Polynomial Functions

Section Preview

Note to the Instructor (click to expand)

The purpose of the material in this section is to introduce the vocabulary needed to work with polynomial functions. Students are introduced to power functions, polynomials, and end behavior but are not asked to graph at this time.

Operational Prerequisites (i.e., "Completing the Homework"): The following prerequisite topics (which have not already been listed as prerequisites in previous sections of this text) are needed to complete the homework.

(click to expand)

• Factoring Techniques
  • Factoring Trinomials
• Simplifying Expressions
  • Laws of Exponents—Power Rules
  • Laws of Exponents—Zero and Negative Exponent Rules
  • Simplifying Expressions Involving Positive Rational Exponents

The following is a list of learning objectives for this section.

Learning Objectives (click to expand)

• Identify power functions.
• Identify end behavior of power functions.
• Identify polynomial functions.
• Identify the degree and leading coefficient of polynomial functions.
• Compute the Difference Quotient for power and polynomial functions.

Figure \(\PageIndex{1}\): credit: Jason Bay, Flickr

Suppose a certain species of bird thrives on a small island. Its population over the last few years is shown in Table \(\PageIndex{1}\).

Table \(\PageIndex{1}\)

Year	2009	2010	2011	2012	2013
Bird Population	800	897	992	1,083	1,169

The population can be estimated using the function \(P(t)=-0.3 t^3 +97t+800\), where \(P(t)\) represents the bird population on the island \(t\) years after 2009. We can use this model to estimate the maximum bird population and when it will occur. We can also use this model to predict when the bird population will disappear from the island. In this section, we will examine functions that we can use to estimate and predict these types of changes.

Identifying Power Functions

In order to better understand the bird problem, we need to understand a specific type of function called a power function.

Definition: Power Function

A power function is a function of the form\[f(x) = a x^n, \nonumber \]where \(a\) and \(n\) are real numbers and \(a \neq 0\). \(a\) is known as the coefficient of the power function, and \(n\) is called its exponent.

As an example, consider functions for area or volume. The function for the area of a circle with radius \(r\) is \(A(r)= \pi r^2\) and the function for the volume of a sphere with radius \(r\) is \(V(r)= \frac{4}{3} \pi r^3\). Both of these are examples of power functions because they consist of a coefficient, \(\pi\) or \(\frac{4}{3} \pi\), multiplied by a variable \(r\) raised to a power.

Q&A

Is \(f(x)= 2^x\) a power function?

No. A power function contains a variable base raised to a fixed power. This function has a constant base raised to a variable power. This is called an exponential function, not a power function. We will deal with these types of functions later.

Example \(\PageIndex{1}\): Identifying Power Functions

Which of the following functions are power functions?\[\begin{array}{ll}
f(x)=1 & \text{Constant function} \\[6pt]
f(x)=x & \text{Identify function} \\[6pt]
f(x)= x^2 & \text{Quadratic function} \\[6pt]
f(x)= x^3 & \text{Cubic function} \\[6pt]
f(x)= \dfrac{1}{x} & \text{Reciprocal function} \\[6pt]
f(x)= \dfrac{1}{x^2} & \text{Reciprocal squared function} \\[6pt]
f(x)=\sqrt{ x } & \text{Square root function} \\[6pt]
f(x)= \sqrt[3]{x} & \text{Cube root function} \\[6pt]
\end{array}\nonumber\]

Solution

All of the listed functions are power functions. The constant and identity functions are power functions because they can be written as \(f(x)= x^0\) and \(f(x)= x^1\), respectively. The quadratic and cubic functions are power functions with whole number powers. The reciprocal and reciprocal squared functions are power functions with negative whole number powers because they can be written as \(f(x)= x^{-1}\) and \(f(x)= x^{-2}\). The square and cube root functions are power functions with fractional powers because they can be written as \(f(x)= x^{1/2}\) or \(f(x)= x^{1/3}\).

Checkpoint \(\PageIndex{1}\)

Which functions are power functions?\[\begin{array}{rcl}
f(x) & = & 2 x^2 \cdot 4 x^3 \\[6pt]
g(x) & = & -x^5 +5 x^3 -4x \\[6pt]
h(x) & = & \dfrac{2 x^5 -1}{3 x^2 +4} \\[6pt]
\end{array}\nonumber\]

The ever-important Difference Quotient is often used in Calculus when working with power function.

Example \(\PageIndex{2}\): Revisiting the Difference Quotient

Compute the Difference Quotient for the given function.\[f(x) = x^4\nonumber\]

Solution

Recall that the Difference Quotient for the function \(f\) is defined to be\[\dfrac{f(x + h) - f(x)}{h}.\nonumber\]Therefore, in this case, we have\[\begin{array}{rclcl}
\dfrac{f(x + h) - f(x)}{h} & = & \dfrac{(x + h)^4 - x^4}{h} & \quad & \left( \text{substituting} \right) \\[6pt]
& = & \dfrac{\left((x + h)^2\right)^2 - x^4}{h} & \quad & \left( \text{Laws of Exponents} \right) \\[6pt]
& = & \dfrac{\left((x + h)(x + h)\right)^2 - x^4}{h} & & \\[6pt]
& = & \dfrac{\left(x^2 + 2hx + h^2\right)^2 - x^4}{h} & \quad & \left( \text{distributing} \right) \\[6pt]
& = & \dfrac{\left(x^2 + 2hx + h^2\right)\left(x^2 + 2hx + h^2\right) - x^4}{h} & & \\[6pt]
& = & \dfrac{x^4 +4x^3 h + 6x^2 h^2 + 4xh^3 + h^4 - x^4}{h} & \quad & \left( \text{distributing} \right) \\[6pt]
& = & \dfrac{4x^3 h + 6x^2 h^2 + 4xh^3 + h^4}{h} & \quad & \left( \text{combining like terms} \right) \\[6pt]
& = & \dfrac{h\left(4x^3 + 6x^2 h + 4xh^2 + h^3\right)}{h} & \quad & \left( \text{factoring out the GCF from the numerator} \right) \\[6pt]
& = & \dfrac{\cancelto{1}{h}\left(4x^3 + 6x^2 h + 4xh^2 + h^3\right)}{\cancelto{1}{h}} & \quad & \left( \text{canceling like factors} \right) \\[6pt]
& = & 4x^3 + 6x^2 h + 4xh^2 + h^3 & \quad & \left( \text{simplifying} \right) \\[6pt]
\end{array}\nonumber\]

In Example \(\PageIndex{2}\), we could have taken the alternative approach of using factoring techniques to simplify the numerator.\[\begin{array}{rclcl}
\dfrac{f(x + h) - f(x)}{h} & = & \dfrac{(x + h)^4 - x^4}{h} & \quad & \left( \text{substituting} \right) \\[6pt]
& = & \dfrac{\left((x + h)^2\right)^2 - \left(x^2\right)^2}{h} & \quad & \left( \text{Laws of Exponents} \right) \\[6pt]
& = & \dfrac{\left((x + h)^2 - x^2\right)\left((x + h)^2 + x^2\right)}{h} & \quad & \left( \text{Difference of Squares} \right) \\[6pt]
& = & \dfrac{\left((x + h) - x\right)\left((x + h) + x\right)\left((x + h)^2 + x^2\right)}{h} & \quad & \left( \text{Difference of Squares} \right) \\[6pt]
& = & \dfrac{\left(h\right)\left(2x + h\right)\left(x^2 + 2hx + h^2 + x^2\right)}{h} & \quad & \left( \text{simplifying} \right) \\[6pt]
& = & \dfrac{ \cancelto{1}{h}\left(2x + h\right)\left(x^2 + 2hx + h^2 + x^2\right)}{\cancelto{1}{h}} & \quad & \left( \text{canceling like factors} \right) \\[6pt]
& = & \left(2x + h\right)\left(2x^2 + 2hx + h^2\right) & \quad & \left( \text{simplifying} \right) \\[6pt]
& = & 4x^3 + 6x^2 h + 4xh^2 + h^3 & \quad & \left( \text{distributing} \right) \\[6pt]
\end{array}\nonumber\]While both methods are valid and have their place, a student well-prepared for Calculus will inevitably want to try the second method.

Identifying End Behavior of Power Functions

Figure \(\PageIndex{2}\) shows the graphs of \(f(x)= x^2\), \(g(x)= x^4\), and \(h(x)= x^6\), which are all power functions with even, whole-number powers. Notice that these graphs have similar shapes, very much like that of the quadratic function in the toolkit. However, as the power increases, the graphs flatten somewhat near the origin and become steeper away from the origin.

Figure \(\PageIndex{2}\): Even-power functions

To describe the behavior as numbers become larger and larger, we use the idea of infinity. We use the symbol \(\infty\) for positive infinity and \(- \infty\) for negative infinity. When we say that "\(x\) approaches infinity," which can be symbolically written as \(x \to \infty\), we are describing a behavior; we are saying that \(x\) is increasing without bound.

With the even-power function, as the input increases or decreases without bound, the output values become very large, positive numbers. Equivalently, we could describe this behavior by saying that as \(x\) approaches positive or negative infinity, the \(f(x)\) values increase without bound. In symbolic form, we could write\[\text{as }x \to \pm \infty,\quad f(x) \to \infty.\nonumber\]Figure \(\PageIndex{3}\) shows the graphs of \(f(x)= x^3\), \(g(x)= x^5\), and \(h(x)= x^7\), which are all power functions with odd, whole-number powers. Notice that these graphs look similar to the cubic function in the toolkit. Again, as the power increases, the graphs flatten near the origin and become steeper away from the origin.

Figure \(\PageIndex{3}\): Odd-power functions

These examples illustrate that functions of the form \(f(x)= x^n\) reveal symmetry of one kind or another. First, in Figure \(\PageIndex{2}\) we see that functions of the form \(f(x)= x^n\), where \(n\) is even, are symmetric about the \(y\)-axis. In Figure \(\PageIndex{3}\), we see that functions of the form \(f(x)= x^n\), where \(n\) is odd, are symmetric about the origin.

For these odd power functions, as \(x\) approaches negative infinity, \(f(x)\) decreases without bound. As \(x\) approaches positive infinity, \(f(x)\) increases without bound. In symbolic form, we write\[\text{as }x \to - \infty,\quad f(x) \to - \infty \quad \text{ and as }x \to \infty,\quad f(x) \to \infty.\nonumber\]The behavior of the graph of a function as the input values get very negative ( \(x \to - \infty\) ) and get very large ( \(x \to \infty\) ) is referred to as the end behavior of the function. We can use words or symbols to describe end behavior.

Definition: End Behavior of a Function

The end behavior of a function refers to the way the graph of the function behaves as the input (typically the \( x \)-value) approaches positive or negative infinity (denoted \( x \to \infty \) or \( x \to -\infty \)).

Figure \(\PageIndex{4}\) shows the end behavior of power functions in the form \(f(x)=k x^n\), where \(n\) is a non-negative integer depending on the power and the constant.

Figure \(\PageIndex{4}\)

Interactive Element: End Behavior of a Power Function

The following Interactive Element might help you visualize how the exponent and the coefficient of a power function \(f(x)=k\,x^n\) control its end behavior.

Interact: Drag the red point along the horizontal axis to change the exponent \(n\) (whole numbers), and drag the orange point up or down to change the coefficient \(k\).

Observation: When \(n\) is even, the graph is symmetric about the \(y\)-axis and both ends point the same way; when \(n\) is odd, it is symmetric about the origin and the ends point in opposite directions. Making \(k\) negative reflects the graph vertically, reversing both ends.

Example \(\PageIndex{3}\): Identifying the End Behavior of a Power Function

Describe the end behavior of the graph of \(f(x)= x^8\).

Solution

The coefficient is 1 (positive) and the exponent of the power function is 8 (an even number). As \(x\) approaches infinity, the output (value of \(f(x)\) ) increases without bound. Therefore, as \(x \to \infty\), \(f(x) \to \infty\). As \(x\) approaches negative infinity, the output increases without bound. In symbolic form, as \(x \to - \infty\), \(f(x) \to \infty\). We can graphically represent the function as shown in Figure \(\PageIndex{5}\).

Figure \(\PageIndex{5}\)

Example \(\PageIndex{4}\): Identifying the End Behavior of a Power Function.

Describe the end behavior of the graph of \(f(x)= -x^9\).

Solution

The exponent of the power function is 9 (an odd number). Because the coefficient is \(-1\) (negative), the graph is the reflection about the \(x\)-axis of the graph of \(f(x)= x^9\). Figure \(\PageIndex{6}\) shows that as \(x\) approaches infinity, the output decreases without bound. As \(x\) approaches negative infinity, the output increases without bound. In symbolic form, we would write\[\text{as }x \to - \infty,\,f(x) \to \infty;\,\text{as } x \to \infty,\,f(x) \to - \infty.\nonumber\]

Figure \(\PageIndex{6}\)

We can check our work by using the table feature on a graphing utility.

Table \(\PageIndex{2}\)

\(x\)	\(f(x)\)
-10	1,000,000,000
-5	1,953,125
0	0
5	-1,953,125
10	-1,000,000,000

We can see from Table \(\PageIndex{2}\) that, when we substitute very small values for \(x\), the output is very large, and when we substitute very large values for \(x\), the output is very small (meaning that it is a very large negative value).

Checkpoint \(\PageIndex{4}\)

Describe in words and symbols the end behavior of \(f(x)=-5 x^4\).

Identifying Polynomial Functions

An oil pipeline bursts in the Gulf of Mexico, causing an oil slick in a roughly circular shape. The slick is currently 24 miles in radius, but that radius is increasing by 8 miles each week. We want to write a formula for the area covered by the oil slick by combining two functions. The radius, \(r\), of the spill depends on the number of weeks, \(w\), that have passed. This relationship is linear.\[r(w) =24+8w.\nonumber\]We can combine this with the formula for the area \(A\) of a circle, \(A(r) = \pi r^2\). Composing these functions gives a formula for the area in terms of weeks.\[A(w)=A(r(w)) =A(24+8w) = \pi (24+8w)^2.\nonumber\]Distributing gives the formula\[A(w)=576 \pi +384 \pi w+64 \pi w^2.\nonumber\]This formula is an example of a polynomial function.

Definition: Polynomial Function

A polynomial function is a function that can be written in the form\[ f(x)= a_n x^n + a_{n - 1} x^{n - 1} + \ldots + a_2 x^2 + a_1 x + a_0, \nonumber \]where \( n \in \mathbb{N} \). This is called the general form of a polynomial function. Each \( a_i \) is a coefficient and can be any real number. Each product \( a_i x_i \) is called a term of the function. The largest \( n \) for which \( a_n \neq 0 \) is called the degree of the polynomial function. The leading term is the term containing the highest power of the variable, or the term with the highest degree. The leading coefficient is the coefficient of the leading term.

Example \(\PageIndex{5}\): Identifying Polynomial Functions

Which of the following are polynomial functions?\[\begin{array}{rcl}
f(x) & = & 2 x^3 \cdot 3x+4 \\[6pt]
g(x) & = & -x(x^2 -4) \\[6pt]
h(x) & = & 5\sqrt{ x } +2 \\[6pt]
\end{array}\nonumber\]

Solution

The first two functions are examples of polynomial functions because they can be written in the form \(f(x)= a_n x^n + a_{n - 1} x^{n - 1} + \ldots + a_2 x^2 + a_1 x + a_0\) where the powers are non-negative integers and the coefficients are real numbers.

• \(f(x)\) can be written as \(f(x)=6 x^4 +4\).
• \(g(x)\) can be written as \(g(x)=- x^3 +4x\).
• \(h(x)\) cannot be written in this form and is therefore not a polynomial function.

Example \(\PageIndex{6}\): Revisiting the Difference Quotient

Find and simplify the Difference Quotient for the following function.\[P(x) = 2x^3 + x - 4\nonumber\]

Solution

\[\begin{array}{rclcl}
\dfrac{P(x + h) - P(x)}{h} & = & \dfrac{2(x + h)^3 + (x + h) - 4 - \left( 2x^3 + x - 4 \right)}{h} & \quad & \left( \text{substituting} \right) \\[6pt]
& = & \dfrac{2(x + h)^3 + (x + h) - 4 - 2x^3 - x + 4}{h} & \quad & \left( \text{distributing the negative} \right) \\[6pt]
& = & \dfrac{2(x^3 + 3x^2h + 3xh^2 + h^3) + x + h - 4 - 2x^3 - x + 4}{h} & \quad & \left( \text{distributing} \right) \\[6pt]
& = & \dfrac{2x^3 + 6x^2h + 6xh^2 + 2h^3 + x + h - 4 - 2x^3 - x + 4}{h} & \quad & \left( \text{distributing} \right) \\[6pt]
& = & \dfrac{6x^2h + 6xh^2 + 2h^3 + h}{h} & \quad & \left( \text{combining like terms} \right) \\[6pt]
& = & \dfrac{h\left(6x^2 + 6xh + 2h^2 + 1\right)}{h} & \quad & \left( \text{factoring out the GCF from the numerator} \right) \\[6pt]
& = & \dfrac{\cancelto{1}{h}\left(6x^2 + 6xh + 2h^2 + 1\right)}{\cancelto{1}{h}} & \quad & \left( \text{canceling like factors} \right) \\[6pt]
& = & 6x^2 + 6xh + 2h^2 + 1 & \quad & \left( \text{simplifying} \right) \\[6pt]
\end{array}\nonumber\]

Because of the form of a polynomial function, we can see an infinite variety in the number of terms and the power of the variable. Although the order of the terms in the polynomial function is not important for performing operations, we typically arrange the terms in descending order of power (this is what we call, in the definition of the polynomial function, general form).

Example \(\PageIndex{7}\): Identifying the Degree and Leading Coefficient of a Polynomial Function

Identify the degree, leading term, and leading coefficient of the following polynomial functions.\[\begin{array}{rcl}
f(x) & = & 3+2 x^2 -4 x^3 \\[6pt]
g(t) & = & 5 t^5 -2 t^3 +7t \\[6pt]
h(p) & = & 6p- p^3 -2 \\[6pt]
\end{array}\nonumber\]

Solution

For the function \(f(x)\), the highest power of \(x\) is 3, so the degree is 3. The leading term is the term containing that degree, \(-4 x^3\). The leading coefficient is the coefficient of that term, \(-4\).

For the function \(g(t)\), the highest power of \(t\) is \(5\), so the degree is \(5\). The leading term is the term containing that degree, \(5 t^5\). The leading coefficient is the coefficient of that term, \(5\).

For the function \(h(p)\), the highest power of \(p\) is \(3\), so the degree is \(3\). The leading term is the term containing that degree, \(-p^3;\) the leading coefficient is the coefficient of that term, \(-1\).

Checkpoint \(\PageIndex{7}\)

Identify the degree, leading term, and leading coefficient of the polynomial \(f(x)=4 x^2 - x^6 +2x-6\).

Identifying End Behavior of Polynomial Functions

Knowing the degree of a polynomial function is useful in helping us predict its end behavior. Recall the following definition.

Definition: End Behavior of a Function

The end behavior of a function refers to the way the graph of the function behaves as the input (typically the \( x \)-value) approaches positive or negative infinity (denoted \( x \to \infty \) or \( x \to -\infty \)).

To determine its end behavior, look at the leading term of the polynomial function. Because the power of the leading term is the highest, that term will grow significantly faster than the other terms as \(x\) gets very large or very small, so its behavior will dominate the graph. We state this in the following theorem along with a formal proof of the result.

Theorem: End Behavior of a Polynomial Function

Let\[ P(x) = a_n x^n + a_{n - 1}x^{n - 1} + a_{n - 2}x^{n - 2} + \cdots + a_2 x^2 + a_1 x + a_0, \nonumber \]where \( a_n \neq 0 \) and \(n \geq 1\). The end behavior of the graph of \( P(x) \) matches the end behavior of the graph of \( y = a_n x^n \). Hence,

1. If \(n\) is even and \(a_n \gt 0\), then \(P(x) \to \infty\) as \(x \to -\infty\) and as \(x \to \infty\).
2. If \(n\) is even and \(a_n \lt 0\), then \(P(x) \to -\infty\) as \(x \to -\infty\) and as \(x \to \infty\).
3. If \(n\) is odd and \(a_n \gt 0\), then \(P(x) \to -\infty\) as \(x \to -\infty\), and \(P(x) \to \infty\) as \(x \to \infty\).
4. If \(n\) is odd and \(a_n \lt 0\), then \(P(x) \to \infty\) as \(x \to -\infty\), and \(P(x) \to -\infty\) as \(x \to \infty\).

Proof

Let\[ P(x) = a_n x^n + a_{n - 1}x^{n - 1} + a_{n - 2}x^{n - 2} + \cdots + a_2 x^2 + a_1 x + a_0, \nonumber \]where \( a_n \neq 0 \). Then\[ \begin{array}{rclcl}
P(x) & = & a_n x^n + a_{n - 1}x^{n - 1} + a_{n - 2}x^{n - 2} + \cdots + a_2 x^2 + a_1 x + a_0 & & \\[6pt]
& = & a_n x^n\left(1 + \dfrac{a_{n - 1}x^{n - 1}}{a_n x^n} + \dfrac{a_{n - 2}x^{n - 2}}{a_n x^n} + \cdots + \dfrac{a_2 x^2}{a_n x^n} + \dfrac{a_1 x}{a_n x^n} + \dfrac{a_0}{a_n x^n}\right) & \quad & \left( \text{factoring out }a_n x^n \right) \\[6pt]
& = & a_n x^n\left(1 + \dfrac{a_{n - 1}}{a_n x} + \dfrac{a_{n - 2}}{a_n x^2} + \cdots + \dfrac{a_2 }{a_n x^{n - 2}} + \dfrac{a_1}{a_n x^{n - 1}} + \dfrac{a_0}{a_n x^n}\right) & \quad & \left( \text{Laws of Exponents} \right) \\[6pt]
\end{array} \nonumber \]Since \( a_n \neq 0 \), the fractions \( \frac{a_k}{a_n} \) are finite constants. For notational efficiency, let's denote \( \frac{a_k}{a_n} \) as \( b_k \). Therefore,\[ P(x) = a_n x^n\left(1 + b_{n - 1} \cdot \dfrac{1}{x} + b_{n - 2} \cdot \dfrac{1}{x^2} + \cdots + b_{2} \cdot \dfrac{1}{x^{n - 2}} + b_1 \cdot \dfrac{1}{x^{n - 1}} + b_0 \dfrac{1}{x^n}\right). \nonumber \](The next step is proved in Calculus, but can be understood here as a strong possibility.) As long as the power on \( x \) is not negative, as \( x \to \pm \infty \), \( \frac{1}{x^p} \to 0 \). Since all the powers on \( x \) in\[ P(x) = a_n x^n\left(1 + b_{n - 1} \cdot \dfrac{1}{x} + b_{n - 2} \cdot \dfrac{1}{x^2} + \cdots + b_{2} \cdot \dfrac{1}{x^{n - 2}} + b_1 \cdot \dfrac{1}{x^{n - 1}} + b_0 \dfrac{1}{x^n}\right) \nonumber \]are not negative, all terms involving \( x \) in the denominator must tend to 0 as \( x \to \pm \infty \). That is, as \( x \to \pm \infty \), \( P(x) \to a_n x^n (1 + 0 + 0 + \cdots + 0 + 0 + 0) = a_n x^n \). Hence, the end behavior of \( P(x) \) is \( y = a_n x^n \).

Table \(\PageIndex{3}\) illustrates some examples.

Table \(\PageIndex{3}\)

Polynomial Function	Leading Term	Graph of Polynomial Function
\(f(x)=5 x^4 +2 x^3 -x-4\)	\(5 x^4\)
\(f(x)=-2 x^6 - x^5 +3 x^4 + x^3\)	\(-2 x^6\)
\(f(x)=3 x^5 -4 x^4 +2 x^2 +1\)	\(3 x^5\)
\(f(x)=-6 x^3 +7 x^2 +3x+1\)	\(-6 x^3\)

Example \(\PageIndex{8}\): Identifying End Behavior and Degree of a Polynomial Function

Describe the end behavior and determine a possible degree of the polynomial function in Figure \(\PageIndex{7}\).

Figure \(\PageIndex{7}\)

Solution

As the input values \(x\) get very large, the output values \(f(x)\) increase without bound. As the input values \(x\) get very negative, the output values \(f(x)\) decrease without bound. We can describe the end behavior symbolically by writing\[\text{as }x \to - \infty,\,f(x) \to - \infty;\,\text{as } x \to \infty,\,f(x) \to \infty.\nonumber\]In words, we could say that as \(x\) values approach infinity, the function values approach infinity, and as \(x\) values approach negative infinity, the function values approach negative infinity.

We can tell this graph has the shape of an odd degree power function that has not been reflected, so the degree of the polynomial creating this graph must be odd and the leading coefficient must be positive.

Checkpoint \(\PageIndex{8}\)

Describe the end behavior, and determine a possible degree of the polynomial function in Figure \(\PageIndex{8}\).

Figure \(\PageIndex{8}\)

Example \(\PageIndex{9}\): Identifying End Behavior and Degree of a Polynomial Function

Given the function \(f(x)=-3 x^2 (x-1)(x+4)\), express the function as a polynomial in general form, and determine the leading term, degree, and end behavior of the function.

Solution

Obtain the general form by expanding the given expression for \(f(x)\).\[\begin{array}{rcl}
f(x) & = & -3 x^2 (x-1)(x+4) \\[6pt]
& = & -3 x^2 (x^2 +3x-4) \\[6pt]
& = & -3 x^4 -9 x^3 +12 x^2 \\[6pt]
\end{array}\nonumber\]The general form is \(f(x)=-3 x^4 -9 x^3 +12 x^2\). The leading term is \(-3 x^4\); therefore, the degree of the polynomial is 4. The degree is even (4) and the leading coefficient is negative (-3), so the end behavior is\[\text{as } x \to - \infty,\,f(x) \to - \infty;\,\text{as } x \to \infty,\,f(x) \to - \infty.\nonumber\]

Checkpoint \(\PageIndex{9}\)

Given the function \(f(x)=0.2(x-2)(x+1)(x-5)\), express the function as a polynomial in general form and determine the leading term, degree, and end behavior of the function.

Identifying Local Behavior of Polynomial Functions

In addition to the end behavior of polynomial functions, we are also interested in what happens in the "middle" of the function. In particular, we are interested in locations where graph behavior changes. At first glance, most people think we are immediately going to find the turning points of the function.

Definition: Turning Point

A turning point of a graph is a point at which the graph changes from increasing to decreasing, or from decreasing to increasing.

Unfortunately, determining the exact locations of the turning points requires Calculus (however, we will soon be able to determine the maximum number of turning points for any given polynomial!).

While any further discussion of turning points must be placed on hold, we can (at the very least) concern ourselves with the intercepts. As with all functions, the \(y\)-intercept is the point at which the graph intersects the vertical axis. The point corresponds to the coordinate pair in which the input value is zero. Because a polynomial is a function, only one output value corresponds to each input value so there can be only one \(y\)-intercept, \((0,a_0)\). The \(x\)-intercepts occur at the input values that correspond to an output value of zero. It is possible to have more than one \(x\)-intercept. See Figure \(\PageIndex{9}\).

Figure \(\PageIndex{9}\)

Example \(\PageIndex{10}\): Determining the Intercepts of a Polynomial Function

Given the polynomial function \(f(x)=(x-2)(x+1)(x-4)\), written in factored form for your convenience, determine the \(y\)- and \(x\)-intercepts.

Solution

The \(y\)-intercept occurs when the input is zero so substitute 0 for \(x\).\[\begin{array}{rcl}
f(0) & = & (0-2)(0+1)(0-4) \\[6pt]
& = & (-2)(1)(-4) \\[6pt]
& = & 8 \\[6pt]
\end{array}\nonumber\]The \(y\)-intercept is \((0,8)\).

The \(x\)-intercepts occur when the output is zero.\[0=(x-2)(x+1)(x-4)\nonumber\]\[\begin{array}{rccccc}
\implies & x-2=0 & \text{or} & x+1=0 & \text{or} & x-4=0 \\[6pt]
\implies & x=2 & \text{or} & x=-1 & \text{or} & x=4 \\[6pt]
\end{array}\nonumber\]The \(x\)-intercepts are \((2,0)\), \((-1,0)\), and \((4,0)\).

We can see these intercepts on the graph of the function shown in Figure \(\PageIndex{10}\).

Figure \(\PageIndex{10}\)

Example \(\PageIndex{11}\): Determining the Intercepts of a Polynomial Function with Factoring

Given the polynomial function \(f(x)= x^4 -4 x^2 -45\), determine the \(y\)- and \(x\)-intercepts.

Solution

The \(y\)-intercept occurs when the input is zero.\[\begin{array}{rcl}
f(0) & = & (0)^4 -4 (0)^2 -45 \\[6pt]
& = & -45 \\[6pt]
\end{array}\nonumber\]The \(y\)-intercept is \((0,-45)\).

The \(x\)-intercepts occur when the output is zero. To determine when the output is zero, we will need to factor the polynomial.\[f(x)= x^4 -4 x^2 -45 =(x^2 -9)(x^2 +5) =(x-3)(x+3)(x^2 +5).\nonumber\]Therefore, we solve\[0=(x-3)(x+3)(x^2 +5).\nonumber\]This mean \(x-3=0\) or \(x+3=0\) (\(x^2 +5=0\) would require \(x^2 = -5\), which has no real solution). Hence, \(x=3\) or \(x=-3\). The \(x\)-intercepts are \((3,0)\) and \((-3,0)\).

We can see these intercepts on the graph of the function shown in Figure \(\PageIndex{11}\). We can see that the function is even because \(f(x)=f(-x)\).

Figure \(\PageIndex{11}\)

Checkpoint \(\PageIndex{11}\)

Given the polynomial function \(f(x)=2 x^3 -6 x^2 -20x\), determine the \(y\)- and \(x\)-intercepts.

Comparing Smooth and Continuous Graphs

We already know that the degree of a polynomial function helps us to determine its end behavior; however, the degree also helps us to determine the number of \(x\)-intercepts and, as such, a rough approximation to the number of turning points. A polynomial function of \(n^{\text{th}}\) degree is the product of \(n\) factors, so it will have at most \(n\) zeros, or \(x\)-intercepts. Therefore, the graph of the polynomial function of degree \(n\) must have at most \(n-1\) turning points. This means the graph has at most one fewer turning point than the degree of the polynomial or one fewer than the number of factors.

Theorem: Number of Turning Points for a Polynomial Function

The graph of a polynomial function of degree \(n \geq 1\) will have, at most, \(n-1\) turning points and \( n \) horizontal intercepts.

Proof

This proof must wait until Calculus.

The proof of this theorem requires more mathematical "technology" than we currently have access to, so we hold off until Calculus.

While not technically true, we will, for the sake of simplicity, define a continuous function to be one that has no breaks in its graph: the graph can be drawn without lifting the pen from the paper. A smooth curve is a graph that has no sharp corners. The turning points of a smooth graph must always occur at rounded curves. The graphs of polynomial functions are both continuous and smooth.

Example \(\PageIndex{12}\): Determining the Number of Intercepts and Turning Points of a Polynomial

Without graphing the function, determine the local behavior of the function by finding the maximum number of \(x\)-intercepts and turning points for \(f(x)=-3 x^{10} +4 x^7 - x^4 +2 x^3\).

Solution

The polynomial has a degree of \(10\), so there are at most \(10\) \(x\)-intercepts and at most \(10-1=9\) turning points.

Checkpoint \(\PageIndex{12}\)

Without graphing the function, determine the maximum number of \(x\)-intercepts and turning points for \(f(x)=108-13 x^9 -8 x^4 +14 x^{12} +2 x^3\).

Example \(\PageIndex{13}\): Drawing Conclusions about a Polynomial Function from the Graph

What can we conclude about the polynomial represented by the graph shown in Figure \(\PageIndex{12}\) based on its intercepts and turning points?

Figure \(\PageIndex{12}\)

Solution

The end behavior of the graph tells us this is the graph of an even-degree polynomial. See Figure \(\PageIndex{13}\).

Figure \(\PageIndex{13}\)

The graph has 2 \(x\)-intercepts, suggesting a degree of 2 or greater, and 3 turning points, suggesting a degree of 4 or greater. Based on this, it would be reasonable to conclude that the degree is even and at least 4.

Checkpoint \(\PageIndex{13}\)

What can we conclude about the polynomial represented by the graph shown in Figure \(\PageIndex{14}\) based on its intercepts and turning points?

Figure \(\PageIndex{14}\)

Example \(\PageIndex{14}\): Drawing Conclusions about a Polynomial Function from the Factors

Given the function \(f(x)=-4x(x+3)(x-4)\), determine the local behavior.

Solution

The \(y\)-intercept is found by evaluating \(f(0)\).\[\begin{array}{rcl}
f(0) & = & -4(0)(0+3)(0-4) \\[6pt]
& = & 0 \\[6pt]
\end{array}\nonumber\]The \(y\)-intercept is \((0,0)\).

The \(x\)-intercepts are found by determining the zeros of the function.\[0=-4x(x+3)(x-4) \implies x=0 \text{ or } x+3=0 \text{ or } x-4=0 \implies x=0 \text{ or } x=-3 \text{ or } x=4.\nonumber\]The \(x\)-intercepts are \((0,0),(-3,0)\), and \((4,0)\).

The degree is 3 so the graph has at most 2 turning points.

Checkpoint \(\PageIndex{14}\)

Given the function \(f(x)=0.2(x-2)(x+1)(x-5)\), determine the local behavior.

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Hidden Resources: CalFresh

CalFresh (formerly known as Food Stamps) is a nutrition program that helps students and their families buy healthy foods.

How Does CalFresh Work?

CalFresh benefits are issued on an electronic benefit transfer (EBT) card that works like a debit card and can be used at most grocery stores and many farmers' markets to buy food. You can even use CalFresh to purchase food online through Amazon and Walmart.com.

If you are on a limited income and are struggling to buy the healthy foods that you need, then CalFresh may be able to help! Plus, CalFresh benefits do not impact your financial aid, and you do not need your parents' tax return information to apply.

For more information, please see this page on the CRC website.

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Success in Mathematics: Why Study Mathematics? (Part 1)

Until modern times, only a tiny percent of people had the opportunity to be "liberally educated." These people were not studying to learn a job or trade - they wanted to learn what other humans had thought and created. They were also trying to develop new knowledge. Menaechmus (350 B.C.E) was this kind of student. As he studied geometry, he discovered the properties of conic sections. Others built on his knowledge. Apollonius (350 B.C.E) wrote an eight-chapter book about conic sections.

Today we use our knowledge of conic sections to predict orbits, build telescopes, locate ships at sea, make "eyes" for robots, and collect solar energy. But we also study conic sections as an accomplishment of our human ancestors. In developing their equations, we also develop logic and creative problem-solving skills. Think about the football coach who has his team doing agility drills and running through tires. Why? When has a football team ever had to run through tires during a game? Similarly, with the right attitude, you can develop your ability to reason and think in your life outside of college by doing mathematics. You are responsible for recognizing that learning and studying may be valuable even when it is not immediately practical.

One problem with teaching mathematics in the K-12 system - and I see it as a major difficulty - is that there is virtually nothing the pupils learn that has a non-trivial application in today's world. The most a teacher can tell a student who inquires, entirely reasonably, "How is this useful?" is that almost all mathematics finds uses, in many cases important ones, and that what they learn in school leads on to mathematics that definitely is used.

- Keith Devlin, Standford University

By "non-trivial," Dr. Devlin means that many real-life applications require mathematics beyond what is studied in high school.

1. Younger students often ask, " When will I use this math?" Their teachers often reply, "You need to learn this math to go on to the next course. In college, you can take math that is very powerful for doing all kinds of practical things." Do you think this is the best reply to give to the student? Explain.
2. I've often pondered the best mixture of application and theory in a mathematics course. My love of mathematics lies in theory; however, my research in mathematics is in applications. As a student, which do you prefer? Explain.