Power Functions and Polynomial Functions

By the end of this section, you will be able to:

Identify power functions
Identify end behavior of power functions
Identify polynomial functions
Identify the degree and leading coefficient of polynomial functions

Suppose a certain species of bird thrives on a small island. Its population over the last few years is shown in the table below.

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

The population can be estimated using the function $P(t)=-0.3t^3+97t+800$ , 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. A power function is a function with a single term that is the product of a real number, a coefficient, and a variable raised to a fixed real number. (A number that multiplies a variable raised to an exponent is known as a coefficient.)

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)=\tfrac{4}{3}\pi r^3

Both of these are examples of power functions because they consist of a coefficient, $\pi$ or $\tfrac{4}{3}\pi$ , multiplied by a variable $r$ raised to a power.

Power function. A power function is a function that can be represented in the form

f(x)=kx^p

where $k$ and $p$ are real numbers, and $k$ is known as the coefficient.

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.

Example. Which of the following functions are power functions?

\begin{array}{ll} f(x)=1 & \text{Constant function} \\ f(x)=x & \text{Identity function} \\ f(x)=x^2 & \text{Quadratic function} \\ f(x)=x^3 & \text{Cubic function} \\ f(x)=\tfrac{1}{x} & \text{Reciprocal function} \\ f(x)=\tfrac{1}{x^2} & \text{Reciprocal squared function} \\ f(x)=\sqrt{x} & \text{Square root function} \\ f(x)=\sqrt[3]{x} & \text{Cube root function} \end{array}

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 $f(x)=x^2$ and $f(x)=x^3$ .

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 $f(x)=x^{1/3}$ .

Which of the following functions is a power function? $f(x)=2x^2\cdot 4x^3$ , $g(x)=-x^5+5x^3-4x$ , or $h(x)=\tfrac{2x^5-1}{3x^2+4}$

Source note. The source module’s Try It justifies this answer with “

f(x)=8x^4

”. Multiplying

2x^2

4x^3

adds the exponents, so the single term is

8x^5

: at

x=2

the product is

2(4)\cdot4(8)=256

, which is

8x^5

and not

8x^4=128

. This page writes

8x^5

. The conclusion the solution reaches — that

f

is the only power function of the three — is unchanged.

Identifying end behavior of power functions

Figure: Three curves on one grid: h of x equals x squared, g of x equals x to the fourth, and f of x equals x to the sixth, all passing through the origin and rising steeply on both sides, arrows at the top left and top right. The x-axis runs from negative 3 to 3 and the y-axis from negative 1 to 4.

The graph above shows $f(x)=x^2$ , $g(x)=x^4$ , and $h(x)=x^6$ , which are all power functions with even, positive integer 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.

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,\ f(x)\to\infty

Figure: Three curves on one grid: f of x equals x cubed, g of x equals x to the fifth, and h of x equals x to the seventh, all passing through the origin, symmetric about it, flattening near the origin and steepening away from it, with arrows at the top right and bottom left. The x-axis runs from negative 3 to 3 and the y-axis from negative 4 to 4.

The graph above shows $f(x)=x^3$ , $g(x)=x^5$ , 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.

These examples illustrate that functions of the form $f(x)=x^n$ reveal symmetry of one kind or another. First, even functions of the form $f(x)=x^n$ , $n$ even, are symmetric about the $y$ -axis. Odd functions of the form $f(x)=x^n$ , $n$ 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

\begin{array}{lrcl} \text{as } x\to-\infty, & f(x) &\to& -\infty \\[4pt] \text{as } x\to\infty, & f(x) &\to& \infty \end{array}

The behavior of the graph of a function as the input values get very small ( $x\to-\infty$ ) 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.

The four graphs below show the end behavior of power functions in the form $f(x)=kx^n$ where $n$ is a non-negative integer, for each combination of even or odd degree and positive or negative leading coefficient.

Even degree, positive coefficient.

Figure: The graph of y equals x squared: an even-degree power function with positive leading coefficient, opening upward with arrows at both ends and passing through the origin. As x approaches negative infinity, f(x) approaches positive infinity, and as x approaches positive infinity, f(x) approaches positive infinity. The x-axis runs from negative 3 to 3 and the y-axis from negative 4 to 4.

Odd degree, positive coefficient.

Figure: The graph of y equals x cubed: an odd-degree power function with positive leading coefficient, rising from lower left to upper right with arrows at both ends and passing through the origin. As x approaches negative infinity, f(x) approaches negative infinity, and as x approaches positive infinity, f(x) approaches positive infinity. The x-axis runs from negative 3 to 3 and the y-axis from negative 4 to 4.

Even degree, negative coefficient.

Figure: The graph of y equals negative x squared: an even-degree power function with negative leading coefficient, opening downward with arrows at both ends and passing through the origin. As x approaches negative infinity, f(x) approaches negative infinity, and as x approaches positive infinity, f(x) approaches negative infinity. The x-axis runs from negative 3 to 3 and the y-axis from negative 4 to 4.

Odd degree, negative coefficient.

Figure: The graph of y equals negative x cubed: an odd-degree power function with negative leading coefficient, falling from upper left to lower right with arrows at both ends and passing through the origin. As x approaches negative infinity, f(x) approaches positive infinity, and as x approaches positive infinity, f(x) approaches negative infinity. The x-axis runs from negative 3 to 3 and the y-axis from negative 4 to 4.

How to: given a power function $f(x)=kx^n$ where $n$ is a positive integer, identify the end behavior.

Determine whether the power is even or odd.
Determine whether the constant is positive or negative.
Use the comparison graphs above to identify the end behavior.

Example. 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. We write 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 below.

Figure: The graph of f of x equals x to the eighth power: a steep U shape symmetric about the y-axis, touching the origin and rising sharply on both sides past y equals 6 near x equals negative 1 and x equals 1. The x-axis runs from negative 3 to 3 and the y-axis from negative 1 to 6.

Example. 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 $f(x)=x^9$ . The figure below 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

\begin{array}{lrcl} \text{as } x\to-\infty, & f(x) &\to& \infty \\[4pt] \text{as } x\to\infty, & f(x) &\to& -\infty \end{array}

Figure: The graph of f of x equals negative x to the ninth power: a steeply falling curve from upper left to lower right, flat near the origin, dropping past y equals negative 10 just after x equals 1 and rising past y equals 10 just before x equals negative 1. The x-axis runs from negative 5 to 5 and the y-axis from negative 10 to 10.

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

$x$	$f(x)$
$-10$	$1{,}000{,}000{,}000$
$-5$	$1{,}953{,}125$
$0$	$0$
$5$	$-1{,}953{,}125$
$10$	$-1{,}000{,}000{,}000$

We can see from the table 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).

Describe the end behavior of $f(x)=-5x^4$ .

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

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.

\begin{array}{lrcl} & A(w) &=& A(r(w)) \\[4pt] & &=& A(24+8w) \\[4pt] & &=& \pi(24+8w)^2 \end{array}

Multiplying gives the formula.

A(w)=576\pi+384\pi w+64\pi w^2

This formula is an example of a polynomial function. A polynomial function consists of either zero or the sum of a finite number of non-zero terms, each of which is a product of a number, called the coefficient of the term, and a variable raised to a non-negative integer power.

Polynomial functions. Let $n$ be a non-negative integer. A polynomial function is a function that can be written in the form

f(x)=a_nx^n+a_{n-1}x^{n-1}+\ldots+a_2x^2+a_1x+a_0

This is called the general form of a polynomial function. Each $a_i$ is a coefficient and in this section can only be a real number, but $a_n\ne 0$ . Each product $a_ix^i$ is a term of a polynomial function.

Example. Which of the following are polynomial functions?

\begin{array}{l} f(x)=2x^3\cdot 3x+4 \\ g(x)=-x(x^2-4) \\ h(x)=5\sqrt{x}+2 \end{array}

Solution. The first two functions are examples of polynomial functions because they can be written in the form $f(x)=a_nx^n+a_{n-1}x^{n-1}+\ldots+a_2x^2+a_1x+a_0$ where the powers are non-negative integers and the coefficients are real numbers.

Source note. The source module prints this closing term with a duplicated subscript,

\ldots+a_1x+a_1

, where the general form should end in the constant term

a_0

. The same module writes

a_0

correctly three other times — the Polynomial Functions definition box, the Terminology of Polynomial Functions diagram, and the glossary. This page writes

a_0

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

Identifying the degree and leading coefficient of a polynomial function

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, or in general form. The degree of the polynomial is the highest power of the variable that occurs in the polynomial; it is the power of the first variable if the function is in general form. 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.

Terminology of polynomial functions. We often rearrange polynomials so that the powers are descending,

f(x)=\underbrace{a_n}_{\text{leading coefficient}}\underbrace{x^n}_{\text{degree}}+\ldots+a_2x^2+a_1x+a_0

, with

\underbrace{a_nx^n}_{\text{leading term}}

the term containing the highest power. When a polynomial is written this way, we say that it is in general form.

How to: given a polynomial function, identify the degree and leading coefficient.

Find the highest power of $x$ to determine the degree of the function.
Identify the term containing the highest power of $x$ to find the leading term.
Identify the coefficient of the leading term.

Example. Identify the degree, leading term, and leading coefficient of the following polynomial functions.

\begin{array}{l} f(x)=3+2x^2-4x^3 \\ g(t)=5t^5-2t^3+7t \\ h(p)=6p-p^3-2 \end{array}

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, $-4x^3$ . 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, $5t^5$ . 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, $-1$ .

Identify the degree of the polynomial $f(x)=4x^2-x^6+2x-6$ .

Identify the leading term of that same polynomial, $f(x)=4x^2-x^6+2x-6$ .

Identify the leading coefficient of that same polynomial, $f(x)=4x^2-x^6+2x-6$ .

Identifying end behavior of polynomial functions

Knowing the degree of a polynomial function is useful in helping us predict its end behavior. 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. For any polynomial, the end behavior of the polynomial will match the end behavior of the term of highest degree. See the table below.

Polynomial function	Leading term
$f(x)=5x^4+2x^3-x-4$	$5x^4$
$f(x)=-2x^6-x^5+3x^4+x^3$	$-2x^6$
$f(x)=3x^5-4x^4+2x^2+1$	$3x^5$
$f(x)=-6x^3+7x^2+3x+1$	$-6x^3$

Figure: The graph of f of x equals 5x to the fourth plus 2x cubed minus x minus 4: a steep W-shaped curve with two dips near x equals negative 1 and x equals 1, rising past y equals 6 on both sides. The x-axis runs from negative 5 to 5 and the y-axis from negative 6 to 6.

Figure: The graph of f of x equals negative 2x to the sixth minus x to the fifth plus 3x to the fourth plus x cubed: a curve with two dips near x equals negative 1 and x equals 1 that falls toward negative infinity on both sides, exiting the bottom of the frame past y equals negative 6. The x-axis runs from negative 5 to 5 and the y-axis from negative 6 to 6.

Figure: The graph of f of x equals 3x to the fifth minus 4x to the fourth plus 2x squared plus 1: a curve falling steeply from the bottom left, leveling near the origin, and rising steeply past y equals 6 just after x equals 1. The x-axis runs from negative 5 to 5 and the y-axis from negative 6 to 6.

Figure: The graph of f of x equals negative 6x cubed plus 7x squared plus 3x plus 1: a curve rising from the bottom left to a hump above the x-axis near x equals 1, then falling steeply past y equals negative 6 before x equals 2. The x-axis runs from negative 5 to 5 and the y-axis from negative 6 to 6.

Example. Describe the end behavior and determine a possible degree of the polynomial function graphed below.

Figure: A curve falling from the top left, exiting the bottom of the frame near x equals negative 3, rising to a local maximum, dipping to a local minimum, then rising again and exiting the top of the frame near x equals 5. The x-axis runs from negative 6 to 6 and the y-axis from negative 5 to 6.

Solution. As the input values $x$ get very large, the output values $f(x)$ increase without bound. As the input values $x$ get very small, the output values $f(x)$ decrease without bound. We can describe the end behavior symbolically by writing

\begin{array}{lrcl} \text{as } x\to-\infty, & f(x) &\to& -\infty \\[4pt] \text{as } x\to\infty, & f(x) &\to& \infty \end{array}

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.

Figure: A dome-shaped curve with a single hump peaking near x equals 0.5, falling away steeply on both sides and exiting the bottom of the frame near x equals negative 2 and x equals 3. The x-axis runs from negative 6 to 6 and the y-axis from negative 6 to 6.

What can we conclude about the end behavior and shape of the polynomial function graphed above?

Example. Given the function $f(x)=-3x^2(x-1)(x+4)$ , 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}{lrcl} & f(x) &=& -3x^2(x-1)(x+4) \\[4pt] & &=& -3x^2(x^2+3x-4) \\[4pt] & &=& -3x^4-9x^3+12x^2 \end{array}

The general form is $f(x)=-3x^4-9x^3+12x^2$ . The leading term is $-3x^4$ ; 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

\begin{array}{lrcl} \text{as } x\to-\infty, & f(x) &\to& -\infty \\[4pt] \text{as } x\to\infty, & f(x) &\to& -\infty \end{array}

Given the function $f(x)=0.2(x-2)(x+1)(x-5)$ , express the function as a polynomial in general form.

What is the end behavior of that same function, $f(x)=0.2(x-2)(x+1)(x-5)$ ?

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. A turning point is a point at which the function values change from increasing to decreasing or decreasing to increasing.

We are also interested in 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 the figure below.

Figure: A schematic cubic-shaped curve illustrating vocabulary: a small hump above the x-axis is marked as a turning point, a deeper dip below the x-axis is marked as a second turning point, the three points where the curve crosses the x-axis are marked as x-intercepts, and the point where the curve crosses the y-axis, below the axis, is marked as the y-intercept. No axis numbers are shown.

Intercepts and turning points of polynomial functions. A turning point of a graph is a point at which the graph changes direction from increasing to decreasing or decreasing to increasing. The

y

-intercept is the point at which the function has an input value of zero. The

x

-intercepts are the points at which the output value is zero.

How to: given a polynomial function, determine the intercepts.

Determine the $y$ -intercept by setting $x=0$ and finding the corresponding output value.
Determine the $x$ -intercepts by solving for the input values that yield an output value of zero.

Example. 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}{lrcl} & f(0) &=& (0-2)(0+1)(0-4) \\[4pt] & &=& (-2)(1)(-4) \\[4pt] & &=& 8 \end{array}

The $y$ -intercept is $(0,8)$ .

The $x$ -intercepts occur when the output is zero.

\begin{array}{lrcl} & 0 &=& (x-2)(x+1)(x-4) \\[4pt] x-2=0 \quad\text{or}\quad x+1=0 \quad\text{or}\quad x-4=0 & & &\\[4pt] x=2 \quad\text{or}\quad x=-1 \quad\text{or}\quad x=4 & & & \end{array}

The $x$ -intercepts are $(2,0)$ , $(-1,0)$ , and $(4,0)$ .

We can see these intercepts on the graph of the function shown below.

Figure: The graph of f of x equals the quantity x minus 2, times the quantity x plus 1, times the quantity x minus 4. The curve crosses the x-axis at negative 1, 2, and 4, crosses the y-axis at 8, and has a local maximum above the axis between negative 1 and 2 and a local minimum below the axis between 2 and 4. The x-axis runs from negative 5 to 5 and the y-axis from negative 4 to 9.

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

Solution. The $y$ -intercept occurs when the input is zero.

\begin{array}{lrcl} & f(0) &=& (0)^4-4(0)^2-45 \\[4pt] & &=& -45 \end{array}

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.

\begin{array}{lrcl} & f(x) &=& x^4-4x^2-45 \\[4pt] & &=& (x^2-9)(x^2+5) \\[4pt] & &=& (x-3)(x+3)(x^2+5) \end{array}

0=(x-3)(x+3)(x^2+5)

x-3=0 \quad\text{or}\quad x+3=0 \quad\text{or}\quad x^2+5=0

x=3 \quad\text{or}\quad x=-3 \quad\text{or}\quad (\text{no real solution})

The $x$ -intercepts are $(3,0)$ and $(-3,0)$ .

We can see these intercepts on the graph of the function shown below. We can see that the function is even because $f(x)=f(-x)$ .

Figure: The graph of f of x equals x to the fourth minus 4x squared minus 45. The curve crosses the x-axis at negative 3 and 3, crosses the y-axis at negative 45, and is symmetric about the y-axis with a broad dip in the middle and steep rises past both x equals negative 3 and x equals 3. The x-axis runs from negative 5 to 5 and the y-axis from negative 120 to 120.

Given the polynomial function $f(x)=2x^3-6x^2-20x$ , what is the y-intercept?

Now find the x-intercepts of that same function, $f(x)=2x^3-6x^2-20x$ . Enter the three x-coordinates, from least to greatest, separated by commas.

Comparing smooth and continuous graphs

The degree of a polynomial function helps us to determine the number of $x$ -intercepts and the number of turning points. A polynomial function of $n$ th degree is the product of $n$ factors, so it will have at most $n$ roots or zeros, or $x$ -intercepts. 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.

A continuous function 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.

Intercepts and turning points of polynomials. A polynomial of degree

n

will have, at most,

n

x

-intercepts and

n-1

turning points.

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

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

Without graphing, find the maximum number of x-intercepts for the polynomial function $f(x)=108-13x^9-8x^4+14x^{12}+2x^3$ .

Now find the maximum number of turning points for that same function, $f(x)=108-13x^9-8x^4+14x^{12}+2x^3$ .

Example. What can we conclude about the polynomial represented by the graph shown below based on its intercepts and turning points?

Figure: An even-degree polynomial graph with a small local maximum near the origin sitting below the x-axis, flanked by two local minima, and two arms rising steeply out of the top of the frame near x equals negative 2 and x equals 2. The curve crosses the x-axis twice. The x-axis runs from negative 5 to 5 and the y-axis from negative 5 to 4.

Solution. The end behavior of the graph tells us this is the graph of an even-degree polynomial. 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.

Figure: An odd-degree polynomial graph entering from the top left, falling to a local minimum, rising to a local maximum, then falling steeply out of the bottom of the frame near x equals 1. The curve crosses the x-axis three times. The x-axis runs from negative 5 to 5 and the y-axis from negative 10 to 10.

What can we conclude about the polynomial represented by the graph above, based on its intercepts and turning points?

Example. 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}{lrcl} & f(0) &=& -4(0)(0+3)(0-4) \\[4pt] & &=& 0 \end{array}

The $y$ -intercept is $(0,0)$ .

The $x$ -intercepts are found by determining the zeros of the function.

0=-4x(x+3)(x-4)

x=0 \quad\text{or}\quad x+3=0 \quad\text{or}\quad x-4=0

x=0 \quad\text{or}\quad x=-3 \quad\text{or}\quad x=4

The $x$ -intercepts are $(0,0)$ , $(-3,0)$ , and $(4,0)$ .

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

Given the function $f(x)=0.2(x-2)(x+1)(x-5)$ , what is the y-intercept?

What is the maximum number of turning points for that same function, $f(x)=0.2(x-2)(x+1)(x-5)$ ?

Key equations

General form of a polynomial function	$f(x)=a_nx^n+a_{n-1}x^{n-1}+\ldots+a_2x^2+a_1x+a_0$

Source note. The source module prints this closing term with a duplicated subscript,

\ldots+a_1x+a_1

, where the general form should end in the constant term

a_0

. The same module writes

a_0

correctly three other times — the Polynomial Functions definition box, the Terminology of Polynomial Functions diagram, and the glossary. This page writes

a_0

Key concepts

A power function is a variable base raised to a number power.
The behavior of a graph as the input decreases beyond bound and increases beyond bound is called the end behavior.
The end behavior depends on whether the power is even or odd.
A polynomial function is the sum of terms, each of which consists of a transformed power function with positive whole number power.
The degree of a polynomial function is the highest power of the variable that occurs in a polynomial. The term containing the highest power of the variable is called the leading term. The coefficient of the leading term is called the leading coefficient.
The end behavior of a polynomial function is the same as the end behavior of the power function represented by the leading term of the function.
A polynomial of degree $n$ will have at most $n$ $x$ -intercepts and at most $n-1$ turning points.

Key terms

coefficient — a nonzero real number multiplied by a variable raised to an exponent. continuous function — a function whose graph can be drawn without lifting the pen from the paper because there are no breaks in the graph. degree — the highest power of the variable that occurs in a polynomial. end behavior — the behavior of the graph of a function as the input decreases without bound and increases without bound. leading coefficient — the coefficient of the leading term. leading term — the term containing the highest power of the variable. polynomial function — a function that consists of either zero or the sum of a finite number of non-zero terms, each of which is a product of a number, called the coefficient of the term, and a variable raised to a non-negative integer power. power function — a function that can be represented in the form $f(x)=kx^p$ where $k$ is a constant, the base is a variable, and the exponent, $p$ , is a constant. smooth curve — a graph with no sharp corners. term of a polynomial function — any $a_ix^i$ of a polynomial function in the form $f(x)=a_nx^n+a_{n-1}x^{n-1}+\ldots+a_2x^2+a_1x+a_0$ . turning point — the location at which the graph of a function changes direction.

Practice

Identify power functions

Which statement correctly distinguishes the coefficient from the degree in the power function $f(x)=kx^p$ ?

Identify $f(x)=(x^2)^3$ as a power function, a polynomial function, both, or neither.

Identify end behavior of power functions

Determine the end behavior of $f(x)=x^4$ .

Determine the end behavior of $f(x)=-x^4$ .

In general, what is the end behavior of a polynomial with odd degree if the leading coefficient is positive?

Identify polynomial functions

Identify $f(x)=\tfrac{x^2}{x^2-1}$ as a power function, a polynomial function, both, or neither.

Identify $f(x)=3^{x+1}$ as a power function, a polynomial function, both, or neither.

Identify the degree and leading coefficient of polynomial functions

Find the degree of the polynomial $7-2x^2$ .

Find the leading coefficient of that same polynomial, $7-2x^2$ .

Find the degree of the polynomial $x(4-x^2)(2x+1)$ .

Find the leading coefficient of that same polynomial, $x(4-x^2)(2x+1)$ .

What can we conclude if, in general, the graph of a polynomial function exhibits this end behavior: as $x$ approaches negative infinity, $f(x)$ approaches negative infinity, and as $x$ approaches positive infinity, $f(x)$ approaches negative infinity?

This section is adapted from Precalculus 2e, Section 3.3: Power Functions and Polynomial Functions by Jay Abramson and OpenStax, © OpenStax, licensed under CC BY-NC-SA 4.0. Access the original for free at openstax.org. Changes: recreated every graph as an accessible inline SVG generated from its exact or fitted equation, matching the source’s window, arrows, and marked points — the even-power family $x^2,x^4,x^6$ and the odd-power family $x^3,x^5,x^7$ on shared grids; the source’s four-panel end-behavior schematic (even/odd power against positive/negative coefficient), restored as four separate figures in the source’s own reading order, each drawn from the simplest representative of its case ( $x^2$ , $x^3$ , $-x^2$ , $-x^3$ ) since the source’s cells carry no formula of their own; $x^8$ ; $-x^9$ ; the four Key Equations table graphs from their exact polynomials ( $5x^4+2x^3-x-4$ , $-2x^6-x^5+3x^4+x^3$ , $3x^5-4x^4+2x^2+1$ , $-6x^3+7x^2+3x+1$ ); $(x-2)(x+1)(x-4)$ and $x^4-4x^2-45$ with their intercepts marked; and fitted an explicit cubic or quartic formula, recorded in the source ledger, for the four schematic figures that had no source formula (the turning-point/intercept vocabulary diagram, the “describe the end behavior” generic odd-degree curve, the generic even-degree dome, and the two “conclude the degree” schematic curves) — never traced as a freeform spline; condensed the paired plain and annotated versions of the turning-point example’s graph into one labeled figure; presented the source’s tabular data (the bird population, and the polynomial/leading-term table) as Markdown tables; replaced the “Terminology of Polynomial Functions” annotated artwork (arrows from labels into the general-form equation) with the equivalent underbrace annotation directly in the KaTeX; corrected a duplicated subscript in the source’s general-form formula, $\ldots+a_1x+a_1$ , to $\ldots+a_1x+a_0$ in both the Example about identifying polynomial functions and the Key Equations table, each flagged with a visible Source note callout — the correct $a_0$ form already appears elsewhere in the same module (the Polynomial Functions definition box, the Terminology diagram, and the glossary), and the duplicated-subscript version is not a graded answer, just uncorrected exposition; omitted the decorative photograph of birds on a cliff, which carries no mathematics; omitted the Media callout’s external links; converted the practice problems (“Try Its”) into interactive exercises with instant feedback, using multiple choice for every end-behavior and shape judgement (never gradeable as free-response math) and splitting each multipart item into one component per part; and adapted 12 selected end-of-section exercises — a coefficient-versus-degree conceptual question, a power/polynomial/neither classification, two more power/polynomial/neither classifications, two pure-power end-behavior judgements, a general odd-degree end-behavior judgement, two degree-and-leading-coefficient pairs (four fill-ins), and an end-behavior-to-degree-and-coefficient conclusion — into 12 interactive components in a closing Practice block, one group per objective. One correction: where the source’s Try It justifies its answer with “ $f(x)=8x^4$ ”, this page writes $8x^5$ , since $2x^2\cdot4x^3=8x^{2+3}$ ; the conclusion the solution reaches — that $f$ is the only power function of the three — is unchanged.

Quadratic Functions Graphs of Polynomial Functions