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Power Functions and Polynomial Functions

Identify power functions.

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

\[V(r)=\frac{4}{3}\pi {r}^{3}\]

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.

Example

Try it.

Which of the following functions are power functions?

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

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

Identifying End Behavior of Power Functions

shows the graphs of \(f(x)={x}^{2},\ g(x)={x}^{4}\) and \(\text{and}\ h(x)={x}^{6},\) 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

\[\text{as }x\to \pm \infty ,\ f(x)\to \infty\]

shows the graphs of \(f(x)={x}^{3},\ g(x)={x}^{5},\ \text{and}\ h(x)={x}^{7},\) 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, in we see that even functions of the form \(f(x)={x}^{n}\text{, }n\text{ even,}\) are symmetric about the \(y\text{-}\) axis. In we see that odd functions of the form \(f(x)={x}^{n}\text{, }n\text{ odd,}\) 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

\[\begin{array}{l}\text{as}\ x\to -\infty ,\ f(x)\to -\infty \ \\ \text{as}\ x\to \infty ,\ f(x)\to \infty \end{array}\]

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.

Condensed — the full section is in OpenStax Precalculus 2e.

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.

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

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.

Example

Try it.

Which of the following are polynomial functions?

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

Solution

The first two functions are examples of polynomial functions because they can be written in the form \(f(x)={a}_{n}{x}^{n}+{a}_{n-1}{x}^{n-1}+...+{a}_{2}{x}^{2}+{a}_{1}x+{a}_{1}\) 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.

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.

Example

Try it.

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

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

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, \(-4{x}^{3}.\) 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, \(-1.\)

Condensed — the full section is in OpenStax Precalculus 2e.

Key Concepts

A power function is a variable base raised to a number power. See .
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. See and .
A polynomial function is the sum of terms, each of which consists of a transformed power function with positive whole number power. See .
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. See .
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. See and .
A polynomial of degree \(n\) will have at most \(n\) x-intercepts and at most \(n-1\) turning points. See , , , , and .

Practice (40)

Try each one on paper first. Reveal the answer to check; verified ones can be opened in the solver for every step.

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

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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 functions are power functions?
\(\begin{array}{l}f(x)=2{x}^{2}⋅4{x}^{3} \\ g(x)=-{x}^{5}+5{x}^{3}-4x \\ h(x)=\frac{2{x}^{5}-1}{3{x}^{2}+4}\end{array}\)

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\(f(x)\) is a power function because it can be written as \(f(x)=8{x}^{4}.\) The other functions are not power functions.
Describe the end behavior of the graph of \(f(x)={x}^{8}.\)

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

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The exponent of the power function is 9 (an odd number). Because the coefficient is \(-1\) (negative), the graph is the reflection about the \(x\text{-}\) axis of the graph of \(f(x)={x}^{9}.\) 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
\[\begin{array}{l}\text{as}\ x\to -\infty ,\ f(x)\to \infty \ \\ \text{as}\ x\to \infty ,\ f(x)\to -\infty \end{array}\]
Describe in words and symbols the end behavior of \(f(x)=-5{x}^{4}.\)

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As \(x\) approaches positive or negative infinity, \(f(x)\) decreases without bound: as \(x\to \pm \infty\), \(f(x)\to -\infty\) because of the negative coefficient.
Which of the following are polynomial functions?
\[\begin{array}{l}\begin{array}{l} \\ f(x)=2{x}^{3}⋅3x+4\end{array} \\ g(x)=-x({x}^{2}-4) \\ h(x)=5\sqrt{x}+2\end{array}\]
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The first two functions are examples of polynomial functions because they can be written in the form \(f(x)={a}_{n}{x}^{n}+{a}_{n-1}{x}^{n-1}+...+{a}_{2}{x}^{2}+{a}_{1}x+{a}_{1}\) 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.
Identify the degree, leading term, and leading coefficient of the following polynomial functions.
\[\begin{array}{l}\begin{array}{l} \\ f(x)=3+2{x}^{2}-4{x}^{3}\end{array} \\ g(t)=5{t}^{5}-2{t}^{3}+7t \\ h(p)=6p-{p}^{3}-2\end{array}\]

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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, \(-1.\)
Identify the degree, leading term, and leading coefficient of the polynomial \(f(x)=4{x}^{2}-{x}^{6}+2x-6.\)

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The degree is 6. The leading term is \(-{x}^{6}.\) The leading coefficient is \(-1.\)
Describe the end behavior and determine a possible degree of the polynomial function in .

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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
\[\begin{array}{l}\text{as}\ x\to -\infty ,\ f(x)\to -\infty \ \\ \text{as}\ x\to \infty ,\ f(x)\to \infty \end{array}\]
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.
Describe the end behavior, and determine a possible degree of the polynomial function in .

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As \(x\to \infty ,\ f(x)\to -\infty ;\ as\ x\to -\infty ,\ f(x)\to -\infty .\) It has the shape of an even degree power function with a negative coefficient.
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.

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Obtain the general form by expanding the given expression for \(f(x).\)
\[\begin{array}{l}\begin{array}{l} \\ f(x)=-3{x}^{2}(x-1)(x+4)\end{array} \\ =-3{x}^{2}({x}^{2}+3x-4) \\ =-3{x}^{4}-9{x}^{3}+12{x}^{2}\end{array}\]
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
\[\begin{array}{l}\text{as}\ x\to -\infty ,\ f(x)\to -\infty \ \\ \text{as}\ x\to \infty ,\ f(x)\to -\infty \end{array}\]
Given the function \(f(x)=0.2(x-2)(x+1)(x-5),\) express the function as a polynomial in general form and determine the leading term, degree, and end behavior of the function.

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The leading term is \(0.2{x}^{3},\) so it is a degree 3 polynomial. As \(x\) approaches positive infinity, \(f(x)\) increases without bound; as \(x\) approaches negative infinity, \(f(x)\) decreases without bound.
Given the polynomial function \(f(x)=(x-2)(x+1)(x-4),\) written in factored form for your convenience, determine the \(y\text{-}\) and \(x\text{-}\) intercepts.

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The y-intercept occurs when the input is zero so substitute 0 for \(x.\)
\[\begin{array}{l}f(0)=(0-2)(0+1)(0-4) \\ =(-2)(1)(-4) \\ =8\end{array}\]
The y-intercept is (0, 8).
The x-intercepts occur when the output is zero.
\[\begin{array}{lllllllllllllllll}\ 0=(x-2)(x+1)(x-4) \\ \begin{array}{lllllllll}x-2=0 & & \text{or} & & x+1=0 & & \text{or} & & x-4=0 \\ x=2 & & \text{or} & & \ x=-1 & & \text{or} & & \ x=4\end{array}\end{array}\]
The \(x\text{-}\) intercepts are \((2,0),(-1,0),\) and \((4,0).\)
We can see these intercepts on the graph of the function shown in .
Given the polynomial function \(f(x)={x}^{4}-4{x}^{2}-45,\) determine the \(y\text{-}\) and \(x\text{-}\) intercepts.

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The y-intercept occurs when the input is zero.
\[\begin{array}{l}\begin{array}{l} \\ f(0)={(0)}^{4}-4{(0)}^{2}-45\end{array} \\ =-45\end{array}\]
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.
\[\begin{array}{l}f(x)={x}^{4}-4{x}^{2}-45 \\ =({x}^{2}-9)({x}^{2}+5) \\ =(x-3)(x+3)({x}^{2}+5)\end{array}\]\[0=(x-3)(x+3)({x}^{2}+5)\]\[\begin{array}{lllll}x-3=0 & \text{or} & x+3=0 & \text{or} & {x}^{2}+5=0 \\ x=3 & \text{or} & \ x=-3 & \text{or} & \text{(no real solution)}\end{array}\]
The x-intercepts are \((3,0)\) and \((-3,0).\)
We can see these intercepts on the graph of the function shown in . We can see that the function is even because \(f(x)=f(-x).\)
Given the polynomial function \(f(x)=2{x}^{3}-6{x}^{2}-20x,\) determine the \(y\text{-}\) and \(x\text{-}\) intercepts.

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y-intercept \((0,0);\) x-intercepts \((0,0),(-2,0),\) and \((5,0)\)
Without graphing the function, determine the local behavior of the function by finding the maximum number of \(x\text{-}\) intercepts and turning points for \(f(x)=-3{x}^{10}+4{x}^{7}-{x}^{4}+2{x}^{3}.\)

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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 the function, determine the maximum number of \(x\text{-}\) intercepts and turning points for \(f(x)=108-13{x}^{9}-8{x}^{4}+14{x}^{12}+2{x}^{3}\)

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There are at most 12 \(x\text{-}\) intercepts and at most 11 turning points.
What can we conclude about the polynomial represented by the graph shown in based on its intercepts and turning points?

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The end behavior of the graph tells us this is the graph of an even-degree polynomial. See .
The graph has 2 \(x\text{-}\) 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.
What can we conclude about the polynomial represented by the graph shown in based on its intercepts and turning points?

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The end behavior indicates an odd-degree polynomial function; there are 3 \(x\text{-}\) intercepts and 2 turning points, so the degree is odd and at least 3. Because of the end behavior, we know that the lead coefficient must be negative.
Given the function \(f(x)=-4x(x+3)(x-4),\) determine the local behavior.

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The \(y\text{-}\) intercept is found by evaluating \(f(0).\)
\[\begin{array}{l}f(0)=-4(0)(0+3)(0-4) \\ =0\end{array}\]
The \(y\text{-}\) intercept is \((0,0).\)
The \(x\text{-}\) intercepts are found by determining the zeros of the function.
\[\begin{array}{lllllllllllllllll}0=-4x(x+3)(x-4) \\ \begin{array}{lllllllll}x=0 & & \text{or} & & x+3=0 & & \text{or} & & x-4=0 \\ x=0 & & \text{or} & & \ x=-3 & & \text{or} & & \ x=4\end{array}\end{array}\]
The \(x\text{-}\) 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 \(f(x)=0.2(x-2)(x+1)(x-5),\) determine the local behavior.

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The \(x\text{-}\) intercepts are \((2,0),(-1,0),\) and \((5,0),\) the y-intercept is \((0,\text{2}),\) and the graph has at most 2 turning points.
Explain the difference between the coefficient of a power function and its degree.

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The coefficient of the power function is the real number that is multiplied by the variable raised to a power. The degree is the highest power appearing in the function.
If a polynomial function is in factored form, what would be a good first step in order to determine the degree of the function?
In general, explain the end behavior of a polynomial with odd degree if the leading coefficient is positive.

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As \(x\) decreases without bound, so does \(f(x).\) As \(x\) increases without bound, so does \(f(x).\)
What is the relationship between the degree of a polynomial function and the maximum number of turning points in its graph?
What can we conclude if, in general, the graph of a polynomial function exhibits the following end behavior? As \(x\to -\infty ,\ f(x)\to -\infty\) and as \(x\to \infty ,\ f(x)\to -\infty .\)

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The polynomial function is of even degree and leading coefficient is negative.
\(f(x)={x}^{5}\)
\(f(x)={({x}^{2})}^{3}\)

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\(f(x)\) is a power function because it contains a variable base raised to a fixed power. It is also a polynomial, with all coefficients except one equal to zero.
\(f(x)=x-{x}^{4}\)
\(f(x)=\frac{{x}^{2}}{{x}^{2}-1}\)

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Neither
\(f(x)=2x(x+2){(x-1)}^{2}\)
\(f(x)={3}^{x+1}\)

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Neither
\(7-2{x}^{2}\)

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Degree = 2, Coefficient = –2
\(-2{x}^{2}-\ 3{x}^{5}+\ x-6\)
\(x(4-{x}^{2})(2x+1)\)

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Degree =4, Coefficient = –2
\({x}^{2}{(2x-3)}^{2}\)
\(f(x)={x}^{4}\)

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As \(x\to \infty\), \(f(x)\to \infty\), as \(x\to -\infty\), \(f(x)\to \infty\)
\(f(x)={x}^{3}\)
\(f(x)=-{x}^{4}\)

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As \(x\to -\infty\), \(f(x)\to -\infty\), as \(x\to \infty\), \(f(x)\to -\infty\)
\(f(x)=-{x}^{9}\)

Symbols used here

\sqrt{x},\ \sqrt[n]{x}

square root, n-th root
The non-negative number whose square (n-th power) is x.

\infty

infinity
Not a number: "grows without bound" in limits and intervals.

\pi

pi
Ratio of a circle's circumference to its diameter, 3.14159…

imaginary unit
i² = −1.

\pm

plus or minus
Both signs at once: x = 3 ± 2 means 5 and 1.

\neq

not equal
The two sides are different.

\sup,\ \inf

supremum, infimum
Least upper bound, greatest lower bound.

|x|

absolute value / modulus
Distance from zero: |−3| = 3. For a complex number, distance from the origin.

\binom{n}{k}

binomial coefficient, "n choose k"
Number of k-element subsets of n things: n!/(k!(n−k)!).

Euler's number
2.71828…, the base whose exponential is its own derivative.

\theta

theta
The usual name for an angle.

\log_b x,\ \ln x

logarithm, natural log
The exponent b must be raised to for x; ln uses base e.

\mathbf{v},\ \vec{v}

vector
A quantity with magnitude and direction; a column of numbers.

How to: Power Functions and Polynomial Functions

Identify power functions.
Identify end behavior of power functions.
Identify polynomial functions.
Identify the degree and leading coefficient of polynomial functions.
Determine whether the power is even or odd.
Determine whether the constant is positive or negative.
Use

Questions people ask

What is a function, really?

A rule that assigns exactly one output to each input. The vertical-line test on a graph is the same idea: no input may have two outputs.

Why do we need complex numbers?

Because x² + 1 = 0 has no real solution, and allowing one new number i with i² = −1 makes every polynomial equation solvable. They then turn out to describe rotation, waves and alternating current more naturally than real numbers do.

Tente o seu próprio

Parts of this page are adapted from OpenStax Precalculus 2e (CC BY-NC-SA 4.0). Condensed and re-explained here; errors are ours.

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