The standard form of a nonconstant polynomial function is given by f\left(x\right)=a_nx^n+a_{n-1}x^{n-1}+a_{n-2}x^{n-2}+\ldots+a_2x^2+a_1x+a_0 where n is a positive integer, a_n,a_{n-1},a_{n-2},\ldots,a_2,a_1,a_0 are real numbers, and a_n \neq 0.

n is the degree of the polynomial. The leading term is a_n x^n and the leading coefficient is a_n.

Constant functions are also considered polynomial functions. Constant functions have a degree of 0.

degree

the value of the highest exponent on a variable in the polynomial expression

leading term

the term in a polynomial expression with the highest exponent of the variable

leading coefficient

the coefficient on the leading term in a polynomial expression

The degree and leading coefficient together determine the end behavior of the polynomial function (how the function behaves as the input values approach positive or negative infinity).

The degree (specifically whether it is even or odd) dictates whether the function's ends have the same or opposite directions, while the leading coefficient determines whether the function increases or decreases as the input values approach positive or negative infinity.

Degree	Leading Coefficient	End Behavior	Graph of the function
\text{even}	\text{positive}	f(x) \to + \infty, \text{as } x \to -\infty \\ f(x) \to + \infty, \text{as } x \to +\infty	\text{rises to the left and} \\ \text{to the right}
\text{even}	\text{negative}	f(x) \to - \infty, \text{as } x \to -\infty \\ f(x) \to - \infty, \text{as } x \to +\infty	\text{falls to the left and} \\ \text{to the right}
\text{odd}	\text{positive}	f(x) \to -\infty, \text{as } x \to -\infty \\ f(x) \to + \infty, \text{as } x \to +\infty	\text{falls to the left and} \\ \text{rises to the right}
\text{odd}	\text{negative}	f(x) \to + \infty, \text{as } x \to -\infty \\ f(x) \to - \infty, \text{as } x \to + \infty	\text{rises to the left and} \\ \text{falls to the right}

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{\text{degree}=2 \text{ (even),}} and {\text{leading coefficient}=\dfrac{1}{2} \text{ (positive)}}

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{\text{degree}=4 \text{ (even),}} and {\text{leading coefficient}=-3 \text{ (negative)}}

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{\text{degree}=5 \text{ (odd),}} and {\text{leading coefficient}=2 \text{ (positive)}}

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{\text{degree}=3 \text{ (odd),}} and {\text{leading coefficient}=-4 \text{ (negative)}}

Examples

Example 1

State whether f\left(x\right)= x^{2} + 2x^{\frac{1}{2}} + 3 is a polynomial function or not.

Worked Solution

Create a strategy

To know whether a function is a polynomial function, check whether all variables are raised to non-negative integer powers.

Apply the idea

The variables in the function are raised to the following powers: 2, \dfrac{1}{2} and 0. Since \dfrac{1}{2} is not a positive integer, f\left(x\right)= x^{2} + 2x^{\frac{1}{2}} + 3 is not a polynomial function.

Example 2

Identify the degree, leading term, and leading coefficient of: p\left(x\right)=-5x^2+4x+2x^3-3

Worked Solution

Create a strategy

First we need to rewrite the polynomial in descending order, or the term with the highest exponent to the term with the smallest exponent.

Apply the idea

Writing in descending order: p\left(x\right)=2x^3-5x^2+4x-3

Degree: 3

Leading term: 2x^3

Leading coefficient: 2

Example 3

Given the polynomial function below:

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Find the average rate of change over the following intervals:

-3\leq x\leq-2
-2\leq x\leq 0
0\leq x\leq 2
2\leq x\leq 3

Worked Solution

Create a strategy

The average rate of change is found by \dfrac{f\left(b\right)-f\left(a\right)}{b-a} where a is one endpoint of the interval and b is the other endpoint of the interval.

Apply the idea

-3\leq x\leq-2

\displaystyle \dfrac{f\left(b\right)-f\left(a\right)}{b-a}	\displaystyle =	\displaystyle \dfrac{f\left(-2\right)-f\left(-3\right)}{-2-\left(-3\right)}	Substitute a=-3 and b=-2
	\displaystyle =	\displaystyle \dfrac{-20-70}{-2-\left(-3\right)}	Substitute f\left(-3\right)=70 and f\left(-2\right)=-20
	\displaystyle =	\displaystyle -90	Evaluate the subtraction and division

-2\leq t\leq 0

\displaystyle \dfrac{f\left(b\right)-f\left(a\right)}{b-a}	\displaystyle =	\displaystyle \dfrac{f\left(0\right)-f\left(-2\right)}{0-\left(-2\right)}	Substitute a=-2 and b=0
	\displaystyle =	\displaystyle \dfrac{0-\left(-20\right)}{0-\left(-2\right)}	Substitute f\left(-2\right)=-20 and f\left(0\right)=0
	\displaystyle =	\displaystyle 10	Evaluate the subtraction and division

0\leq t\leq 2

\displaystyle \dfrac{f\left(b\right)-f\left(a\right)}{b-a}	\displaystyle =	\displaystyle \dfrac{f\left(2\right)-f\left(0\right)}{2-0}	Substitute a=0 and b=2
	\displaystyle =	\displaystyle \dfrac{20-0}{2-0}	Substitute f\left(0\right)=0 and f\left(2\right)=20
	\displaystyle =	\displaystyle 10	Evaluate the subtraction and division

2\leq t\leq 3

\displaystyle \dfrac{f\left(b\right)-f\left(a\right)}{b-a}	\displaystyle =	\displaystyle \dfrac{f\left(3\right)-f\left(2\right)}{3-2}	Substitute a=2 and b=3
	\displaystyle =	\displaystyle \dfrac{-70-20}{3-2}	Substitute f\left(2\right)=20 and f\left(3\right)=-70
	\displaystyle =	\displaystyle -90	Evaluate the subtraction and division

Reflect and check

The average rate of change is the same for the intervals -3\leq x\leq 2 and 2\leq x\leq 3, and it is the same again for the intervals -2\leq x\leq 0 and 0\leq x\leq 2. However, the intervals around the point of inflection are much slower than the others.

Determine the end behavior of the function.

Worked Solution

Create a strategy

We want to determine what happens to the y-values when the x-values are very small and very large.

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Apply the idea

As x gets very small, y gets very large. So, as x\to -\infty, y\to \infty.

As x gets very large, y gets very small. So, as x\to \infty, y\to -\infty.

State the domain and range of the function.

Worked Solution

Create a strategy

The domain represents the x-values of the function, and the range represents the y-values of the function. Recall that the graph of a polynomial function is one smooth curve over a continuous domain.

Apply the idea

As a polynomial is defined for any real number input x, the domain is \left(-\infty ,\infty\right).This can be seen in the graph where the function is a smooth continuous curve and continues to be defined to the left, as the x-values get small, and the right, as the x-values get large.

Since the y-values continue to get increasingly small and increasingly large indefinitely, the range is also \left(-\infty ,\infty\right).

Idea summary

A nonconstant polynomial function has the form:

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

degree: \left(n\right) the highest exponent on a variable

Leading term: \left(a_n x^n\right) the term with the highest exponent

Leading coefficient: \left(a_n \right) the coefficient of the leading term

Degree	Leading Coeff.	End Behavior	Graph of the function
\text{even}	\text{positive}	f(x) \to + \infty, \text{as } x \to -\infty \\ f(x) \to + \infty, \text{as } x \to +\infty	\text{rises to the left and} \\ \text{to the right}
\text{even}	\text{negative}	f(x) \to - \infty, \text{as } x \to -\infty \\ f(x) \to - \infty, \text{as } x \to +\infty	\text{falls to the left and} \\ \text{to the right}
\text{odd}	\text{positive}	f(x) \to -\infty, \text{as } x \to -\infty \\ f(x) \to + \infty, \text{as } x \to +\infty	\text{falls to the left and} \\ \text{rises to the right}
\text{odd}	\text{negative}	f(x) \to + \infty, \text{as } x \to -\infty \\ f(x) \to - \infty, \text{as } x \to + \infty	\text{rises to the left and} \\ \text{falls to the right}

Local and global extrema

Extrema are the highest and lowest points of a function within a specific region or overall. A local maximum or local minimum is the highest or lowest output value within a small interval of the function, whereas a global maximum or global minimum is the highest or lowest output value across the entire function. Under this definition, every global extrema is also a local extrema. Extrema occur where the function switches between increasing and decreasing or at the included endpoint of a polynomial with a restricted domain.

A function on a coordinate plane with all of the low points labeled local minima but only the lowest point labled global minimum and all of the high points labeled local maxima but only the highest point labled global maximum

Between any two distinct, real zeros of a nonconstant polynomial function, there will be at least one local maximum or minimum.

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Examples

Example 4

The graph of a quartic polynomial is shown.

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Identify the zeros of g\left(x\right).

Worked Solution

Create a strategy

The zeros coincide with the x-intercepts on the graph.

Apply the idea

The zeros are x=-4, \text{ } x=0, \text{ and } x=2

Reflect and check

The zero x=0 has a multiplicity of 2. This is called a double zero.

Label all local maxima and minima.

Worked Solution

Create a strategy

Local maxima and minima are the turning points of the graph.

Apply the idea

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Point A local minimum, approximate coordinates: \left(-2.9,-4.5\right)

Point B local minimum, coordinates: \left(0,0\right)

Point C local minimum, approximate coordinates: \left(1.4,-0.65\right)

Label all global maxima and minima.

Worked Solution

Create a strategy

We will look at all of the local extrema we identified in part (a) and determine if any of them are the highest or lowest points on the entire function.

Apply the idea

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Only point A is a global extrema. Because it is the lowest function value it is a global minimum.

Reflect and check

Remember, all extrema are local but only some are global.

Is the leading coefficient of g\left(x\right) positive or negative? Explain.

Worked Solution

Create a strategy

We can determine the leading coefficient based on the degree and the end behavior. This function has an even degree (of 4) and both ends are pointing up toward positive infinity.

Apply the idea

The leading coefficient is positive.

Idea summary

A local maximum is higher relative to the points around it. A global maximum is the highest local maximum.

A local minimum is lower relative to the points around it. A global minimum is the lowest local minimum.

There is at least one local maximum or minimum point between two real zeros.

Points of inflection

A point of inflection is a point on the graph of a polynomial function where the curvature of the graph changes. These points occur where the rate of change of the function switches from increasing to decreasing or from decreasing to increasing. In other words, this is where the graph of a polynomial function changes from concave up to concave down or vice versa.

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caption

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caption

Examples

Example 5

The graph of p\left(x\right) is shown:

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Identify the point of inflection of p\left(x\right).

Worked Solution

Create a strategy

Points of inflection occur where the concavity of the graph changes.

Apply the idea

We can see that there is a point of inflection at exactly \left(2,4\right).

There is another point of inflection to the left of that point as well. We will need to approximate those coordinates. They appear to be approximately \left(0.77,2\right).

Describe the concavity of the graph.

Worked Solution

Create a strategy

The concavity changes at a point of inflection, so we can use the points of inflection to divide the graphs into sections and determine the concavity of each section.

Apply the idea

Start by plotting the points of inflection identified in part (a).

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The points of inflection divide the graph into three sections. The left and right are concave up and the middle is concave down.

In interval notation that is:

Concave up \left(-\infty,0.77\right)\cup \left(2,\infty \right)

Concave down \left(0.77,2\right)

Idea summary

Points of inflection occur when the concavity of a graph changes.

Outcomes

1.4.A

Identify key characteristics of polynomial functions related to rates of change.

1.4 Polynomial functions and rates of change

Introduction

Ideas

Polynomial functions

Examples

Example 1

Example 2

Example 3

Local and global extrema

Examples

Example 4

Points of inflection

Examples

Example 5

Outcomes

1.4.A

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