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3. Polynomial Expressions, Equations, & Functions
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Algebra 2

3.04 Characteristics of polynomial functions

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Characteristics of polynomial functions

A polynomial function is a function that involves variables raised to non-negative integer powers.

The standard form of a polynomial function is given by f\left(x\right)=a_{n}x^{n}+a_{n-1}x^{n-1}+a_{n-2}x^{n-2}+\ldots + a_{2} x^{2}+a_{1}x+a_{0} where n is a positive integer and a_{n},\,a_{n-1},\,a_{n-2},\,\ldots,\,a_{2},\,a_{1},\,a_{0} are constant coefficients.

The domain of every polynomial function is \left(-\infty,\,\infty\right).

The degree, n, of a polynomial function can be found by identifying the highest exponent on the independent variable. The degree tells us information about the key features, such as the range, end behavior, number of x-intercepts, and turning points.

A polynomial of degree n can have at most n x-intercepts, with those of odd degree having at least one.

Absolute (global) maximum

The largest value over the domain of a function

Absolute (global) minimum

The smallest value over the domain of a function

Relative (local) maximum

A point where a function changes from increasing to decreasing

Relative (local) minimum

A point where a function changes from decreasing to increasing

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Even degree polynomial

  • Range depends on the minimum or maximum and leading coefficient

  • Move in the same direction at the ends

  • Has an absolute minimum or maximum

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Odd degree polynomial

  • Range: \left(-\infty,\,\infty\right)

  • Move in opposite directions at the ends

  • No absolute minimum or maximum

The rate of change of a polynomial function is variable, meaning it changes over the course of the domain.

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In some polynomials, the function increases (or decreases) at a fast rate, then the rate of change slows around a point called an inflection point, or turning point. In other words, the function continues increasing (or decreasing), but the rate is slower around the point of inflection.

This is one type of a point of inflection, sometimes referred to as a horizontal point of inflection, not all points of inflection will see the rate slow around the point.

A polynomial of degree n can have up to n-1 turning points, with those of even degree having at least one.

Examples

Example 1

Given this polynomial function:

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a

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.

b

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).

c

Determine if the degree of the function is even or odd.

Worked Solution
Create a strategy

The function moves in opposite directions at the extremities, and does not have a global maximum or minimum.

Apply the idea

The function has an odd degree.

Reflect and check

We can use other key features about the graph of the polynomial to help us determine its equation and other information as needed.

Example 2

This graph shows f\left(x\right)=x^{4}+2x^{3}-7x^{2}-8x+12.

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a

Determine f\left(2\right).

Worked Solution
Create a strategy

We want to find the value of the function f\left(x\right) when x=2, which corresponds on the graph to the ordered pair \left(2,\,y\right).

Apply the idea

When x=2, the function crosses the x-axis, which represents the point \left(2,\,0\right).

Therefore, f\left(2\right)=0.

Reflect and check

The x-intercepts of a graph can be used to find the roots and binomial factors of the original function.

This function has x-intercepts at x = -3,\, -2,\, 1, and 2, so we know that \left(x+3\right)\left(x+2\right)\left(x-1\right)\\ \left(x-2\right) are part of the polynomial, which simplifies to x^{4}+2x^{3}-7x^{2}-8x+12.

Some polynomials have a vertical stretch or compression in the form of a numerical greatest common factor, which does not apply in this case.

b

Name the type of extrema seen on the graph.

Worked Solution
Create a strategy

Since the function has an even degree, the function will have a global minimum or maximum.

While the global minimum or maximum may occur for multiple x-values, the y-value of these points will be the same.

Apply the idea

The graph of the function opens up, so the function has a global (absolute) minimum at y=-4.

Reflect and check

The leading coefficient of the polynomial also tells us which direction the graph of the function will open.

x^{4}+2x^{3}-7x^{2}-8x+12 has a leading coefficient of 1. A polynomial with a positive leading coefficient opens up, and will therefore have a global minimum.

While there is only one global minimum there are two x-values that have this minimum y-value,

There is also a local maximum, but we cannot see it on the part of the graph we are shown.

c

Determine the range, writing your answer in set notation.

Worked Solution
Create a strategy

The range of an even function is determined by its extrema. In the case of an even degree polynomial that opens up, all of the y-values of the function will be increasing from the y-value of the global minimum.

Apply the idea

The global minimum has a y-value of -4.

In set notation, this will be written as \left\{y \middle\vert y \geq -4\right\} .

Example 3

The graphs of two functions are shown:

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a

Identify and compare the intervals where each function is increasing, decreasing, or constant.

Worked Solution
Create a strategy

Identify the intervals by observing the graph. Note where it rises (indicating increasing behavior), falls (indicating decreasing behavior), or remains flat (indicating constant behavior). Then compare where the two graphs have a similar pattern.

Apply the idea

For f\left(x\right), we can observe the following:

  • f\left(x\right) is increasing on \left(-\infty, 0\right) and \left(2,\infty \right)

  • f\left(x\right) is decreasing on \left(0,2\right)

For g\left(x\right), we can observe the following:

  • g\left(x\right) is increasing on \left(-\infty, 0\right) and \left(2,3 \right)

  • g\left(x\right) is decreasing on \left(0,2\right)

  • g\left(x\right) is constant on \left(3,\infty\right)

The two functions are both increasing on \left(-\infty,0\right) and \left(2,3\right), but g\left(x\right) is constant on \left(3,\infty\right) while f\left(x\right) increases steadily after x=3. They are also both decreasing on \left(0,2\right).

b

Identify and compare the x and y-intercepts of each function.

Worked Solution
Create a strategy

The x-intercepts are where the graph crosses the x-axis, and the y-intercept is where the graph intersects the y-axis.

Apply the idea

For f\left(x\right):

  • x-intercept: \left(-1,0\right), \left(2,0\right)

  • y-intercept: \left(0,4\right)

For g\left(x\right):

  • x-intercept: \left(-2,0\right), \left(2,0\right)

  • y-intercept: \left(0,4\right)

The two fuctions have the same y-intercept of \left(0,4\right) and x-intercept of \left(2,0\right). However, they each have a second x-intercep,t and these are at different points.

Idea summary

The degree, n, of a polynomial function is the same as the highest exponent on the variable. It tells us information about the key features, such as the:

  • range

  • number of x-intercepts

  • end behavior

  • number of turning points

The domain of every polynomial function is (-\infty,\,\infty).

Outcomes

A2.F.2

The student will investigate and analyze characteristics of square root, cube root, rational, polynomial, exponential, logarithmic, and piecewise-defined functions algebraically and graphically.

A2.F.2a

Determine and identify the domain, range, zeros, and intercepts of a function presented algebraically or graphically, including graphs with discontinuities.

A2.F.2b

Compare and contrast the characteristics of square root, cube root, rational, polynomial, exponential, logarithmic, and piecewise-defined functions.

A2.F.2c

Determine the intervals on which the graph of a function is increasing, decreasing, or constant.

A2.F.2d

Determine the location and value of absolute (global) maxima and absolute (global) minima of a function.

A2.F.2e

Determine the location and value of relative (local) maxima or relative (local) minima of a function.

A2.F.2f

For any value, x, in the domain of f, determine f(x) using a graph or equation. Explain the meaning of x and f(x) in context, where applicable.

A2.F.2g

Describe the end behavior of a function.

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