PrecalculusMathematics and Statistics
Graphs of Polynomial Functions
The revenue in millions of dollars for a fictional cable company from 2006 through 2013 is shown in [link].
The revenue can be modeled by the polynomial function
whererepresents the revenue in millions of dollars andrepresents the year, withcorresponding to 2006. Over which intervals is the revenue for the company increasing? Over which intervals is the revenue for the company decreasing? These questions, along with many others, can be answered by examining the graph of the polynomial function. We have already explored the local behavior of quadratics, a special case of polynomials. In this section we will explore the local behavior of polynomials in general.
Recognizing Characteristics of Graphs of Polynomial Functions
Polynomial functions of degree 2 or more have graphs that do not have sharp corners; recall that these types of graphs are called smooth curves. Polynomial functions also display graphs that have no breaks. Curves with no breaks are called continuous. [link] shows a graph that represents a polynomial function and a graph that represents a function that is not a polynomial.
Which of the graphs in [link] represents a polynomial function?
The graphs ofandare graphs of polynomial functions. They are smooth and continuous.
The graphs ofandare graphs of functions that are not polynomials. The graph of functionhas a sharp corner. The graph of functionis not continuous.
Do all polynomial functions have as their domain all real numbers?
Yes. Any real number is a valid input for a polynomial function.
Using Factoring to Find Zeros of Polynomial Functions
Recall that ifis a polynomial function, the values offor whichare called zeros ofIf the equation of the polynomial function can be factored, we can set each factor equal to zero and solve for the zeros.
We can use this method to findintercepts because at theintercepts we find the input values when the output value is zero. For general polynomials, this can be a challenging prospect. While quadratics can be solved using the relatively simple quadratic formula, the corresponding formulas for cubic and fourth-degree polynomials are not simple enough to remember, and formulas do not exist for general higher-degree polynomials. Consequently, we will limit ourselves to three cases in this section:
- The polynomial can be factored using known methods: greatest common factor and trinomial factoring.
- The polynomial is given in factored form.
- Technology is used to determine the intercepts.
Given a polynomial functionfind the x-intercepts by factoring.
- If the polynomial function is not given in factored form:
- Factor out any common monomial factors.
- Factor any factorable binomials or trinomials.
- Set each factor equal to zero and solve to find theintercepts.
Find the x-intercepts of
We can attempt to factor this polynomial to find solutions for
This gives us fiveintercepts:andSee [link]. We can see that this is an even function.
Find theintercepts of
Find solutions for
There are threeintercepts:andSee [link].
Find the and x-intercepts of
The y-intercept can be found by evaluating
So the y-intercept is
The x-intercepts can be found by solving
So theintercepts areand
Find theintercepts of
This polynomial is not in factored form, has no common factors, and does not appear to be factorable using techniques previously discussed. Fortunately, we can use technology to find the intercepts. Keep in mind that some values make graphing difficult by hand. In these cases, we can take advantage of graphing utilities.
Looking at the graph of this function, as shown in [link], it appears that there are x-intercepts atand
We can check whether these are correct by substituting these values forand verifying that
Eachintercept corresponds to a zero of the polynomial function and each zero yields a factor, so we can now write the polynomial in factored form.
Find the and x-intercepts of the function
Identifying Zeros and Their Multiplicities
Graphs behave differently at variousintercepts. Sometimes, the graph will cross over the horizontal axis at an intercept. Other times, the graph will touch the horizontal axis and bounce off.
Suppose, for example, we graph the function
Notice in [link] that the behavior of the function at each of theintercepts is different.
Theintercept is the solution of equationThe graph passes directly through theintercept atThe factor is linear (has a degree of 1), so the behavior near the intercept is like that of a line—it passes directly through the intercept. We call this a single zero because the zero corresponds to a single factor of the function.
Theinterceptis the repeated solution of equationThe graph touches the axis at the intercept and changes direction. The factor is quadratic (degree 2), so the behavior near the intercept is like that of a quadratic—it bounces off of the horizontal axis at the intercept.
The factor is repeated, that is, the factorappears twice. The number of times a given factor appears in the factored form of the equation of a polynomial is called the multiplicity. The zero associated with this factor,has multiplicity 2 because the factoroccurs twice.
Theinterceptis the repeated solution of factorThe graph passes through the axis at the intercept, but flattens out a bit first. This factor is cubic (degree 3), so the behavior near the intercept is like that of a cubic—with the same S-shape near the intercept as the toolkit functionWe call this a triple zero, or a zero with multiplicity 3.
For zeros with even multiplicities, the graphs touch or are tangent to theaxis. For zeros with odd multiplicities, the graphs cross or intersect theaxis. See [link] for examples of graphs of polynomial functions with multiplicity 1, 2, and 3.
For higher even powers, such as 4, 6, and 8, the graph will still touch and bounce off of the horizontal axis but, for each increasing even power, the graph will appear flatter as it approaches and leaves theaxis.
For higher odd powers, such as 5, 7, and 9, the graph will still cross through the horizontal axis, but for each increasing odd power, the graph will appear flatter as it approaches and leaves theaxis.
If a polynomial contains a factor of the formthe behavior near theinterceptis determined by the powerWe say thatis a zero of multiplicity
The graph of a polynomial function will touch theaxis at zeros with even multiplicities. The graph will cross the x-axis at zeros with odd multiplicities.
The sum of the multiplicities is the degree of the polynomial function.
Given a graph of a polynomial function of degreeidentify the zeros and their multiplicities.
- If the graph crosses the x-axis and appears almost linear at the intercept, it is a single zero.
- If the graph touches the x-axis and bounces off of the axis, it is a zero with even multiplicity.
- If the graph crosses the x-axis at a zero, it is a zero with odd multiplicity.
- The sum of the multiplicities is
Use the graph of the function of degree 6 in [link] to identify the zeros of the function and their possible multiplicities.
The polynomial function is of degreeThe sum of the multiplicities must be
Starting from the left, the first zero occurs atThe graph touches the x-axis, so the multiplicity of the zero must be even. The zero ofhas multiplicity
The next zero occurs atThe graph looks almost linear at this point. This is a single zero of multiplicity 1.
The last zero occurs atThe graph crosses the x-axis, so the multiplicity of the zero must be odd. We know that the multiplicity is likely 3 and that the sum of the multiplicities is likely 6.
Use the graph of the function of degree 5 in [link] to identify the zeros of the function and their multiplicities.
The graph has a zero of –5 with multiplicity 1, a zero of –1 with multiplicity 2, and a zero of 3 with even multiplicity.
Determining End Behavior
As we have already learned, the behavior of a graph of a polynomial function of the form
will either ultimately rise or fall as increases without bound and will either rise or fall as decreases without bound. This is because for very large inputs, say 100 or 1,000, the leading term dominates the size of the output. The same is true for very small inputs, say –100 or –1,000.
Recall that we call this behavior the end behavior of a function. As we pointed out when discussing quadratic equations, when the leading term of a polynomial function,is an even power function, asincreases or decreases without bound,increases without bound. When the leading term is an odd power function, asdecreases without bound,also decreases without bound; asincreases without bound,also increases without bound. If the leading term is negative, it will change the direction of the end behavior. [link] summarizes all four cases.
Understanding the Relationship between Degree and Turning Points
In addition to the end behavior, recall that we can analyze a polynomial function’s local behavior. It may have a turning point where the graph changes from increasing to decreasing (rising to falling) or decreasing to increasing (falling to rising). Look at the graph of the polynomial functionin [link]. The graph has three turning points.
This function is a 4th degree polynomial function and has 3 turning points. The maximum number of turning points of a polynomial function is always one less than the degree of the function.
A turning point is a point of the graph where the graph changes from increasing to decreasing (rising to falling) or decreasing to increasing (falling to rising).
A polynomial of degreewill have at mostturning points.
Find the maximum number of turning points of each polynomial function.
First, rewrite the polynomial function in descending order:
Identify the degree of the polynomial function. This polynomial function is of degree 5.
The maximum number of turning points is
First, identify the leading term of the polynomial function if the function were expanded.
Then, identify the degree of the polynomial function. This polynomial function is of degree 4.
The maximum number of turning points is
Graphing Polynomial Functions
We can use what we have learned about multiplicities, end behavior, and turning points to sketch graphs of polynomial functions. Let us put this all together and look at the steps required to graph polynomial functions.
Given a polynomial function, sketch the graph.
- Find the intercepts.
- Check for symmetry. If the function is an even function, its graph is symmetrical about theaxis, that is, If a function is an odd function, its graph is symmetrical about the origin, that is,
- Use the multiplicities of the zeros to determine the behavior of the polynomial at theintercepts.
- Determine the end behavior by examining the leading term.
- Use the end behavior and the behavior at the intercepts to sketch a graph.
- Ensure that the number of turning points does not exceed one less than the degree of the polynomial.
- Optionally, use technology to check the graph.
Sketch a graph of
This graph has twointercepts. Atthe factor is squared, indicating a multiplicity of 2. The graph will bounce at thisintercept. Atthe function has a multiplicity of one, indicating the graph will cross through the axis at this intercept.
The y-intercept is found by evaluating
Additionally, we can see the leading term, if this polynomial were multiplied out, would be so the end behavior is that of a vertically reflected cubic, with the outputs decreasing as the inputs approach infinity, and the outputs increasing as the inputs approach negative infinity. See [link].
To sketch this, we consider that:
- Asthe functionso we know the graph starts in the second quadrant and is decreasing toward theaxis.
- Since is not equal tothe graph does not display symmetry.
- Atthe graph bounces off of theaxis, so the function must start increasing.
Atthe graph crosses theaxis at theintercept. See [link].
Somewhere after this point, the graph must turn back down or start decreasing toward the horizontal axis because the graph passes through the next intercept atSee [link].
Asthe functionso we know the graph continues to decrease, and we can stop drawing the graph in the fourth quadrant.
Sketch a graph of
Using the Intermediate Value Theorem
In some situations, we may know two points on a graph but not the zeros. If those two points are on opposite sides of the x-axis, we can confirm that there is a zero between them. Consider a polynomial functionwhose graph is smooth and continuous. The Intermediate Value Theorem states that for two numbersandin the domain ofifandthen the functiontakes on every value betweenand We can apply this theorem to a special case that is useful in graphing polynomial functions. If a point on the graph of a continuous functionatlies above theaxis and another point atlies below theaxis, there must exist a third point betweenandwhere the graph crosses theaxis. Call this pointThis means that we are assured there is a solutionwhere
In other words, the Intermediate Value Theorem tells us that when a polynomial function changes from a negative value to a positive value, the function must cross theaxis. [link] shows that there is a zero between and
Let be a polynomial function. The Intermediate Value Theorem states that if and have opposite signs, then there exists at least one value between and for which
Show that the function has at least two real zeros between and
As a start, evaluate at the integer valuesSee [link].
We see that one zero occurs atAlso, sinceis negative andis positive, by the Intermediate Value Theorem, there must be at least one real zero between 3 and 4.
We have shown that there are at least two real zeros between and
Show that the functionhas at least one real zero betweenand
Becauseis a polynomial function and sinceis negative andis positive, there is at least one real zero betweenand
Writing Formulas for Polynomial Functions
Now that we know how to find zeros of polynomial functions, we can use them to write formulas based on graphs. Because a polynomial function written in factored form will have anintercept where each factor is equal to zero, we can form a function that will pass through a set ofintercepts by introducing a corresponding set of factors.
If a polynomial of lowest degree has horizontal intercepts at then the polynomial can be written in the factored form: where the powers on each factor can be determined by the behavior of the graph at the corresponding intercept, and the stretch factor can be determined given a value of the function other than the x-intercept.
Given a graph of a polynomial function, write a formula for the function.
- Identify the x-intercepts of the graph to find the factors of the polynomial.
- Examine the behavior of the graph at the x-intercepts to determine the multiplicity of each factor.
- Find the polynomial of least degree containing all the factors found in the previous step.
- Use any other point on the graph (the y-intercept may be easiest) to determine the stretch factor.
Write a formula for the polynomial function shown in [link].
This graph has threeintercepts:andTheintercept is located atAtand the graph passes through the axis linearly, suggesting the corresponding factors of the polynomial will be linear. Atthe graph bounces at the intercept, suggesting the corresponding factor of the polynomial will be second degree (quadratic). Together, this gives us
To determine the stretch factor, we utilize another point on the graph. We will use theinterceptto solve for
The graphed polynomial appears to represent the function
Using Local and Global Extrema
With quadratics, we were able to algebraically find the maximum or minimum value of the function by finding the vertex. For general polynomials, finding these turning points is not possible without more advanced techniques from calculus. Even then, finding where extrema occur can still be algebraically challenging. For now, we will estimate the locations of turning points using technology to generate a graph.
Each turning point represents a local minimum or maximum. Sometimes, a turning point is the highest or lowest point on the entire graph. In these cases, we say that the turning point is a global maximum or a global minimum. These are also referred to as the absolute maximum and absolute minimum values of the function.
A local maximum or local minimum at(sometimes called the relative maximum or minimum, respectively) is the output at the highest or lowest point on the graph in an open interval aroundIf a function has a local maximum atthenfor allin an open interval aroundIf a function has a local minimum atthenfor allin an open interval around
A global maximum or global minimum is the output at the highest or lowest point of the function. If a function has a global maximum at thenfor allIf a function has a global minimum atthenfor all
We can see the difference between local and global extrema in [link].
Do all polynomial functions have a global minimum or maximum?
No. Only polynomial functions of even degree have a global minimum or maximum. For example,has neither a global maximum nor a global minimum.
An open-top box is to be constructed by cutting out squares from each corner of a 14 cm by 20 cm sheet of plastic then folding up the sides. Find the size of squares that should be cut out to maximize the volume enclosed by the box.
We will start this problem by drawing a picture like that in [link], labeling the width of the cut-out squares with a variable,
Notice that after a square is cut out from each end, it leaves acm bycm rectangle for the base of the box, and the box will becm tall. This gives the volume
Notice, since the factors areandthe three zeros are 10, 7, and 0, respectively. Because a height of 0 cm is not reasonable, we consider the only the zeros 10 and 7. The shortest side is 14 and we are cutting off two squares, so valuesmay take on are greater than zero or less than 7. This means we will restrict the domain of this function toUsing technology to sketch the graph ofon this reasonable domain, we get a graph like that in [link]. We can use this graph to estimate the maximum value for the volume, restricted to values forthat are reasonable for this problem—values from 0 to 7.
From this graph, we turn our focus to only the portion on the reasonable domain,We can estimate the maximum value to be around 340 cubic cm, which occurs when the squares are about 2.75 cm on each side. To improve this estimate, we could use advanced features of our technology, if available, or simply change our window to zoom in on our graph to produce [link].
From this zoomed-in view, we can refine our estimate for the maximum volume to about 339 cubic cm, when the squares measure approximately 2.7 cm on each side.
Use technology to find the maximum and minimum values on the intervalof the function
The minimum occurs at approximately the pointand the maximum occurs at approximately the point
Access the following online resource for additional instruction and practice with graphing polynomial functions.
- Polynomial functions of degree 2 or more are smooth, continuous functions. See [link].
- To find the zeros of a polynomial function, if it can be factored, factor the function and set each factor equal to zero. See [link], [link], and [link].
- Another way to find theintercepts of a polynomial function is to graph the function and identify the points at which the graph crosses theaxis. See [link].
- The multiplicity of a zero determines how the graph behaves at theintercepts. See [link].
- The graph of a polynomial will cross the horizontal axis at a zero with odd multiplicity.
- The graph of a polynomial will touch the horizontal axis at a zero with even multiplicity.
- The end behavior of a polynomial function depends on the leading term.
- The graph of a polynomial function changes direction at its turning points.
- A polynomial function of degree has at most turning points. See [link].
- To graph polynomial functions, find the zeros and their multiplicities, determine the end behavior, and ensure that the final graph has at most turning points. See [link] and [link].
- Graphing a polynomial function helps to estimate local and global extremas. See [link].
- The Intermediate Value Theorem tells us that if have opposite signs, then there exists at least one value between and for which See [link].
What is the difference between anintercept and a zero of a polynomial function
Theintercept is where the graph of the function crosses theaxis, and the zero of the function is the input value for which
If a polynomial function of degree has distinct zeros, what do you know about the graph of the function?
Explain how the Intermediate Value Theorem can assist us in finding a zero of a function.
If we evaluate the function at and at and the sign of the function value changes, then we know a zero exists between and
Explain how the factored form of the polynomial helps us in graphing it.
If the graph of a polynomial just touches theaxis and then changes direction, what can we conclude about the factored form of the polynomial?
There will be a factor raised to an even power.
For the following exercises, find the or t-intercepts of the polynomial functions.
For the following exercises, use the Intermediate Value Theorem to confirm that the given polynomial has at least one zero within the given interval.
and Sign change confirms.
between and .
and Sign change confirms.
and Sign change confirms.
For the following exercises, find the zeros and give the multiplicity of each.
0 with multiplicity 2, with multiplicity 5, 4 with multiplicity 2
0 with multiplicity 2, –2 with multiplicity 2
with multiplicity 2, 0 with multiplicity 3
For the following exercises, graph the polynomial functions. Note andintercepts, multiplicity, and end behavior.
x-intercepts, with multiplicity 2, with multiplicity 1,intercept .
x-interceptswith multiplicity 3,with multiplicity 2,intercept
x-intercepts with multiplicity 1,intercept
For the following exercises, use the graphs to write the formula for a polynomial function of least degree.
For the following exercises, use the graph to identify zeros and multiplicity.
–4, –2, 1, 3 with multiplicity 1
–2, 3 each with multiplicity 2
For the following exercises, use the given information about the polynomial graph to write the equation.
Degree 3. Zeros at andy-intercept at
Degree 3. Zeros at andy-intercept at
Degree 5. Roots of multiplicity 2 at and , and a root of multiplicity 1 at y-intercept at
Degree 4. Root of multiplicity 2 atand a roots of multiplicity 1 atandy-intercept at
Degree 5. Double zero atand triple zero at Passes through the point
Degree 3. Zeros atandy-intercept at
Degree 3. Zeros at and y-intercept at
Degree 5. Roots of multiplicity 2 at and and a root of multiplicity 1 at
Degree 4. Roots of multiplicity 2 atand roots of multiplicity 1 atand
Double zero at and triple zero at Passes through the point
For the following exercises, use a calculator to approximate local minima and maxima or the global minimum and maximum.
local max local min
For the following exercises, use the graphs to write a polynomial function of least degree.
For the following exercises, write the polynomial function that models the given situation.
A rectangle has a length of 10 units and a width of 8 units. Squares of by units are cut out of each corner, and then the sides are folded up to create an open box. Express the volume of the box as a polynomial function in terms of
Consider the same rectangle of the preceding problem. Squares of by units are cut out of each corner. Express the volume of the box as a polynomial in terms of
A square has sides of 12 units. Squares by units are cut out of each corner, and then the sides are folded up to create an open box. Express the volume of the box as a function in terms of
A cylinder has a radius of units and a height of 3 units greater. Express the volume of the cylinder as a polynomial function.
A right circular cone has a radius of and a height 3 units less. Express the volume of the cone as a polynomial function. The volume of a cone is for radius and height
- Linear Functions
- Polynomial and Rational Functions
- Exponential and Logarithmic Functions
- Trigonometric Functions
- Periodic Functions
- Trigonometric Identities and Equations
- Further Applications of Trigonometry
- Systems of Equations and Inequalities
- Introduction to Systems of Equations and Inequalities
- Systems of Linear Equations: Two Variables
- Systems of Linear Equations: Three Variables
- Systems of Nonlinear Equations and Inequalities: Two Variables
- Partial Fractions
- Matrices and Matrix Operations
- Solving Systems with Gaussian Elimination
- Solving Systems with Inverses
- Solving Systems with Cramer's Rule
- Analytic Geometry
- Sequences, Probability and Counting Theory
- Introduction to Calculus