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Graphing functions is essential in different fields of research. This has led many people to develop programs and technologies that draw accurate graphs, such as graphing calculators. However, knowing how to draw a graph by hand is also essential. After all, that is how they did it not too many years ago!

Unfortunately, graphing an arbitrary function usually requires knowing advanced algebraic techniques. However, when the function is a polynomial, some key features of the graph can be derived directly from the function rule, leading to an accurate graph. As such, this lesson aims to teach how to graph polynomials.

Catch-Up and Review

Here are a few recommended readings before getting started with this lesson.

Challenge

Investigating the End Behavior

In order to improve her graphing polynomial skills, Emily decided to try graphing the fourth-degree polynomial function To start, she made a table of values.

Next, she plotted all the points in a coordinate plane and connected them with a smooth curve. She noticed that the left part of the curve seemed to be going up while the right part seemed to be going down. She decided to extend the curve based on this intuition.
Curve connecting the points from the table. The left part of the curve is extended upwards and the right-part is extended downwards.
However, Emily was not so sure about her reasoning. Is Emily's graph correct? Is there a way she can be sure about the left and right parts of the graph?
Discussion

Polynomial Functions and End Behavior

Functions are usually named after the algebraic expression that defines them. For example, if the function rule is a real number, the function is called a constant function. Something similar happens when the function rule is a linear, quadratic, or exponential expression.

Function Rule Name
Constant Function
Linear Function
Quadratic Function
Exponential Function

The same holds true to those functions whose function rule is a polynomial.

Concept

Polynomial Function

A polynomial function is a function whose rule is a polynomial. In general, a polynomial function has the following form.


Here, all exponents are whole numbers and all the coefficients are real numbers with Unless explicitly stated, the domain of a polynomial function is all real numbers and its range depends on the end behavior.
Concept

End Behavior

The end behavior of a function is the value to which tends as extends to the left or the right infinitely. If keeps increasing without bound, it is said to tend to positive infinity. The end behavior of this case is stated as up.
Conversely, if keeps decreasing without bound, it is said to tend to negative infinity. In this case, the end behavior is stated as down.
For example, consider the graph of a function
Graph of g(x)=x^3+(3*x^2)-1
From the arrows on the graph, it can be seen that the left end of the graph extends downward, while the right end extends upward. The end behavior of can then be expressed as follows.
To state the end behavior of a function in words, begin by stating the left-end behavior, then state the right-end behavior. A dash can also be used to separate the words. For instance, the end behavior of the graph of can be written as down and up or as down-up.

End Behavior of Polynomial Functions

When the function is a polynomial function, the end behavior can be determined from the function rule.
Particularly, the end behavior is given by the sign of the leading coefficient and the degree of the polynomial.
Leading Coeff. positive and degree even: Up-Up; Leading Coeff. positive and degree odd: Down-Up; Leading Coeff. negative and degree even: Down-Down; Leading Coeff. negative and degree even: Up-Down;
Note that when the degree of the polynomial is even, both ends have the same behavior, which depends on the sign of the leading coefficient. By contrast, when the degree is odd, both ends have opposite behaviors.
Pop Quiz

Determining the End Behavior

Determine the end behavior of the given polynomial function.

Either the function rule or the graph of a polynomial function.
Example

Image Editor Running Time

Emily loves to record every trip she takes on her blog, including amazing photos. However, since she does not have a professional camera, some of her photos need some retouching. To help with this, she plans to buy an image editing software. She is torn between two options whose running times to process an pixels image are given by the following polynomials.

Running Time (nanoseconds)
Software
Software
a Emily used online software to plot both polynomials in an effort to clear things up before deciding which software to purchase.
Graph of both polynomials between x=0 and x=325.
From the graph, Emily noticed that the right-end behavior of the graph of is down, while the right-end behavior of the graph of is up. Therefore, she concluded that for images with realistic dimensions, Software is faster and plans to buy it. Should Emily buy Software
b What is the end behavior of the graph of
c What is the end behavior of the graph of

Hint
a Evaluate both polynomials at different values and compare the results. In general, images have sides of at least pixels. In order to study the end behavior of the curves, the graph should be plotted in a large enough domain.
b The end behavior of a polynomial graph can be determined by studying the sign of the leading coefficient and the degree of the polynomial rule. When the degree is odd, both ends of the graph have opposite behaviors.
c When the degree is even, both ends of the graph have the same behavior.

Solution
a Before analyzing Emily's graph, evaluate the polynomial functions at different values to compare algebraically the running times of both software.
Image Size (pixels) Running Times (nanoseconds) Conclusion
-by-
Software is faster
-by-
Similar Running Times
-by-
Software is faster
-by-
Software is faster
-by-
Software is faster
-by-
Software is faster

The table suggests the following conclusions.

  • For very small images, Software is faster than Software
  • For images of about pixels, both software have similar running times.
  • For images with more realistic dimensions, pixels images or larger, Software is faster than Software

Emily's graph also suggests the first two conclusions. However, that graph corresponds only to a small portion of the domain. Therefore, before making any decisions about end behavior, graph both polynomials in a bigger domain.

Graph of both polynomials between x=0 and x=2300.

It can be seen that the right-end behavior of the graph of is up, and not down as Emily concluded. Additionally, for images with pixel sides or bigger, the running time of Software is considerably shorter than the running time of Software

Conclusion
In general, Software is faster.

Although the right-end behavior of both graphs is up, the graph of increases way much faster than the graph of Therefore, Emily should buy Software

b Although the behavior of the right end of the graph of was studied in Part A, it can also be determined using the function rule.
Start by writing the end behavior of a polynomial graph based on the leading coefficient and degree.
Leading Coefficient Degree End Behavior
Positive Even Up-Up
Positive Odd Down-Up
Negative Odd Up-Down
Negative Even Down-Down

The leading term of is which means that the leading coefficient is and the degree is Since the leading coefficient is positive and the degree is odd, the end behavior of the graph of is Down-Up.

Graph of P(x) between x=-1800 and x=1800.

Despite the axis being measured in seconds rather than nanoseconds, the conclusion about the end behavior or the fastest software remains the same.

c As in Part B, the end behavior of the graph of can be obtained from its function rule.
Here, the leading term is so the leading coefficient is and the degree is Since the leading coefficient is positive and the degree is even, the end behavior of the graph of is Up-Up.
Graph of Q(x) between x=-3000 and x=3000.

Again, the axis is measured in seconds, not in nanoseconds.

Discussion

Alternative Methods for Finding the intercepts

When graphing polynomial functions, the intercepts is a key feature. Such information can sometimes be derived from the function rule, but it is not always self-evident. Consider the following polynomial.
At first glance, it is not clear whether the graph of intercepts the axis, and if so, where. However, if such a point existed, then it would have the form which means that
A portion of a graph crossing the x-axis at the point (k,0)
Note that the fact that has two implications — that is a zero of the polynomial function and that is a solution to the equation
Consequently, finding the intercepts of the graph of is equivalent to finding the real zeros of the polynomial which in turn is equivalent to finding the real solutions to the equation The following diagram explains what this equivalence means.
Let f(x) be a polynomial function. (a) k is a zero of f(x) if and only if k is a solution to the equation f(x)=0. (b) A real number k is a zero of f(x) if and only if k is an x-intercept of the graph of f(x). (c) k is a real solution to the equation f(x)=0 if and only if k is an x-intercept of the graph of f(x).
Remember, in an equivalence, both the conditional statement and its converse are true.
Example

Graphing a Polynomial Function

Consider the following polynomial function.
a List all the values where the graph of intercepts the axis.
b List the missing numbers in the following table of values. Write them in the order they should be in the table from left to right.
c What is the end behavior of the graph of
d Which of the following is the graph

Hint
a Finding the intercepts is the same as finding the solutions to the equation Factor the polynomial and use the Zero Product Property.
b Evaluate the polynomial at the missing values.
c Study the sign of the leading coefficient and the degree of the polynomial. When the degree is even, both ends have the same behavior.
d Start by plotting the intercepts obtained in Part A. Next, plot the points from the table of values. Connect all the points using a smooth curve and extend the graph according to the end behavior.

Solution
a Determining the places where the graph of intercepts the axis is equivalent to finding the solutions to the equation
To solve the polynomial equation, factor the polynomial. This can be done by grouping, but some rewrites are needed first.
Substitute these expressions into the polynomial equation.
▼
Factor
SubstituteExpressions

Substitute expressions

CommutativePropAdd

Commutative Property of Addition

FactorOut

Factor out

FactorOut

Factor out

The cubic polynomial within the parentheses can also be factored by groping. As before, some of the terms have to be rewritten first.
Now, substitute the expressions into the cubic polynomial.
▼
Factor
SubstituteII

,

CommutativePropAdd

Commutative Property of Addition

FactorOut

Factor out

FactorOut

Factor out

Once again, factor the quadratic polynomial within the parentheses by grouping. To do so, start by rewriting as
▼
Factor
Rewrite

Rewrite as

FactorOut

Factor out

FactorOut

Factor out

The polynomial has been completely factored. Next, by the Zero Product Property, at least one of the four factors is equal to zero. This produces four linear equations that can be solved for
Consequently, the solutions to the polynomial equation, and therefore the intercepts, are and
b To find the values missing from the table, evaluate at the corresponding value. For example, begin by substituting
▼
Substitute for and simplify
Substitute

CalcPowProd

Calculate power and product

Multiply

Multiply

AddSubTerms

Add and subtract terms

The remaining missing values can be found in a similar way.
Result

Consequently, the complete table of values looks as follows.

c To determine the end behavior of the graph of a polynomial, start by identifying the leading term.
Since the degree is which is an even number, both ends of the graph have the same behavior. Additionally, since the leading coefficient is which is a positive number, the end behavior of the graph of is up-up.
d To determine which of the graphs is correct, plot the graph of by using the information obtained in the previous parts.
  1. Begin by plotting the intercepts from Part A.
  2. Next, graph the points from the table of values in Part B.
  3. Connect the points using a smooth curve.
  4. Extend both ends of the graph upwards, since the end behavior of the graph of was found to be up-up in Part C.
Highlighting the erroneous parts from the given graphs
Comparing the graph obtained with the given ones, it can be concluded that the graph of is Graph
Highlighting the erroneous parts from the given graphs
Discussion

Locating the Intercepts of a Graph

Given a polynomial function, the end behavior of the graph can be determined directly from the function rule without doing anything to it. For example, consider the following function.
f(x)=8x^3 - 12x^2 - 26x + 15
The end behavior of the graph of is down-up. In contrast, the zeros are not easy to identify simply by looking at the function rule. Usually, the polynomial must be factored first. However, not all polynomials have a suitable factorization. Therefore, it is worth considering whether there is a simple way to estimate the zeros of a function. The following principle answers this.
Rule

The Location Principle

Let be a polynomial function and and be two real numbers. If and have opposite signs, then there is at least one zero between and

A polynomial graph having one x-intercept. A point of the curve being above the x-axis is marked and another point of the curve below the x-axis is marked. The x-intercept is highlighted.

Notice that if and have opposite signs, the principle guarantees one zero between and but there could be more. Conversely, the fact that and have the same sign does not imply that there are no zeros between and

Proof

 Informal Justification

Let and be two real numbers with and let be a polynomial function such that and A sketch of the graph of around and around could look as follows.

A short curve passing through (a,f(a)) and another short curve passing through (b,f(b)).
Since graphs of polynomial functions are continuous — that is, they have no gaps — the points and must be connected by a smooth curve that will cross the axis at least once, implying that there is at least one zero between and
Connecting (a,f(a)) with (b,f(b)) with a curve that crosses the x-axis.

For it can be found that and Notice that and have opposite signs. Consequently, has at least one zero between and

A coordinate plane. Segment between x=0 and x=1 highlighted.

The interval where the zero is located can be reduced by considering closer testing values.

A coordinate plane. Segment between x=0.25 and x=0.75 highlighted.
All real zeros of a polynomial can be estimated in this fashion.
Pop Quiz

Locating Zeros

Use the Location Principle to indicate whether the given polynomial function is guaranteed to have a zero in the indicated interval.

Random Polynomial Functions and random testing values
Discussion

Maximum and Minimum Points of a Graph

Sometimes knowing that a graph passes through a point does not give enough information about how the curve is before and after the point. However, when the point is either a maximum or a minimum, there is a clear view of the behavior of the graph around it.

Concept

Relative Minimum and Maximum

The value is a relative minimum, or local minimum, of a function if is the least output of around Likewise, the value is a relative maximum, or local maximum, of a function if is the greatest output of around

Relative extrema of a cubic function
If the function is continuous, the function switches from increasing to decreasing at a relative maximum or from decreasing to increasing at a relative minimum.
Sometimes the phrase relative extrema is used to refer to both relative maximums and relative minimums. Note that a function can have one or more relative extrema, or none at all.

When a relative extrema is greater or lower than any other point in the graph, it has a particular name.

Concept

Absolute Minimum and Maximum

The absolute minimum, or global minimum, of a function is the least output in its whole domain.

Absolute minimum of a quartic function 0.3*(x+3)*(x+2)*(x+1)*(x-1) located at (0.326345,-2.07423)

The absolute maximum, or global maximum, of a function is defined in a similar way. It is the greatest output of the function in its whole domain.

Absolute minimum of a quartic function -0.3*(x+2.5)*(x+1)*(x)*(x-2) located at (1.3,2.38602)

The absolute maximum of a function is also a relative maximum, and the absolute minimum is also a relative minimum. If a function increases indefinitely, it does not have an absolute maximum. Likewise, if a function decreases indefinitely, it does not have an absolute minimum. The function might still have relative extrema.

Different functions with different extrema
Discussion

Turning Points of Polynomial Functions

Relative or absolute extrema are very useful when graphing functions, but the method for finding them is beyond the scope of this lesson. That said, when working with polynomial functions, there is a way to estimate the maximum number of relative extrema by using the turning points of the graph.

Concept

Turning Point

A turning point is a point where the graph of a function changes from increasing to decreasing or vice versa. In other words, the turning points correspond to the relative extrema of the graph of a function.

Graph of f(x)=x^5 + 0.5 x^4 - 23.5 x^3 + 7 x^2 + 60 x

The graph corresponds to a fifth-degree polynomial with four turning points. When dealing with polynomial functions, there is a close connection between the degree, the zeros, and the turning points.

  1. The graph of a polynomial function of degree can have at most turning points.
  2. If a polynomial function has distinct real zeros, then its graph has exactly turning points.
Pop Quiz

Determining the Maximum Number of Turning Points

Consider the given polynomial function. Determine either the maximum or the exact number of turning points that its graph may have.

Either the function rule or the graph of a polynomial function.
Example

Graphing a Fifth-Degree Polynomial Function

After purchasing the image editing software, Emily was curious about the online software she used to plot the polynomials and wanted to try graphing a polynomial by hand. To challenge herself, she chose the polynomial function So far, she has created the following table of values.

She next plans to find the zeros of the polynomial and determine the end behavior of the graph. She also knows that and are turning points. Emily thinks it is enough information to make an accurate sketch of the graph. Follow Emily's plan and graph

a Find all the zeros of the polynomial function.
b What is the end behavior of the graph of
c Sketch of the graph of

Answer
a Zeros:
b End behavior: Up-Down
c Graph:
Graph of f(x)= -0.5 x^5 - 3.5x^4 + 91x^3 + 374x^2 - 3556x - 7840

Hint
a Use the Location Principle.
b Take a look at the degree and the sign of the leading coefficient of the polynomial.
c Use the fact that the zeros are the intercepts of the graph. Plot the turning points. Connect all the points with a smooth curve and extend it according to the end behavior.

Solution
a One way of finding the zeros of the function is by factoring the given polynomial. However, it does not seem easy to factor. Alternatively, the Location Principle can help locate where the zeros could be. In the given table, highlight the positive values with one color and the negative values with a different color.
The first sign change occurs between and Therefore, there is a zero between those two values. For simplicity, test the integer values within this interval. For example, start with
▼
Substitute for and evaluate
Substitute

CalcPowProd

Calculate power and product

Multiply

Multiply

AddSubTerms

Add and subtract terms

Since it means that is a zero of From the table, the second sign change happens between and As before, test the integer values between and

The polynomial has a second zero at The remaining zeros can be found by applying a similar reasoning.

  • The third sign change occurs between and
  • The fourth sign change occurs between and
  • The fifth sign change occurs between and

After testing some integers that belong to each of the mentioned intervals, the remaining zeros can be found at and

Zeros

Since the polynomial has degree five, it has no more zeros.

b The end behavior of the graph of a polynomial function can be determined by the sign of the leading coefficient and the degree of the polynomial.
Leading Coefficient Degree End Behavior
Positive Even Up-Up
Positive Odd Down-Up
Negative Odd Up-Down
Negative Even Down-Down

For the given polynomial, the leading term is The leading coefficient is negative and the degree is odd. Consequently, the end behavior of the graph of is Up-Down.

c Since the zeros of a polynomial are the intercepts, start by plotting these points on the coordinate plane. It also may help to make a sketch reflecting the end behavior of the graph.
x-intercepts and two arrows reflecting the end behavior.

Now, graph the points from the table.

Points from the table, x-intercepts, and two arrows reflecting the end behavior.

Before connecting the points with a smooth curve, plot the turning points and At these points, the graph changes from increasing to decreasing or vice versa.

Points from the table, x-intercepts, and turning points plotted. Two arrows reflecting the end behavior.

According to the distribution of the points, from left to right, the first turning point is a relative minimum, the second one a relative maximum, the third another relative minimum, and the fourth and final one another relative maximum. There are no absolute extrema. With all this information, the graph can finally be drawn.

Graph of f(x) = -0.5 x^5 - 3.5x^4 + 91x^3 + 374x^2 - 3556x - 7840
Discussion

Even and Odd Functions

Some functions have the property that their graphs look the same on both sides of the axis, as if the axis were a mirror. Such a property can be easily identified once the graph is drawn but, can it be deduced algebraically? The next concepts can answer this question.

Concept

Even Function

An even function is a function for which the value of is equal to the value of for all the values in its domain. That is, opposite inputs have the same output.


The graph of an even function is symmetric about the axis. The functions  and are two examples of even functions.
Graphs of f(x)=x^2 and graph of g(x)=2|x|
Notice that if a function is even and the point is on the graph, then the point is also on the graph.

Extra

 Determining If a Function is Even
To determine algebraically whether a function is even, substitute into the function rule and simplify. If the resulting expression is equal to then the function is even; otherwise, it is not. For example, consider the following function.
Substitute for and simplify.
Substitute

NegBaseToPosPow

NegBaseToPosBase

Substitute

Since the given function is even.

Some other functions seem to be reflected across the origin. Functions with this property are called odd functions.

Concept

Odd Function

An odd function is a function for which the value of is equal to the value of for all the values in its domain. It is like if the function allows moving the negative sign from the input to the output.


The graph of an odd function is symmetric about the origin, meaning that the graph looks the same after a rotation about the origin. The functions  and are two examples of odd functions.
Graphs of f(x)=x and graph of g(x)=x^3
Notice that if a function is odd and the point is on the graph, then the point is also on the graph.

Extra

 Determining If a Function is Odd
To determine algebraically whether a function is odd, substitute into the function rule and simplify. If the resulting expression is equal to then the function is odd; otherwise, it is not. For example, consider the following function.
Substitute for and simplify.
Substitute

NegBaseToPosBase

MoveNegNumToFrac

Put minus sign in front of fraction

Substitute

Since the given function is odd.

Be aware that these concepts are exclusive — in other words, if a function is even, it cannot be odd, and vice versa. However, unlike integers, a function may be neither even nor odd.

f(x) can be either even, odd, or neither
Notice that if a function is even or odd, it is enough to make the graph only for positive values because the graph for negative values will be its reflection either across the axis or across the origin.
Pop Quiz

Classifying Functions

Determine if the given function is even, odd, or neither.

Either the function rule or the graph of a polynomial function.
Closure

Verifying End Behavior

At the beginning of the lesson, Emily graphed the polynomial function using a table of values, but she was not so sure about the end behavior of the graph. She concluded that the end behavior was "Up-Down."

Curve connecting the points from the table. The left part of the curve is extended upwards and the right-part is extended downwards.
Since Emily learned that the end behavior of the graph of a polynomial can be deduced from the function rule, she can now verify whether her gut was correct or not.
The leading coefficient is positive and the degree is even. Therefore, the end behavior is up-up. Consequently, Emily's graph was not correct. She had to extend the right part upward, and not downward.
Graph of f(x)=x^4 - 4x^3 - 39x^2 + 46x + 80.
Notice that if Emily had determined the end behavior of the graph from the function rule first, she probably would have made a table of values that included whole numbers greater than This would have led her to graph correctly in her first attempt.


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