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Analyzing Graphs of Polynomials

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Analyzing Graphs of Polynomials

Given a graph of a polynomial function, we are able to observe several properties. For example, we may be able to determine any zeros or turning points the function may have. Moreover, we may be interested in determining the end behaviour of the function, or whether it is an odd or even function. In this article, we will go through the steps involved in analysing the graphs of polynomials.

End Behaviour of Polynomial Functions

Given a polynomial function, the end behaviour is what happens to the graph as x goes towards the boundaries of the domain. If we sketch the graph of a polynomial function, the end behaviour is what happens to the graph as we approach the "ends" of the real axis.

End Behaviour of Polynomials, Graph Showing End Behaviour, GeogebraExample Showing End Behaviour of Polynomials, Jordan Madge- StudySmarter Originals

Above is the graph of . Here we can see that as x gets larger and larger, the graph goes up. We say that as x tends to infinity, the function tends to infinity. Similarly, we say that as x approaches negative infinity, the function approaches positive infinity since as x gets smaller and smaller the graph also goes up.

End Behaviour of Polynomial Functions, Graph Showing End Behaviour, GeogebraGraph showing End Behaviour of cubic polynomial, Jordan Madge- StudySmarter Originals

Above is the graph of . Here we can see that as x approaches positive infinity, the function approaches positive infinity, however as x approaches negative infinity, the graph approaches negative infinity.

A shorthand version of writing "x tends towards infinity" is. So, in the above example, we could instead write: as, and as,

What Determines the End Behaviour of a Polynomial Function?

The degree of a polynomial is the highest power that a polynomial has. For example, the polynomial function is a degree 5 polynomial. The leading coefficient of a polynomial function is the term with the highest degree in the polynomial. So, for the polynomial, the leading coefficient is 7 and the degree is 5.

As x gets really big or really small (as or ), the leading coefficient becomes significant because that term will take over and grow significantly faster compared with the other terms. Therefore, to determine the end behaviour of a polynomial, we only need to look at the degree and leading coefficient to draw a conclusion. There are four possible scenarios.

CaseDegreeLeading CoefficientEnd Behaviour Example
1Even Positive As , As ,

End Behaviour of Polynomials, Even function, Geogebra End Behaviour of Even Function, Jordan Madge- StudySmarter Originals.

2EvenNegativeAs , As ,

End Behaviour of Polynomials, Even function, Geogebra

End Behaviour of Even Function, Jordan Madge- StudySmarter Originals.
3OddPositiveAs , As ,

End Behaviour of Polynomials, Odd function, Geogebra

End Behaviour of Odd Function, Jordan Madge- StudySmarter Originals.
4OddNegativeAs , As ,

End Behaviour of Polynomials, Odd function, Geogebra

End Behaviour of Odd Function, Jordan Madge- StudySmarter Originals.

Determine the end behaviour of the polynomial function.

Solution:

Here, the degree is 2 which is even and the leading coefficient is 1 which is positive. Therefore, we have case 1 and so as and as, .

Determine the end behaviour of the polynomial.

Solution:

Here, the degree is 4 which is even and the leading coefficient is -1 which is negative.

Therefore, we have case 2 and so as, and as, .

Determine the end behaviour of the polynomial.

Solution:

Here, the degree is 3 which is odd and the leading coefficient is 2 which is positive. Therefore, we have case 3 and so as , and as, .

Determine the end behaviour of the polynomial

Solution:

Here, the degree is 5 which is odd and the leading coefficient is -7 which is negative. Therefore, we have case 4 and so as , and as, .

Zeros of Polynomial Functions

The zeros of a polynomial are the x values that make the polynomial equal to zero. Often, we denote a polynomial function using the notation p(x). Therefore, the zeros can be found by equating p(x) to zero and solving for x. The zeros are also the x-intercepts of the function.

Find the zeros of the polynomial function

Solution:

Equating p(x) to zero, we obtain.

Solving for x, we get .

Find the zeros of the polynomial function.

Solution:

Equating p(x) to zero, we obtain.

Therefore,

Locating zeros

Suppose we have the polynomial function. If we work out , we getwhich is positive. If we work out , we get which is negative.

The location principle states that for the polynomial function, if and , then there must be a zero between a and b.

Locating Zeroes, Example, Geogebra Locating Zeroes Example, Jordan Madge- StudySmarter Originals

Above is the graph of . If we look at we see it is negative. If we look at f(0), we see it is positive. Clearly, there must be a zero between and because the graph must cross the x-axis at some point in order to go from being negative to positive. This is the theory behind the location principle. It is really useful for graphs where it may be more difficult to locate the zeros using conventional methods for solving, such as quartics (order 4 polynomials), quintics (order 5 polynomials) or higher-order polynomials.

Use the location principle to show that for the function , there is a root between and

Solution:

Graphing Polynomial Functions

In this section, we will put some of what we have already discussed together to graph polynomial functions. To graph any polynomial function, there are four main steps:

Step 1: Determine any zeros that the graph may have.

Step 2: Draw up a table of values.

Step 3: Determine the end behaviour of the polynomial function.

Step 4: Use the above information to draw out the graph.

Sketch the graph of the polynomial function .

Solution:

Step 1: Determine any zeros.

Equating the function to zero, we obtain . We can factorise this to get and we, therefore, get as the only real zero.

Step 2: Draw up a table of values.

x-10123
f(x)-2-10726

Above I have chosen some values of x and worked out the corresponding values of f(x).

Step 3: Determine the end behaviour of the polynomial function.

This polynomial has an odd degree and the leading polynomial is 1 which is positive. Therefore, we have as , and as, .

Step 4: Use the above information to draw out the graph.

Graphing Polynomials, Example, Geogebra

Graphing Polynomials Example, Jordan Madge- StudySmarter Originals

Odd and Even Functions

Once we have drawn the graph of our polynomial function, we may wish to determine whether it is an odd or even function. An even function occurs when we have symmetry about the y-axis. In other words, a function is even when for all values of x. An odd function occurs when .

Odd and Even Funcitons, Example, Geogebra Odd and Even Functions Example, Jordan Madge- StudySmarter Originals

Above is the graph of . We can see that it is an even function because there is symmetry about the y-axis. Whenever we have symmetry about the y-axis, we have that .

Odd and Even Functions, Example,  Geogebra

Odd and Even Functions Example, Jordan Madge- StudySmarter Originals

Above is the graph of . We can see that it is an odd function because for all values of x.

Turning Points of Polynomial Functions

A turning point on a graph is a point where the graph goes from increasing to decreasing. In other words, the graph will change from going “upwards“ to “downwards” or vise versa. The local maxima are specifically when the graph changes from increasing to decreasing. The local minima are specifically when the graph changes from decreasing to increasing.

The plural for maximum is maxima and the plural for the minimum is minima. The umbrella term for the two words is extrema.

Turning Points, Example, Geogebra

Turning Points Example, Jordan Madge- StudySmarter Originals

On the above graph, we can see a turning point that occurs at At this point, the graph goes from increasing to decreasing and so it is a local maximum.

Analyzing Graphs of Polynomials - Key takeaways

  • The end behaviour of a polynomial function is what happens to the graph as x approaches positive or negative infinity.
  • We can determine the end behaviour of any polynomial by looking at the leading coefficient and degree of the polynomial.
  • A zero of a polynomial function is the point where it crosses the x-axis. It can be determined by equating the polynomial function to zero and solving for x.
  • We can use the location principal to determine values that zeros lie in between.
  • Even functions are functions that have symmetry about the x-axis and
  • Odd functions are functions where
  • A turning point is a point when a function goes from increasing to decreasing or vice versa.
  • A local minimum occurs where a function goes from decreasing to increasing.
  • A local maximum occurs where a function goes from increasing to decreasing.

Frequently Asked Questions about Analyzing Graphs of Polynomials

First determine the end behaviour of the graph. Then work out any zeroes. Maybe work out if it is an odd or even function and determine any turning points the graph has. 

It may have turning points, it may be increasing or decreasing, it may have some zeros, as x and y intercepts. 

Quadratic graphs, cubic graphs, quartic graphs and so on. 

Determine any zeros, draw a table of values, determine the end behaviour, draw out the graph using the information gathered. 

Determine turning points, intercepts, zeroes and end behaviour. 

Final Analyzing Graphs of Polynomials Quiz

Question

How can we tell if a line will be dotted or solid for an inequality?

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Answer

If the inequality is just less than or greater than, then the line is dotted. If we have \(\le\) or \(\ge\) then we are looking at a solid line as the inequality can also be equal to.

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Question

What does the variable k govern in the turning point formula?

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Answer

Vertical translation of the graph and the y-coordinate of the turning point vertex.

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Question

What does the variable h govern in the turning point formula?

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Answer

Horizontal translation of the graph and the x-coordinate of the turning point vertex.

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Question

What does the variable \(a\) govern in the turning point formula?

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Answer

concavity and dilation

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Question

What is a cubic function? State the general form of a cubic equation. 

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Answer

A cubic function is a polynomial of degree three. The general form of a cubic equation is ax3 + bx2 + cx + d = 0 where a  0.

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Question

Do cubic functions have axes of symmetry? 

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Answer

No 

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Question

What are the first two steps when plotting cubic functions? 

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Answer

Find the x and y intercepts 

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Question

For the cubic function y = ax3, what does varying the coefficient a do to the graph? 

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Answer

Varying the coefficient a changes the y-direction of the standard cubic graph

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Question

For the cubic function y = x3 + k, what does varying the coefficient k do to the graph? 


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Answer

Varying the coefficient k moves the cubic graph up or down the y-axis by k units. 

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Question

For the cubic function y = (x - h)3, what does varying the coefficient h do to the graph? 



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Answer

Varying the h moves the basic cubic along the x-axis by h units

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Question

For the cubic function y = ax3, when a is large, what happens to the cubic graph? 

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Answer

The cubic graph becomes steeper

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Question

For the cubic function y = ax3, when a is small, what happens to the cubic graph? 


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Answer

The cubic graph becomes flatter

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Question

For the cubic function y = ax3, when a is negative, what happens to the cubic graph? 


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Answer

The cubic graph becomes inverted

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Question

For the cubic function y = x+ k, when k is positive, what happens to the cubic graph? 


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Answer

The cubic graph moves up the y-axis

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Question

For the cubic function y = x+ k, when k is negative, what happens to the cubic graph? 

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Answer

The cubic graph moves down the y-axis

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Question

For the cubic function y = (x - h)3, when h is positive, what happens to the cubic graph? 

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Answer

The cubic graph moves to the right of the x-axis

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Question

For the cubic function y = (x - h)3, when h is negative, what happens to the cubic graph? 

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Answer

The cubic graph moves to the left of the x-axis

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Question

How many turning points do cubic graphs have? What are the properties of these turning points?

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Answer

Cubic graphs have two turning points. A maximum point and a minimum point.

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Question

What are the three steps in plotting cubic functions? 

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Answer

  1. Find the x-intercepts
  2. Locate the y-intercepts
  3. Create a table of values
  4. Identify the minimum and maximum points
  5. Sketch the graph 

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Question

What is the Location Principle in the context of a cubic equation?

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Answer

Suppose y = f(x) is a cubic polynomial function. If a and b are two values such that f(a) < 0 and f(b) > 0, then the function has at least one real zero between a and b.

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