Analyzing Graphs of Polynomials: Steps, Quadratic & Examples

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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.

Example Showing End Behaviour of Polynomials, Jordan Madge- StudySmarter Originals

Above is the graph of $f (x) = x^{2} - 3 x - 4$ . 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, Geogebra Graph showing End Behaviour of cubic polynomial, Jordan Madge- StudySmarter Originals

Above is the graph of $f (x) = x^{3} - 3 x - 2$ . 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 $x \to \infty$ . So, in the above example, we could instead write: as $x \to \infty$ , $f (x) \to \infty$ and as $x \to - \infty$ , $f (x) \to - \infty$

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 $x \to \infty$ or $x \to - \infty$ ), 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.

Case	Degree	Leading Coefficient	End Behaviour	Example
1	Even	Positive	As $x \to \infty$ , $f (x) \to \infty$ As $x \to - \infty$ , $f (x) \to \infty$
2	Even	Negative	As $x \to \infty$ , $f (x) \to - \infty$ As $x \to - \infty$ , $f (x) \to - \infty$	End Behaviour of Even Function, Jordan Madge- StudySmarter Originals.
3	Odd	Positive	As $x \to \infty$ , $f (x) \to \infty$ As $x \to - \infty$ , $f (x) \to - \infty$	End Behaviour of Odd Function, Jordan Madge- StudySmarter Originals.
4	Odd	Negative	As $x \to \infty$ , $f (x) \to - \infty$ As $x \to - \infty$ , $f (x) \to \infty$	End Behaviour of Odd Function, Jordan Madge- StudySmarter Originals.

Determine the end behaviour of the polynomial function $f (x) = x^{2} + 3 x + 2$ .

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 $x \to \infty$ $f (x) \to \infty$ and as $x \to - \infty$ , $f (x) \to \infty$ .

Determine the end behaviour of the polynomial $f (x) = - x^{4} + 3 x^{3} + x - 2$ .

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 $x \to \infty$ , $f (x) \to - \infty$ and as $x \to - \infty$ , $f (x) \to - \infty$ .

Determine the end behaviour of the polynomial $f (x) = 2 x^{3} + 7 x^{2}$ .

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 $x \to \infty$ , $f (x) \to \infty$ and as $x \to - \infty$ , $f (x) \to - \infty$ .

Determine the end behaviour of the polynomial $f (x) = - 7 x^{5} + 3 x^{2} - 1$

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 $x \to \infty$ , $f (x) \to - \infty$ and as $x \to - \infty$ , $f (x) \to \infty$ .

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 $p (x) = (x - 1) (x - 2) (x - 3)$

Solution:

Equating p(x) to zero, we obtain $(x - 1) (x - 2) (x - 3) = 0$ .

Solving for x, we get $x = 1, x = 2, x = 3$ .

Find the zeros of the polynomial function $p (x) = (x + 2) (x + 1)$ .

Solution:

Equating p(x) to zero, we obtain $(x + 2) (x + 1) = 0$ .

Therefore, $x = - 2, - 1$

Locating zeros

Suppose we have the polynomial function $p (x) = - x^{2} + 2 x + 1$ . If we work out $p (2)$ , we get $p (2) = {- (2)}^{2} + 2 (2) + 1 = 1$ which is positive. If we work out $p (3)$ , we get $p (3) = - {(3)}^{2} + 2 (3) + 1 = - 2$ which is negative.

The location principle states that for the polynomial function $p (x)$ , if $p (a) < 0$ and $p (b) > 0$ , 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 $f (x) = x^{2} + 3 x + 2$ . If we look at $f (- 1.5),$ we see it is negative. If we look at f(0), we see it is positive. Clearly, there must be a zero between $x = - 1.5$ and $x = 0$ 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 $p (x) = x^{3} + 3 x + 1$ , there is a root between $x = - 1$ and $x = 0 .$

Solution:

$p (- 1) = - 1 - 3 + 1 = - 3,$

$p (0) = 0 + 0 + 1 = 1$

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 $f (x) = x^{3} - 1$ .

Solution:

Step 1: Determine any zeros.

Equating the function to zero, we obtain $x^{3} - 1 = 0$ . We can factorise this to get $(x - 1) (x^{2} + x + 1) = 0$ and we, therefore, get $x = 1$ as the only real zero.

Step 2: Draw up a table of values.

x	-1	0	1	2	3
f(x)	-2	-1	0	7	26

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 $x \to \infty$ , $f (x) \to \infty$ and as $x \to - \infty$ , $f (x) \to - \infty$ .

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 $f (x) = f (- x)$ for all values of x. An odd function occurs when $- f (x) = f (- x)$ .

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

Above is the graph of $y = x^{4} - 3$ . 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 $f (x) = f (- x)$ .

Odd and Even Functions, Example, Geogebra

Odd and Even Functions Example, Jordan Madge- StudySmarter Originals

Above is the graph of $y = \frac{1}{2} x^{5}$ . We can see that it is an odd function because $- f (x) = f (x)$ 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 $(0, 0) .$ 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 $f (x) = f (- x)$
Odd functions are functions where $f (- x) = - f (x)$
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.

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Frequently Asked Questions about Analyzing Graphs of Polynomials

How do you write an analysis of a graph?

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.

What are the characteristics of a polynomial graph?

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

What are the examples of graphs of a polynomial equation?

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

How to sketch graphs of a polynomial equation?

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

How to analyze a graph of a polynomial function?

Determine turning points, intercepts, zeroes and end behaviour.

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Analyzing Graphs of Polynomials

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End Behaviour of Polynomial Functions

What Determines the End Behaviour of a Polynomial Function?