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Understanding Basic Polynomial Graphs

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  • 0:02 Polynomial Graphs
  • 1:44 Exponents: Even of Odd
  • 3:14 Exponents: Increasing
  • 4:24 Leading Coefficients
  • 6:30 Turning Points
  • 8:04 Lesson Summary
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Lesson Transcript
Instructor: Maria Airth

Maria has a Doctorate of Education and over 15 years of experience teaching psychology and math related courses at the university level.

This lesson will cover understanding basic polynomial graphs. The lesson focuses on how exponents and leading coefficients alter the behavior of the graphs.

Polynomial Graphs

Hello and welcome to this lesson on how to mentally prepare for your cross-country run. Wait! What? No! This isn't supposed to be about running? Oh, that's right, this is Understanding Basic Polynomial Graphs. But, you can think of a graph much like a runner would think of the terrain on a long cross-country race. There may be parts that are steep or very flat. One minute you could be running up hill, then the terrain could change direction, and you are suddenly running downhill.

When visualizing the possible route you will run, you know two things for sure. One is that you will not be allowed to stop running on the track (until your race is finished, that is). The track is continuous, meaning there are no breaks in it. The other is that your run will be smooth: there are no sharp corners in the route. Amazingly, all of these things are true of polynomial graphs as well!

As you can see by these graphs of polynomials, some are steep, like a run through high hills, and some have many ups and downs:

Examples of graphs of polynomials
examples of polynomial graphs

But, none have breaks and none have sharp, angular turns. The basic shape of any polynomial function can be determined by its degree (the largest exponent of the variable) and its leading coefficient. In this lesson, we will investigate these two areas of the polynomial to get an understanding of basic polynomial graphs. We'll start with exponents.

Exponents: Even or Odd

Whenever you are dealing with polynomial functions, you will need to know the degree of the function. This is the highest exponent attached to any term. The degree of the polynomial f(x) = x^4 + 2x^3 - 3 is 4. It is called a fourth degree function.

Polynomial graphs behave differently depending on whether the degree is even or odd. In this example, the blue graph is the graph of the equation y = x^2:

Even degree function in blue; odd degree function in green
odd and even polynomial graphs

The graph of the function y = x^3 is drawn in green.

You can see that the even degree function (the blue line) starts and ends on the same side of the axis. This is true for all even degree functions: they start and end on the same side of the x-axis. The opposite is true for the odd degree function; odd degree functions start and end in opposite directions.

Which of these graphs represents an even degree function?

Example graphs
example graphs

Did you choose the top left and bottom right? Good work! It does not matter how many curves a line has, as long as it starts and ends on the same side of the x-axis, it is an even degree polynomial.

Exponents: Increasing

Beyond the even or odd behavior, the degree level of exponents also changes the look of the graph. Take a look at this image of three even degree functions:

Even degree functions
three even degree function graphs

These are the simplest form of functions with just a variable and exponent involved. When x is any other number, the graph behaves differently.

Our foundational function is y = x^2, and this shows the smoothest curve. As the degree increases, the graph flattens at the bottom and once it starts to rise, it does so in an increasingly steep manner (you might even say it steepens exponentially). It is interesting to note that for all of these simple functions, when x is either 1 or -1, y is always 1.

The same is true for odd degree polynomial graphs. In their simplest form, they all share the same coordinates at x = 1 and -1. Outside of these two points, the higher the degree, the flatter the graph around zero and the steeper the rise (or fall).

Leading Coefficients

Exponents are not the only aspects of polynomials that can have an effect on the graph of the function. A leading coefficient (which is a coefficient attached to the degree term of the polynomial) also has a marked impact on the behavior of the graph. The function f(x) = ax^n is called the power function.

So far, we have only seen positive functions because we have been working with just the variables and exponents. This assumes a positive one as the leading coefficient. But, when we start to work with the true power functions and stop assuming a positive one for the leading coefficient we have to wonder, what if the leading coefficient is negative?

In these graphs, we can see that negative coefficients flip the graph:

Graph with negative coefficients
negative coefficient graphs

That is all they do. How else can a leading coefficient impact the graph of a polynomial function?

Well, in this example, you can see our original 'simplest form odd function' in blue:

All simple, odd polynomials go through (1, 1) and (1, -1).
odd degree graphs

Notice the uniform points of (1, 1) and (1, -1). Remember, all simple, odd polynomials go through these points. Now, look more closely at the green line.

The leading coefficient is 1/2, and it does not pass through the uniform points. Instead, it seems a bit stretched out between the x = 1 and x = -1 region compared to our simple function. Conversely, the pink line with a larger coefficient shows a pinched graph, rising closer to the y-axis.

This pattern will repeat with all leading coefficients in both even and odd degree functions: leading coefficients of values between -1 and 1 will result in graphs that rise (or fall) further from the y-axis; and leading coefficients outside of this region, will result in graphs that rise (or fall) increasingly closer to the y-axis.

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