Back To CourseAlgebra II: High School
23 chapters | 203 lessons
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Maria has a Doctorate of Education and over 15 years of experience teaching psychology and math related courses at the university level.
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:
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.
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:
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?
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.
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:
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).
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:
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:
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.
Turning points in a graph are the points at which a graph changes direction. This could mean that a graph curving upwards, begins to go down or vice versa.
This is a basic lesson in understanding polynomial graphs, so I am not going to review exactly how to graph high degree functions. However, I will let you know that a rule of thumb is that a polynomial graph will have at most one less turn than its degree power. This means that a fourth degree polynomial can have at most three turns. Note, that it can have less, just no more than three.
Consider these polynomial function graphs:
Which of these graphs should be made by the function y = 4x^5 + 5x^2 - x - 1? Well, the degree of the function is 5, which means that its graph can have no more than four turns.
The graphs c. and d. are both even degree polynomial graphs, so they can't be right. That leaves a. or b. Graph a. has exactly four turns, while graph b. actually has six (look closely at the section just to the left of the y-axis and you'll see the extra turns).
Six is too many turns, thus the answer must be a. Graph a. could possibly be the graph of the fifth degree function. To find out for sure, you will need to take further lessons on polynomial graphs.
We started this lesson thinking about a cross-country running track. Polynomial graphs resemble a meandering run through the country side with their hills and valleys and turns. In this lesson, we learned that:
So, maybe the next time you go for a cross-country run, you can ask for the map in polynomial graph form! Good luck.
Following this lesson, you should be able to:
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Back To CourseAlgebra II: High School
23 chapters | 203 lessons