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Remedial Precalculus32 chapters | 253 lessons | 1 flashcard set

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Lesson Transcript

Instructor:
*Yuanxin (Amy) Yang Alcocer*

Amy has a master's degree in secondary education and has taught math at a public charter high school.

Watch this video lesson to learn how you can use the graph of a polynomial to help you explain the polynomial function. Learn how to explain the behavior of the polynomial at the origin and at the far ends.

**Polynomial** functions are functions made up of terms composed of constants, variables, and exponents, and they're very helpful. They help us describe events and situations that happen around us. For example, the polynomial function *f*(*x*) = -0.05*x*^2 + 2*x* + 2 describes how much of a certain drug remains in the blood after *x* number of hours. Graphing this medical function out, we get this graph:

Looking at the graph, we see the level of the drug first increases and then decreases over time. After about 41 hours or so, the drug is fully gone from the blood. In math, we have two ways of describing our graph. The first is called **short run behavior**, which is the behavior of the graph around the origin, and the second is called **long run behavior**, which is the behavior of the graph at the far ends of the graph.

Let's look at these now with our medical graph. Mathematicians like to look into the behavior of polynomials at these two areas because it gives them an idea of what happens to functions at their extremes. This helps with making hypothetical predictions about what might happen when an extreme circumstance occurs.

Let's look first at the **short run behavior**. Again, this is the behavior around the origin. Looking at our graph, we see that around the origin, when we have an *x* value of 0, we already have some of the drug in the blood. In terms of what is really happening, at 0 hours, the patient received the shot of the drug.

Theoretically, according to the function, if we go back in time just a bit, the drug level in the blood will decrease to 0. If we go a bit further, the drug level drops to negative. We know in reality that after the drug's level reaches 0, it can't physically drop any farther. We also see that if we go a bit to the right, the drug's level in the blood increases. So, we see that the short run behavior of this function is that it increases over time.

The next behavior is the **long run behavior**. This is the behavior at the far ends of the graph. When I say far ends, I mean the far left and far right. Looking at our graph, we see that as we go to the far left and to the far right, the graph goes down to negative infinity. So, theoretically, the blood level of the drug just keeps dropping. So, we can say that the long run behavior of the drug is that it goes down to negative infinity. This is the kind of analysis that is required when looking at short run and long run behavior. Let's look at a couple more graphs.

This is a graph of the function *f*(*x*) = 3*x* - 4:

Looking close to the origin, which is the point (0, 0), we see that our graph intersects the *y*-axis at -4. We also see that the graph steadily increases around the origin as well. There we have the short run behavior. As for the long run behavior, if we go to the far left, we see that the graph goes down to negative infinity. Going to the far right, we see that the graph goes up to positive infinity.

So, the graph is telling us that when our *x* values are really small, the function's value is also very small; when the *x* values are really large, the function's value is also very large - and there we have the long run behavior. Really, finding the short run and long run behavior of a graph is pretty straightforward after graphing the function.

Let's look at another function. This is the graph of the function *f*(*x*) = *x*^2 - 3:

Around the origin, it dips down to -3 when the *x* value is 0. From this point, the graph rises on both sides. This is the short run behavior. The long run behavior of this graph is that it goes towards positive infinity to the far left and far right - and there we have both the short run and long run behavior of this graph.

After analyzing different graphs, you might begin to see some patterns in the long run behavior of certain functions. You will notice that if we have a quadratic polynomial, a function of degree 2, and if the function is positive, then the long run behavior is that it goes towards positive infinity; if our quadratic polynomial is negative, then the long run behavior goes towards negative infinity. There are more patterns out there, too many to fit into this lesson. Keep analyzing and you will see more patterns emerge.

Let's review what we've learned. We've learned that **polynomial** functions are functions made up of terms composed of constants, variables, and exponents. They are very helpful because they describe events and situations that happen in the real world. The example we used was the blood level of a certain drug after *x* number of hours.

We learned that the **short run behavior** of a polynomial is the behavior around the origin. The **long run behavior** is the behavior at the far edges of the graph, the far left and far right. To analyze this behavior, we look at the graph and describe what we see.

The long run behavior is the one where you will see patterns emerge. One pattern is that for positive quadratic polynomials, the long run behavior goes towards positive infinity; for negative quadratic polynomials, the long run behavior goes towards negative infinity.

By the time this lesson is finished you should be able to:

- Identify a polynomial function
- Describe and identify short run and long run behavior of a graphed polynomial
- Analyze the graph of a polynomial

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Remedial Precalculus32 chapters | 253 lessons | 1 flashcard set

- Terminology of Polynomial Functions 5:57
- How to Evaluate a Polynomial in Function Notation 8:22
- Understanding Basic Polynomial Graphs 9:15
- Finding Intervals of Polynomial Functions 7:16
- Short Run & Long Run Behavior of Polynomials: Definition & Examples 6:05
- Pascal's Triangle: Definition and Use with Polynomials 7:26
- The Binomial Theorem: Defining Expressions 13:35
- Go to Polynomial Functions Basics

- Go to Continuity

- Go to Limits

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