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The Wonderful World of Polynomial Functions

When economic times are bad, people may move out of town, city, or state to seek employment in a different region. On the other hand, when a region experiences an economic boom, people from other areas tend to move there.

We want to analyze polynomial models of how the flow of population into and out of an area might be affected by its economic situation. Assume that the number of people moving away from Georgia in 2003 is modeled by the function E, and the number of people moving into Georgia is modeled by the function I. In each function below, x is the unemployment rate as a percent.

But, wait! We don’t know what a polynomial is! Do you know what a binomial is? How about a trinomial? What do you think polynomial means?______Let’s break down the word: poly- and –nomial. What does “poly” mean?______Okay, we’re halfway there!

A monomial is a numeral, variable, or the product of a numeral and one or more variables, i.e. -1, ½, 3x, 2xy, ___, ___

A constant is ______, i.e. ____, ____, _____

A coefficient is the numerical factor of a monomial or the ______in front of the variable in a monomial. Give some examples of monomials and their coefficients.

The degree of a monomial is the sum of the exponents of its variables. has degree 4. Why?______What is the degree of the monomial 3?_____ Why?______

Do you know what a polynomial is now? Give the definition. ______

Polynomials can be classified by the number of monomials, or terms, as well as by the degree of the polynomial. The degree of the polynomial is the same as the term with the highest degree. Complete the following chart. Make up your own for the last row.

Example / Degree / Name / No. of terms / Name
2 / Constant / monomial
/ Quadratic / binomial
/ Cubic
/ Quartic
/ Quintic / Trinomial

Classify E(x) and I(x) by degree and number of terms. E(x) = ______, I(x) = ______

The unemployment rate for March was 4.6%. What do the models give as the number of people moving out of and into Georgia? (Evaluate the models for x = 4.6.) E(x) = ______, I(x) = ______

What we are really interested in is the number of people left in Georgia once some have left and others have shown up. This “result” is called the net change. Subtract the number of people moving out from the number of people moving in for March. Did we gain more people than we lost? Or did we lose more than we gained? How can you tell?

If we wanted to tell if we were gaining or losing people each month of the year, doing all of these calculations might get old. There has to be a better way! And there is!!! Let’s subtract the polynomials themselves. After you subtract them, put the new polynomial in standard form, that is, the terms are written in descending order of degree.

I(x) – E(x) = (I – E)(x) = ______(Combine like terms.)

Now, let’s find (I – E)(4.6) = ______. Have you seen this number before? What does this represent? ______

Let n(x) = (I – E)(x). Find the value for n(0). ______. What does this represent? ______

Use function n(x) to complete the table below.

x / 0 / 2 / 4 / 6 / 8 / 10 / 12 / 14 / 16
n(x)

How do you know from the numbers if the population increased or decreased?______

For what values does the population increase? ______Decrease? ______

Graph all three of your functions – E(x), I(x), and n(x) – individuallyon your graphing calculator. Sketch each graph below. Clearly label the x- and y-intercepts. Write a few sentences comparing and contrasting the graphs.

Modeling polynomial functions: Actual unemployment from the past year for the state of Georgia is listed below. Use the different regression possibilities to determine which model – linear, quadratic, cubic, or quartic – is most appropriate to predict the unemployment rate from the month. (The closer is to 1, the better the model.) x = 0 represents November 2002. Give the equation of the best model.

x (month) / 0 / 4 / 7 / 8 / 9
Unemployment rate / 4.6 / 4.7 / 5.5 / 5.3 / 4.8

More on graphing polynomial functions

Four of the most basic polynomials are linear, quadratic, cubic, and quartic functions. Use your graphing calculator to help you sketch the graphs of,,, and. Describe the shape of each: line, U-shaped, S-shaped, or W-shaped.

Are all of these functions continuous? _____ Continuity is a property of all polynomial functions. Name at least one type of discontinuous function that we have studied this year. ______

There are a lot of pretty cool patterns that emerge when studying graphs of polynomial functions. Remember – mathematics is the science of patterns! Use your calculator to complete the following. (Suggestion: have just one person do the graphing)

Function / Degree / Number of U-turns
in the graph
/ 2 / 1
  • Make a conjecture (guess) about the degree of a function and the number of U-turns it has.______
  • Test your conjecture on the following 3 functions:, , and . Were you right? If not, make a new conjecture about the degree of a function and the number of U-turns. ______

______

  • In a parabola, the U-turn has a special name, ______. Every ______(same as the last blank) can be classified as either a ______or a ______.
  • Likewise, each U-turn in a polynomial occurs at either a (from last sentence) ______or a ______in regards to the other points around it. The plural for these terms are maxima and minima.
  • These points are also called turning points. At these points, the functions go from going down to going up, which is called (increasing or decreasing?)______, or from going up to going down, ______.
  • is graphed to the left. It has a local maximum and a local minimum. We call them local because they are the highest and lowest points, respectively, of all the points around them. If they were truly the highest and lowest points, they would be called the absolute maximum and the absolute minimum.
  • The minimum is _____ (y-value) and occurs at _____ (x-value).
  • The maximum is _____(y-value) and occurs at _____ (x-value).
  • The graph decreases until it reaches the minimum, then it increases until it reaches the maximum, and then it decreases again. (Think about this: something decreases until it reaches the lowest point. As the y-values increase, we say that the function is increasing.)
  • Intervals of decreasing: and ; interval of increasing: . Rewrite these last statements using interval notation. ______

Question: How can you make sure your calculator tells you the truth?

(Answer is at the bottom of the page.)

Graph the following on your calculator. Make a rough sketch next to each one and answer the following:

  1. Is the highest degree even or odd?
  2. Is the leading coefficient, the coefficient of the term of highest degree, positive or negative?
  3. Does the graph rise or fall on the left? On the right?

1. 7.

2. 8.

3.9.

4. 10.

5. 11.

6. 12.

Write a conjecture about the end behavior, whether it rises or falls at the ends, of a function of the form for each pair of conditions below. Then test your conjectures on the problems above and/or pg. 430 #51 – 58.

  1. when n is even and a > 0
  2. when n is even and a < 0
  3. when n is odd and a > 0
  4. when n is odd and a < 0
  • For the example at the top of the page: Because we know the leading coefficient and its effect on the end behavior, the graph rises on the left and falls on the right OR as , and as , .

Answer: By knowing how graphs are supposed to look and behave. (Did you really think it was going to be a riddle?!)