HintIntroduction
In the modules on graphing functions which follow, you will become much more familiar with polynomial, rational, and other functions. A function, y = f (x), has one and only one output y for each input x. Note that the output can be written either as y or f (x). Since it is desirable to have only one output for a given input, all of the equations in what follows will be functions. The computer is an excellent tool for allowing you to quickly generate the graphs of a large variety of functions. There will be many carefully chosen questions following each graph to direct you toward an understanding of how functions behave and how changing their equations changes their graphs. You will also learn to classify functions by symmetry, shape, and a variety of other properties.
Have Fun!
A polynomial function y = P(x) has non-negative integer exponents and real coefficients, it is written P(x) = anxn + an-1xn-1 + ... + a1x + a0 , where the an , a n-1 , ... , a0 are the coefficients. Polynomials are defined for all real values of x. The operators (+) and (-) separate the terms. The highest power of x in a polynomial is called the degree of the polynomial. The coefficient of the highest degree term is called the leading coefficient. Polynomials have graphs which are smooth and have no breaks in them.
Test your ability to correctly identify a polynomial function DEFINITIONS and then take the TEST.
In this module, you will investigate polynomial functions to determine their
tail behavior
their x- and y-intercepts and
how the degree affects their graphs.
Critical Thinking Questions
1. We will work initially with the graphs of three polynomial functions. These are
y 1 = x3
y2 = x5
y3 = x7
a. Which point(s) do the graphs have in common?
b. On which intervals would y = x9 be above the other graphs, and on which intervals would it be below the other graphs?
2. Now we will work with the graphs of the following polynomial functions
y 1 = x2
y2 = x4
y3 = x6
a. Which point(s) do the graphs have in common?
b. On which intervals would y = x9 be above the other graphs, and on which intervals would it be below the other graphs?
3. Now let
y 1 = x3 + x2
y2 = x3
y3 = -2x3 + x2
a. Determine all those among the functions y1 , y2 and y 3 which point the same way on their left and right ends?
This is called tail behavior.
b. Identify which part of the equation determines this behavior.
4. Let
y 1 = x4 + x3
y2 = -2x4 + x3
y3 = x4
a. Which of y1 , y2 and y3 points the same way on its left and right ends?
b. Identify which part of the equation determines this behavior.
5. Investigate the tail behavior of a polynomial function by trying enough examples so that you can fill in the chart below.
Draw up or down arrows in the table below that show how the right and left tails of a polynomial function point depending on whether the degree is even or odd and on whether the leading coefficient is positive or negative.
| . |
|
|
| Positive leading coefficient | . | . |
| Negative leading coefficient | .. | . |
6. To find the x- and y-intercepts of a polynomial function, what would you do?
7. Find the x-intercepts of y = x4 - 5x2 + 4 .
8. Given the function y = x6 + x3 - 3x2 - 10
a. Can you find the x-intercepts algebraically?
b. How do you think you can find the x-intercepts?
c. What are the x-intercepts?
Skill Exercises
1. A function with even degree and leading coefficient negative, has tails which point _____ , and when the leading coefficient is positive, it has tails which point _____ .
2. A function with odd degree, and leading coefficient positive, has left tail which points _____ and right tail which points _____ ; when its leading coefficient is negative, it has left tail which points _____ and right tail which points _____
3. Give an example of a polynomial function having all of the following properties
degree seven;
four terms;
passes through the origin; and
left tail pointing downward.
What must its right tail do?
4. Give an example of a polynomial function having all of the following properties
degree four;
three terms;
y-intercept - 5; and
right tail points downward.
What must its left tail do?
5. Given f (x) = -x3 - 3x2 + 2x - 1
a. Which way do the tails point?
b. Find f (-1) .
c. Which point do you know is on the graph?
d. Find f (0) .
e. For any polynomial function, f (0) gives what special value?
6. Consider the polynomial function f(x) = 16x2 - x4
a. What are its x- and y-intercepts?
b. Sketch a graph which shows the correct tail behavior and the intercepts.
7. Consider the polynomial function f(x) = x3 + 2x2 - 4x - 8 (Recall factoring by grouping)
What are its x- and y-intercepts?
8. Consider the polynomial function y = x4 - 64x2 + 8
What are its x- and y-intercepts? Hint
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This document was last modified on 11-Sep-00.
