Module 4: POLYNOMIAL FUNCTIONS AND SYMMETRY
Exploring Calculus
Critical Thinking Questions
1. Consider the graphs of the three polynomial functions
y1 = x
y2 = x3 + 3x2
y3 = x3 + 3x
a. Which of the graphs do not display the same symmetry as the others?
b. Identify the part of the equation you think keeps it from having that symmetry.
2. Now let
y1 = x2
y2 = x4 + 4x2
y3 = x4 + 4x
a. Which one does not display the same symmetry as the others?
b. Which part of the equation do you think keeps it from having that symmetry?
3. Consider the following functions
y1 = x3 + 6x5 - 3x
y2 = x5 + 2x3
y3 = 18x
What do you think the graphs of the above functions have symmetry "with respect to"?
4. Now, consider the functions
y1 = x6 + 4x4 - 3x2
y2 = x4 - 3
y3 = 18x2
What do you think the graphs of the above functions have symmetry "with respect to"?
5. Why do you think polynomial functions which have symmetry with respect to the origin are said to have odd symmetry?
6. Why do you think polynomial functions which have symmetry with respect to the y-axis are said to have even symmetry?
Skill Exercises
1. Does y = x3 + 6x + 6 have symmetry with respect to the origin? Why or why not?
2. Does y = x4 + 5x2 + 65 have symmetry with respect to the y-axis? Why or why not?
3. Give an example of a function with at least four terms which is symmetric with respect to the origin and has left tail pointing up.
4. Give an example of a function with at least four terms which is symmetric with respect to the y-axis and has tails which point up.
5. Let f (x) = 2x4 - 3x2 + 1.
a. Find f (-x).
b. Find -f (x).
c. Does f (-x) = f (x)?
d. Does f (-x) = -f (x)?
e. Does f (x) have even, odd, or neither even nor odd symmetry? DEFINITIONS
6. Let g (x) = 4x - x2 .
a. Find g (-x).
b. Find -g (x).
c. Does g (-x) = g (x)?
d. Does g (-x) = -g (x)?
e. Does g (x) have even, odd, or neither even nor odd symmetry?
This document was served by the faculty/staff web server and is not an official college document.
This document was last modified on Saturday, March 27 1999.