Fall 1997, Math 100
Chapter 4

Be sure to check back often, because assignments may change!

Section 4.1 Polynomial Functions

• To read: page 179-middle of page 186
• Be sure to understand:The definition of a polynomial; the distinction and relationship between the zeros of a function and the roots of an equation.
1. Consider the function y=-6x^2+7x-3.
(a) What is the effect of the highest power being 2?
(b) What is the effect of the leading coefficient being -6?
(c) What is the effect of the constant being -3?
2. A quadratic function (a parabola) always has 2 zeros. What are the three ways these could occur, and what graphical significance do they have?
• E-mail subject line: Math 100 Your Name 10/24

Section 4.1 Polynomial Functions

Respond to yesterday's questions, if you didn't already.

Section 4.1 Polynomial Functions Section 4.3 The Roots of Polynomial Equations: Real or Complex?

• To read: page 186-page 189; page 199-page 206
• Be sure to understand:page 199
1. Examine the graph of f(x)=(x+1)(x-2)(x-4)^3 using a grapher.
(a) What is the degree of this polynomial? (Do NOT multiply it all out!)
(b) What happens near the triple root x=4?
(c) What is the end behavior of this polynomial?
2. Give a graphical explanation for why a cubic must have at least one real root whie a quadratic can have none.
• E-mail subject line: Math 100 Your Name 10/29

Happy Halloween!!

Section 4.4 Building New Functions From Old

• To read: Page 208- the middle of page 213
• Be sure to understand:the bottom of page 208, the bottom of page 209, Example 2
Consider R(x)=(2x^2+1)/(x^2-1)=(2x^2+1)/[(x-1)(x+1)].
1. Calculate R(-1.0001), R(-.9999), R(.9999), R(1.0001).
2. What can you conclude happens to the graph of R(x) near -1 and 1?
3. Calculate R(1000), R(100,000), R(-1000), R(-100,000).
4. What can you conclude about the end behavior of the graph of R(x)?
• E-mail subject line: Math 100 Your Name 10/31

Section 4.4 Building New Functions From Old

1. How does 5^x behave when x is very large? How does x^(-1) behave when x is very large? Which behavior dominates in the product x^(-1) 5^x? In other words, which function controls the behavior for large x, in the product?
2. Consider R(x)=(x^2-4)/2(x^2-1).
1. Where does R(x) have zeros?
2. Where does R(x) have vertical asymptotes?
3. What is the end behavior of R(x)?
3. What does it mean to take the composition of two functions?
• E-mail subject line: Math 100 Your Name 11/3

Section 4.4 Building New Functions From Old

• To read: Page 213-Page 219
1. If f(x)=x^2 and g(x)=2x, find f(g(x)) and g(f(x)). Are they the same or different?
2. Does h(h^(-1)(x))=h^(-1)(h(x))? (What do they each equal?)
3. How do j(x)+4, j(x), and j(x+4) differ graphically (for any function j(x))?
• E-mail subject line: Math 100 Your Name 11/5

Section 4.6 Finding Polynomial Patterns

• To read: Page 232-the bottom of page 237
• Be sure to understand:how you tell if a set of points lies on a quadratic or a cubic; Example 1.
1. Show that the points (-3,-35), (-2,-21), (-1,-11), (0,-5), (1,-3), (2,-5) all lie on a parabola.
2. Describe how you would go about finding the equation of the parabola in (1).
• Also be sure to:study for the 2nd gateway exam!
• E-mail subject line: Math 100 Your Name 11/7

Section 4.6 Finding Polynomial Patterns

• To read: bottom of page 237-page 245
• Be sure to understand:Middle of page 242; Example 4.
1. How many (non-colinear) points are needed to uniquely determine an 8th degree polynomial?
2. When is finding an interpolating polynomial a good idea? When is it a bad idea?
• E-mail subject line: Math 100 Your Name 11/10

Project 2

• Be sure to understand:All of it
• Also be sure to:
start thinking about an approach, and to make contact with a partner. I suggest you work with somebody different this time, although as always, several groups may work together.

Here Endeth Chapter 4
Now we move on to Chapter 7

Janice Sklensky
Wheaton College
Department of Mathematics and Computer Science
Science Center, Room 109
Norton, Massachusetts 02766-0930
TEL (508) 286-3973
FAX (508) 285-8278
jsklensk@wheatonma.edu

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