Download Document

Agenda Class 2
Tuesday, June 21
Terms
Definition: An integer is a whole number.
Definition: A real number is an integer or fraction that has a place on the number line. There are
an infinite number of real numbers.
Definition: A rational number is a real number that can be expressed as a fraction where the
numerator and denominator are integers. Whole #’s are rational numbers because 5 = 5/1.
Definition: An irrational number is a real number that is not rational.
Definition: An imaginary number is a number that doesn’t exist on our real number line. 1 is
an imaginary number, and we call it i. It is not possible because the square root symbol means
that we are looking for two numbers that are the same that multiply to give us -1 (Like 9  3
because 3 and 3 are the same and 3 times 3 gives 9). But how can two equivalent numbers

multiplied together result in a negative? Positive times positive is positive, and negative times
negative is positive. So, 1 doesn’t exist in our world of real numbers. But note that

2
i 2  1  1. So 1 is not real, but when I square it, it becomes real.
 


Definition: Numbers
like 5i mean 5 times i, and they are called pure imaginary numbers.
Another example:
i 5  1  5  5 . Math people don’t like leaving the negative

underneath the radical. It makes them uncomfortable. So, they take out the negative and place
an i in front of the radical to indicate that the number is imaginary.

 Numbers like 5 + i are called complex because there is a pure imaginary part and a
Definition:
real part of the number. These numbers are a bit complicated because in order to multiply them,
I have to use FOIL (first, outer, inner, last).
(1  i)(2  i)  2  i  2i  i 2  2  i  2i  1  1  3i
In order to add or subtract them, I add or subtract the real parts and I add or subtract the
imaginary parts. Note: I cannot combine the real with the imaginary.
(5  3i)  (2  7i)  (5  2)  (3i  7i)  7  4i
To divide complex numbers, multiple through by the conjugate of the denominator:
1 2i 1 2i 4  3i (1 2i)(4  3i) 4  3i  8i  6i 2 4  11i  6 2  11i 2 11







 i
4  3i 4  3i 4  3i
16  9i 2
16  9
25
25
25 25

Opening Exercise
1. Is it possible to add or multiply 2 irrational numbers together to get a rational number? If
so (for add or multiply), give example(s).

2. Is it possible to add/multiply 2 rational numbers together to get an irrational number? If
so (for add or multiply), give example(s).
Agenda Class 2
Tuesday, June 21
3.
Is it possible to add/multiply 2 complex numbers together to get a real number? If so
(for add or multiply), give example(s).
4. What kind of number would you get by adding/multiplying a rational number with an
irrational number?
5. What kind of number would you get by adding/multiplying a real number with a complex
number?
Simplifying Complex Numbers
Given that i  1 , re-write the following using i and simplify:
81
18
4 25




Simplify the following:
2
3
i 
i 

i 
4
i 
5
i10 
Board Problem 1:
Pick a complex number of the form a  bi where b is not 0. Call your number z and write it on
the board (z with the bar over it denotes the conjugate of z). Find the following:
1
a) 5z.
b) –iz.
c) z
d) z  z .
e)
z
Board Problem 2:
Simplify i1001 
Section 1.6: Quadratic Equations

Definition of quadratic equation
What does the graph look like?
Agenda Class 2
Tuesday, June 21
Where are the zero’s or the roots of the function on the graph?
How many distinct roots does a quadratic equation have?
Is it possible that some quadratic equations do not cross the x-axis?
Is it possible that some quadratic equations have no roots?
How do we know if the quadratic equation only has 1 real root?
Finding Roots of Quadratic Equations
What are roots?
Method 1: Finding roots by graphing calculator. Example: x 2  7x  6  0
Method 2: Finding roots by factoring.
Examples: x 2  7x  6  0, 2x 2  3x  9  0 , x 2 8x  17  1
Method 3: Quadratic formula (always works and only way to find complex roots).
x 2  x 1  0
Example:



MUST SET EQUATION = 0 BEFORE FINDING ROOTS, GIVEN THE DEFINITION OF A
ROOT!
What is the relationship between the complex roots?
Board Problem 3:
Find the quadratic equation with real number coefficients that has 3i as a solution.
Board Problem 4:
Solve the following equations by factoring or quadratic formula:
(a) x 2  5 x  14  0
(b) 6 x 2  7 x  2  0
(c) 2 x 2  9 x  5
(d) ( x  3) 2  16
(e) ( x  1)( 2 x  2)  4
Board Problem 5:
Solve this equation for x: 2x 2  4 x . How many answers did you get? How many should you
have gotten?
Board Problem 6:

Try this one: 3x(x  2)  2(x  2)2 . Hint: Try factoring.

Agenda Class 2
Tuesday, June 21
Board Problem 7:
Okay one more: 3x  6  2x 1. Plug your answers into the original equation. Do all of them
work?
TIPS:
a) Don’t divide by x’s! You lose solutions.
 squaring, check your answers b/c you may get extraneous solutions that do not work
b) When
when plugged into the original equation.
Quadratic Functions & Their Graphs
Define vertex and axis of symmetry of parabolas
What is the y-intercept of a parabola with equation y  ax 2  bx  c ?
dictates which way it opens?
Opens up vs. opens down. Which variable
X-coordinate of vertex and axis of symmetry at x = -b/2a
Board Problem 8:
Find vertices & axes of symmetry: x 2  7x  6  0, 2x 2  3x  9  0 , y  ( x  3)( x  5)
The easiest way to graph is to find the vertex and then find additional points around vertex.


Board Problem 9:
Graph the parabolas in board problem 8 by choosing points around the vertex.
Modeling with Quadratics
 Parabolas are symmetric and have EITHER a maximum or a minimum
 Can find max/min’s by finding the vertex. MAKE SURE YOU KNOW WHETHER
THE QUESTION IS ASKING FOR THE X VALUE OR FOR THE Y VALUE.
Board Problem 10:
The demand equation for a certain product is p 140  0.0001x where p is the unit price (in
dollars) of a Nalgene bottle and x is the number of bottles produced and sold. The cost equation
for the product is C  80x 150,000 where C is the total cost and x is the number of bottles
produced. The total profit obtained by producing and selling x bottles is Profit = Revenue – Cost
= xp – C. Let’s pretend that you are
 a top executive in the marketing department of the Nalgene
company. You are asked by your superior to determine a price p that will yield a profit of 9
milliondollars. Is this even possible? Why or why not? (Hint: What type of function is this?
How would you find the maximum? What would be a reasonable window for your calculator?)
Agenda Class 2
Tuesday, June 21
BONUS CHALLENGE PROBLEM: Find two numbers whose sum is 23 and the sum of
whose squares is 269.