Mr. Wolf
Monday 10/20/08
Pre-Calculus
Grades 11-12
Unit 2: Polynomial and Rational Functions
Complex Numbers
Materials and Resources:
 Synthetic Division Warm-up (1 per student)
 Complex Numbers Notes sheet (1 per student)
 Complex Numbers Practice sheet (1 per student)
 Review Sections 2.1-2.4 sheet (1 per student)
 Exit ticket (1 per student)
PA Standards Addressed:
2.1.11 A. Use operations with various number forms (real, complex, etc.)
Instructional Objectives:
 Students will be able to add, subtract, multiply, and divide complex numbers and
write the results in standard form.
Time
Activity
Description
10 min
Warm-up
Pass out the warm-up and review solutions.
1 min
Agenda
Review the goals for the day.
45 min
Complex Numbers
Modeling: Present the Complex Numbers Notes.
Notes
Guiding: Help students complete the examples.
Independent Practice: Complex Numbers Practice
Assessment: Review solutions
Modifications:
Students with special needs will be given notes
sheets that are filled in as well as practice
worksheets that are scaffolded to meet their needs.
Advanced students will be given worksheets with
more challenging practice problems.
min
Review of Sections
Modeling: Pass out the review sheet.
2.1-2.4
Guiding: Help students complete the sheet.
Independent Practice: Students will complete the
sheet.
Assessment: Review solutions.
Modifications:
Students with special needs will be given a
textbook to supplant their notes sheets.
Advanced students will not be allowed to use their
notes.
1 min
Agenda
Revisit goals and identify whether they were met.
5 min
Conclusion
Pass out the Exit Ticket and collect at the bell.
Homework:
Pg. 167 #5, 14, 21, 22, 29, 33, 50
Lesson Reflection:
Pre-Calculus Fall 2008
Name: ________________________
Synthetic Division Warm-up
Complete the division:
x  1 2 x 4  3x 3  5 x 2  8 x  1
x  3 5x 3  ____  2 x  8
Pre-Calculus Fall 2008
Name: ________________________
Synthetic Division Warm-up
Complete the division:
x  1 2 x 4  3x 3  5 x 2  8 x  1
x  3 5x 3  ____  2 x  8
Pre-Calculus Fall 2008
Name: ________________________
Complex Numbers Notes
Solve x 2  1  0
Definitions:
i–
Imaginary Numbers –
Complex Numbers –
complex conjugate –
Examples:
Write in complex form:
1)  25 
2)
 64 
3)
14 
4)
 32 
Write the complex conjugate:
1) 9  2i
2) 3  i
3) 12  7i
Operations with Complex Numbers:
Addition:
(a  bi )  (c  di ) 
Example:
(4  7i )  (1  6i ) 
Subtraction:
(a  bi )  (c  di ) 
Example:
(1  2i )  (4  2i ) 
Multiplication: (a  bi )(c  di ) 
Example:
(2  i )( 4  3i ) 
Example:
(2  5i )( 2  5i ) 
Distribution: bi(c  di) 
Example:
3i (4  i ) 
Writing a Complex Fraction in Standard Form:
In order to write a complex fraction in standard form, we multiply the numerator and
denominator by the complex conjugate of the denominator.
a  bi a  bi c  di (a  bi )(c  di )



c  di c  di c  di
c2  d 2
Example:
Write the fraction in standard form:
1)
2  3i

4  2i
2)
6  3i

5i
Pre-Calculus Fall 2008
Name: ________________________
Complex Numbers Practice
Write the complex number in standard form:
1) 4   36 
2) 2   27 
3) 1   8 
Perform the addition, subtraction, or multiplication and write the result in standard form.
1) (5  i )  (6  2i ) 
2) (13   4 )  (5   36 ) 
3) (8  i)  (4  i) 
4) (3   4 )  (6   169 ) 
5) 5i (3  2i ) 
6) (6  2i )( 2  3i ) 
Write the complex conjugate of the number. Then multiply the two quantities.
1) 6  3i
2) 7  12i
3)  3  2i
Write the quotient in standard form.
1)
6  7i
1  2i
2)
8  16i
3i
3)
6  5i
2i
Pre-Calculus Fall 2008
Name: ________________________
Review Sections 2.1-2.4
1) Identify the vertex and axis of symmetry of the graph:
vertex =
axis of symmetry =
2) Identify the vertex, axis of symmetry, and x-intercepts of the functions:
f ( x)  5( x  3) 2  4
f ( x)   x 2  7 x  2
vertex =
vertex =
axis of symmetry =
axis of symmetry =
x-intercept =
x-intercept =
3) Write the standard form of the quadratic function represented by the graph:
4) Write the standard form of the quadratic function with given information:
Vertex = (5, 1)
Point = (0, 6)
5) Find the maximum or minimum value of the function:
f ( x)  2( x  4) 2  3
f ( x)  2 x 2  6 x  1
6) A small local soft-drink manufacturer has daily production cost of
C  70,000  120 x  0.075x 2 , where C is the total cost (in dollars) and x is the
number of units produced. How many units should be produced each day to yield
a minimum cost?
7) Graph the polynomial: f ( x)  2 x 3  6 x 2
(GRAPH ON THE NEXT PAGE)
Step 1: Apply the Leading Coefficient Test
f ( x)  2 x 3  6 x 2
(n = _____, a n  _____ )
n is even or odd
a n is positive or negative
Step 2: Find the Zeros of the Polynomial
f ( x)  2 x 3  6 x 2
 2x 3  6x 2  0
x = _____, _____
x-intercepts @ ( _____, 0) & ( _____, 0)
Step 3: Create a Table
f ( x)  2 x 3  6 x 2
Test Interval
(-∞, _____ )
( _____, _____ )
( _____ , ∞)
x-value
f (x)
Sign of f (x)
Point on
Graph
8) Factor: 5 x 3  10 x 2  15 x
9) Factor:  8 x 2  14 x  4
10) Find the quotient: x  4 3x 3  5x 2  _____  9
Pre-Calculus Fall 2008
Name: ________________________
Exit Ticket
What did you find to be the most difficult problem on the review sheet?
Did you find your notes helpful when completing the problems?
If we were to have a test tomorrow, what topic would you study the most?
How would you study this topic? (practice problems, memorize notes, etc.)
Pre-Calculus Fall 2008
Name: ________________________
Exit Ticket
What did you find to be the most difficult problem on the review sheet?
Did you find your notes helpful when completing the problems?
If we were to have a test tomorrow, what topic would you study the most?
How would you study this topic? (practice problems, memorize notes, etc.)