Wednesday, June 5, 2013

Quadratics Exploration


Topic: Factoring polynomials
Aim: How can we write a quadratic expression from a product of two binomial factors?
Do Now: What is the quadratic expression given the
factors (x+2)(x-5)?
The product in standard form
(x+2)(x-5)
FOIL method:
First: (x) (x) = x2
Outer: (x) (-5) = -5x
Inner: (2) (x) = 2x
Last: (2) (-5) = -10
Standard form of the quadratic expression is:
X2 - 5x + 2x -10 = x2 - 3x -10
Complete a, b, c
Find the quadratic expression in standard form
Finding the factors given the quadratic expression in standard form
x2 + 5x + 6
Use the diamond method:
+5 (add)
  +3  +2
+6 (multiply)
Factors are (x+ 3)(x+2).
Check using FOIL to see if you are correct!
Complete a – f
Get the factors given the quadratic expression
Complete your final project on quadratics
Complete all parts of the assignment!
All questions # 1, 2 (page 25)
All questions #1 – 3 (page 24)
Make sure you show all work clearly and include your name and period on your answer sheets.
Leave in basket in front of room by the time the bell rings
Do NOT leave your work in your folder!!!
Perfect squares
Multiply each of these expressions to get a quadratic expression:
(x + 4)2  = (x+4)(x+4)
F: x2
O: 4x
I: 4x
L: 16
x2 + 8x + 16
Exit slip
How do the factors you found in Question #2 relate to the x-intercepts if you were to graph each quadratic expression?
Recall the graph of a quadratic is a parabola.

Wednesday, May 15, 2013

Explicit and recursive formulae: Patterns


Topic: Mathematical Patterns
Aim: How can we
determine if a sequence is recursive or explicit?
Do Now: Find the first five terms of the sequence: an = ½ (n) (n-1). Use a table   to organize your data
an = ½ (n) (n-1)
Problem 2: Writing a recursive formula for number of given blocks
Count the blocks in each pyramid
1, 3, 6,10, 15, 21
Subtract consecutive terms to find out what happens from one term to the next
a2 – a1 = 3-1 = 2
a3 a2 = 6-3 = 3
a4 a3 = 10 – 6 = 4
a5 a4 = 15 – 10 = 5
an an-1   = n (use n to express the relationship between successive terms)
Write the formula
State the initial condition and the recursive formula:
a1 = 1
an  =  an-1  +  n
Got it?
Problem 2 on page 566
Write a recursive definition for each of the following sequences:
a. 1, 2, 6, 24, 120, 720,…..
b. 1, 5, 14, 30, 55,…….
1,5,14,30,55,…..
A1 = 1
A2 = 5
A3 = 14
A4 = 30
A5 = 55
A2 – a1 = 5 – 1 = 4
A3 – a2 = 14 – 5 = 9
A4 – a3 = 30 – 14 = 16
A5 – a4 = 55 – 30 = 25
an – an-1 = n2
Recursive formula is a1 = 1; an = an-1 + n2
Practice and problem solving exercises
Page 569 handout
# 43, 45 – 52
Share out
Essential Understanding
If the numbers in a list follow a pattern, you may be able to relate each  number in the list to its numerical position in the list with a rule
Vocabulary
Sequence: an ordered list of numbers
Ex. 1,4,7,10,13,16,19,22,25,28
Term of a sequence: each number in the sequence
Ex. 1st term = a1
Explicit formula: describes the nth term of a sequence using the number n
Ex.  an = 3n - 2
Generating a sequence
A sequence has an explicit formula an = 3n -2.
What are the first 10 terms of this sequence?
Ans.
     an = 3n -2 :write the formula
  a1 = 3(1) – 2 = 1 Substitute 1 for n and simplify
     a2 = 3(2) – 2 = 4…….and so on
Use a table to organize your work
Got it?
#1 on page 565
In addition:
1. Find the first ten terms of the sequence in #1: an = 12n + 3. Use a table to organize your work
Exit slip
Provide a real world example of a  sequence.
What are some of its terms and how does the pattern work?
Full moon data from Beirut, Lebanon
Full moon data from Beirut, Lebanon
Create a table of data
Convert times to decimals
Example:
6:39 am = 6 + 39/60 = 6.65
10:26 am = 10 + 26/60 = 10.43
Translating sine and cosine functions
Write a cosine function for the graph. Then write a sine function for the graph
Know:
Y = a cos(bx +c) + d and y = a sin (bx +c) +d
Find the a, b, c and d.
Plan
The amplitude of the graph is 2
The period of the graph is 2π, so b = 1
The vertical shift is -1
Cosine function is: y = 2 cos x -1
Sine function is y = 2 sin (x-π/2) - 1
Group Work
Page 367 handout –
#1-6 and 9 – 11
Additional problems are #12 – 20
Put your name and leave in your folder for
Full credit
Using the graphing calculator to graph a trig function
Y = sin x + x
Domain: 0 to 2π
Y1 = sin x + x
Window:
Xmin = 0
Xmax = 2π
Xscl = 1
Ymin = 0
Ymax = 10
Yscl = 2
Xres = 1 (always)
Graph
Basic sin graph
Basic sin graph shifted up by x
The graph begins at the same point as the basic sine curve because the initial value is x = 0. Then it is shifted up by positive values because x is always a positive real number (can you explain why?)
The graph should be displayed in the first quadrant only (why is this so?)
Is the graph exhibiting increasing or decreasing behavior over the domain?
Practice Problems
       1.  Y = cos x – 2x
Make sure window is correct
Analyze the graph using the information that we used to analyze previous trig function.
          2. Y = sin (x + cos x)
SAT/ACT Prep
Handout
#55, 56, 57
#67 - 71
Exit slip
Can you describe how a shifted sinusoidal function is used in the real world?
Please be specific and write at least one paragraph explaining how the shift clarifies necessary information in your example.

Friday, May 3, 2013

graphing trig functions over the domain 0 to 2pi


Topic: Sinusoidal Functions
Aim: Graphing the sin and
cos function over the domain 0 to 2π
Do Now: Write the equation for a basic sine function (y = sin x) that is shifted 2 units to the left
Using the graphing calculator to graph a trig function
Y = sin x + x
Domain: 0 to 2π
Y1 = sin x + x
Window:
Xmin = 0
Xmax = 2π
Xscl = 1
Ymin = 0
Ymax = 10
Yscl = 2
Xres = 1 (always)
Graph
Basic sin graph
Basic sin graph shifted up by x
The graph begins at the same point as the basic sine curve because the initial value is x = 0. Then it is shifted up by positive values because x is always a positive real number (can you explain why?)
The graph should be displayed in the first quadrant only (why is this so?)
Is the graph exhibiting increasing or decreasing behavior over the domain?
Practice Problems
       1.  Y = cos x – 2x
Make sure window is correct
Analyze the graph using the information that we used to analyze previous trig function.
          2. Y = sin (x + cos x)
SAT/ACT Prep
Handout
#55, 56, 57
#67 - 71
Exit slip
Can you describe how a shifted sinusoidal function is used in the real world?
Please be specific and write at least one paragraph explaining how the shift clarifies necessary information in your example.

Tuesday, April 30, 2013

Creating a sinusoidal model using temperature v days of the year


Topic: Sinusoidal functions

Aim: How can we use the graphing calculator to find the model of a sine function?
Write the equation of a sine function with a >0, amplitude 7.85, and cycles .016 in 2π  
The sine function with shifts
Y = A sin (Bx + C) + D
A = amplitude
B = cycles in 2π
C = horizontal shift
D = vertical shift
Look at problem 27 on pg 368 handout
Modeling temperature on several days of the year
Graph the model
Y1 = = 7.85 sin(.016x – 2.22) + 82.19
Fix the WINDOW for domain and range
GRAPH
Sketch the graph on the xy plane
Extending your thinking
How could we use the sine model to write the cosine model?
Recall how we shift from sine to cosine (by π/2)
Graph the cos model on the same set of axes
Horizontal translations – phase shifts
g(x): horizontal translation of f(x)
g(x) = f (x – h)
Important consideration
The value of h f (x – h):
If h > 0, the shift is to the right
Example: f ( x – 3 ) is a horizontal shift 3 units to the right
If h < 0, the shift is to the left.
Example: cos (x + 4) = cos (x – (-4)), h = -4; the shift is 4 units to the left
Vertical translations
h(x): vertical translation of f(x)
h(x) = f(x) + k
Exit slip
What are the major differences between the cosine and the sine functions?