•Topic:
Mathematical Patterns
Aim: How can we determine if a sequence is recursive or explicit?
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.
•
