Ss Math 12 Presentation 2
Transcript of Ss Math 12 Presentation 2
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Math Review
Jessica Shao, Ken Wei, Sandeep Nagra, Vennison Cu, Masako Kato
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TransformationsDefinitions:-relations: a number of points in a coordinate plane-domain: the set of x-values of all points in a relation
-range: the set of y-values of all points in a relation-functions: a relation where any x value can only have
y value-vertical shifts: y=f(x)+a is shifted up 'a' units
y=f(x)-a is shifted down in 'a' units-horizontal shifts: y=f(x+a) is shifted down 'a' units
y=f(x-a) is shifted up 'a' units
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Kinds of Graphs
quadratic graph: y=x^2BONUS: what is the vertical andhorizontal shift of the graph?
y=(x^2 +1)-4
Square root graphs:y=square root of 2
absolutevaluegraphs:y=IxI
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kinds of graphs cont.
Cubic graphs: y=x^3 reciprocal graphs: y=1/x
Reflections:y=-f(x) is a reflection in the x axisy=f(-x) is a reflection in the y axisy=-f(-x) is a reflection in both the x and y axis
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compression and expansions vertical: y=a(f(x)) is a vertical
expansion if 'a' is
positive. (by a factor of 'a' ) y=a(f(x)) is a vertical
compression if 'a' is between 0 and 1.(by afactor of 'a')
'a' cannot be a negativenumber
horizontal: y=f(ax) is a horizontal
compression if 'a' is
positive(by a factor of 1/'a') y=f(ax) is a horizontal
compression if 'a' is between 0 and 1.(by afactor of 1/'a')
'a' cannot be negative
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Geometric Serieso The sum of terms in a sequenceo
E.g 2 + 4 + 8 + 16. ..o Use r0, 1
Infinite Geometric Serieso A series that is continuouso
E.g 2 + 1 + +
o Use -1< r
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How to use Sigma Notation Sigma means write the sum of Eg. n= upper limit-lower limit+1
t 3 + t 4+ t 5 + t 6 + t 7
1. n= 7-3+1 = 5 terms to add up2. t1= 5t1= 20t2= 5
t2= 403. r= = =2
17
3
25 i
i
Upper limit
Lower limit
General formula
4.
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1. n=12-1+1=12
2. t1=2(-3)1-1
=2t2=2(-3) 2-1=-6
3. r=1=
6
2=-3
4.
S12= -265720
Use Sn formula
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Word problem
YOU TRY!o An equilateral triangle has sides of length 10. If the
midpoints of each side are joined to form another
triangle, and this process is continued, what is theperimeter of the 5 th triangle?
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Logarithms What is a log? Log is the inverse of functions Ex The
inverse of: x 5 = 49 ------> Log x49 = 5
Basic Rules: 1) x4 * x5 = x 9 ------> Log(4)(5) = Log 4 + Log 52) x 4 x 5 = x -1 ------------> Log 4/5 = Log 4 - 53) (x 4)5 = x 10 --------------> Log x -1 = -Log x4) x 0 = 1 ------------------> Log 1 = 0
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Logarithm Examples1) Log75 = Log5 = 0.827
Log7
2) Log 6abc = Log 6 + Loga + Logb + Logc
3) 2Log3x - Log3y =Log3x2 - Log3y =Log3 (x2)
( y )
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Trigonometry
Standard position means the intial sideis along the positive x-asix with thevertex at the origin. Rotating a rayaround the vertex forms an angle withan intial side and a terminal side.
Reference angle - the positive acute angle that isformed with the terminal side of and the x-axis.
A reference angle is 0 90
SOH-CAH-TOA - S ome Old Hippy Caught Another Hippy Tripping On Acid
Stuff you should knowalready. :U
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Degrees to Radians
Radians to Degrees
Radians = Degrees x .
180
Degrees = Radians x 180
Arc Length:
s = r
s => arc lengthr => radius
=> central angleMake sure ismeasured in radians!
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Special AnglesGive the exact value of sin /3 .
Give the exact value of cos5 /4.
3 .2
-1.
2
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IdentitiesPythagorean Identities
Reciprocal and Quotient Identities
Sum and Difference Identities
Double Angle Identities
sin 2
+ cos 2
= 1 1 + tan 2
= sec 2
1 + cot 2 = csc 2
sec = 1/cos csc = 1/sin cot = 1/tan
tan = sin/cos cot = cos/sin
cos( + ) = coscos - sinsin sin( + ) = sincos + cossin cos( - ) = coscos + sinsin sin( - ) = sincos - cossin
cos2 = cos 2 - sin 2 sin2 = 2sincos = 2cos 2 - 1
= 1 - 2sin 2
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Prove the following identities
sin + cos x cot = sec 1 + sec = csc cos x csc sin + tan
cos 2x = 1 - 2sin 2x cscx - sinx . = cotx1 - tan 2x 1 + cosx
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CombinatoricsFundamental Counting Principle
- M ways for 1st item x N ways for 2nd item
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E.X.- 3 dresses, 4 shirts, 2 hats and 6shoes. How many combinations can u have?3x4x2x6=144
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Permutations and Factorial Notation- nP r : permutation for n distinct objects
taken r at a timeE.X.- Arrangements of 5 books in a line
5P5=120
- n! : n(n-1)(n-2)...E.X.- how many words can be made with
the letters from "saskatoon?"
9!/(2!2!2!)=45360
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Combinations- nCr : combinations for n distinct objects
taken r at a timeE.X.- Combinations for choosing 3 people
from a group of 9
9C3=84E.X.- How many hands of 5 cards with at
least 4 hearts can be formed?
39C1x13C4+39C0x13C5=29172
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The Binomial Theorem- (a+b) n
E.X.- Expand (a+b) 4 1a4+4a 3 b+6a 2 b2+4ab 3+b 4
- t (K+1)=NCK (a) (N-K) (b) K : Real Usage-powers of 'a' decrease going to the right-powers of 'b' increase going to the right-the numerical coefficients are similar to
pascal's triangle
pascal's triangle:
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Probability
example: dice: what is the probability of gettinga 5 from rolling a 6-sided die?
in a deck of 52 cards, what is the probablility of getting a red card?
Probability: P(a)=r/n'a' is the favourable event, 'r' is the possible outcomes for
event 'a' and 'n' represents the number of outcomese uall likel to 'r'
p(5)=1/6
p(red)=26/52
f f
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formulas for combinatoricsand probability
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