ss math 12 presentation 2

7/31/2019 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

-

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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ss math 12 presentation 2

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