Math Review Jessica Shao, Ken Wei, Sandeep Nagra, Vennison Cu, - - PowerPoint PPT Presentation

Math Review

Jessica Shao, Ken Wei, Sandeep Nagra, Vennison Cu, Masako Kato

Transformations

Definitions:

• 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

Kinds of Graphs

quadratic graph: y=x^2 BONUS: what is the vertical and horizontal shift of the graph?

y=(x^2 +1)-4 Square root graphs: y=square root of 2

absolute value graphs: y=IxI

kinds of graphs cont.

Cubic graphs: y=x^3 reciprocal graphs: y=1/x

Reflections: y=-f(x) is a reflection in the x axis

y=f(-x) is a reflection in the y axis y=-f(-x) is a reflection in both the x and y axis

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 a factor of 'a')

• 'a' cannot be a negative

number 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 a factor of 1/'a')

• 'a' cannot be negative

• Geometric Series
• The sum of terms in a sequence
• E.g 2 + 4 + 8 + 16...
• Use

r≠0, 1

• Infinite Geometric Series
• A series that is continuous
• E.g 2 + 1 + ½ + ¼
• Use
• 1< r <1

How to use Sigma Notation

• Sigma means “write the sum of”

Eg.

n= upper limit-lower limit+1

t3 + t4+ t5 + t6 + t7

• 1. n= 7-3+1 = 5 terms to add up
• 2. t1= 5

t1= 20 t2= 5 t2= 40

• 3. r= = =2

1 7 3

2 5

 

 

i i

Upper limit Lower limit General formula

4.

• 1. n=12-1+1=12

2.t1=2(-3)1-1=2

t2=2(-3)2-1=-6

• 3. r=

𝑢2 𝑢1= −6 2 =-3

4.

S12= -265720

Use Sn formula

Word problem

• YOU TRY!
• 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 the perimeter of the 5th triangle?

Logarithms

What is a log? Log is the inverse of functionsEx The inverse of: x5 = 49 ------> Logx

49 = 5

Basic Rules:1) x4 * x5 = x9 ------> Log(4)(5) = Log 4 + Log 5 2) x4 ÷ x5 = x-1 ------------> Log 4/5 = Log 4 - 5 3) (x4)5 = x10 --------------> Log x-1 = -Log x 4) x0 = 1 ------------------> Log 1 = 0

Logarithm Examples

1) Log7

5 = Log5 = 0.827

Log7 2) Log6abc = Log6 + Loga + Logb + Logc 3) 2Log3x - Log3y =Log3x2 - Log3y =Log3 (x2) ( y )

Trigonometry

Standard position means the intial side is along the positive x-asix with the vertex at the origin. Rotating a ray around the vertex forms an angle with an intial side and a terminal side.

Reference angle - the positive acute angle that is formed with the terminal side of θ and the x-axis. A reference angle is 0°≤ θ ≤ 90°

SOH-CAH-TOA - Some Old Hippy Caught Another Hippy Tripping On Acid Stuff you should know

• already. :U

Degrees to Radians Radians to Degrees

Radians = Degrees x π . 180° Degrees = Radians x 180° π Arc Length: s = θr s => arc length r => radius θ => central angle Make sure θ is measured in radians!

Special Angles

Give the exact value of sinπ/3. Give the exact value of cos5π/4. √3 . 2

• 1 .

√2

Identities

Pythagorean Identities Reciprocal and Quotient Identities Sum and Difference Identities Double Angle Identities

sin2θ + cos2θ = 1 1 + tan2θ = sec2θ 1 + cot2θ = csc2θ secθ = 1/cosθ cscθ = 1/sinθ cotθ = 1/tanθ tanθ = sinθ/cosθ cotθ = cosθ/sinθ cos(α + β) = cosαcosβ - sinαsinβ sin(α + β) = sinαcosβ + cosαsinβ cos(α - β) = cosαcosβ + sinαsinβ sin(α - β) = sinαcosβ - cosαsinβ cos2θ = cos2θ - sin2θ sin2θ = 2sinθcosθ = 2cos2θ - 1 = 1 - 2sin2θ

Prove the following identities sinθ + cosθ x cotθ = secθ 1 + secθ = cscθ cosθ x cscθ sinθ + tanθ cos2x = 1 - 2sin2x cscx - sinx . = cotx 1 - tan2x 1 + cosx

Combinatorics

Fundamental Counting Principle

• M ways for 1st item x N ways for 2nd item
• E.X.- 3 dresses, 4 shirts, 2 hats and 6
• shoes. How many combinations can u have?

3x4x2x6=144

Permutations and Factorial Notation

• nPr : permutation for n distinct objects

taken r at a time E.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

Combinations

• nCr : combinations for n distinct objects

taken r at a time E.X.- Combinations for choosing 3 people from a group of 9 9C3=84 E.X.- How many hands of 5 cards with at least 4 hearts can be formed? 39C1x13C4+39C0x13C5=29172

The Binomial Theorem

• (a+b)n

E.X.- Expand (a+b)4 1a4+4a3b+6a2b2+4ab3+b4

• 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:

Probability

example: dice: what is the probability of getting a 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 outcomes equally likely to 'r'

p(5)=1/6 p(red)=26/52

formulas for combinatorics and probability