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 Square root graphs: y=square root of 2 quadratic graph: y=x^2 absolute BONUS: what is the vertical and value horizontal shift of the graph? graphs: y=(x^2 +1)-4 y=IxI
kinds of graphs cont. reciprocal graphs: y=1/x Cubic graphs: y=x^3 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: horizontal: • y=a(f(x)) is a vertical • y=f(ax) is a horizontal expansion if 'a' is compression if 'a' is positive. (by a factor of positive(by a factor of 'a' ) 1/'a') • y=a(f(x)) is a vertical • y=f(ax) is a horizontal compression if 'a' is compression if 'a' is between 0 and 1.(by a between 0 and 1.(by a factor of 'a') factor of 1/'a') • 'a' cannot be a negative • 'a' cannot be negative number
• Geometric Series o The sum of terms in a sequence o E.g 2 + 4 + 8 + 16. .. r≠0, 1 o Use • Infinite Geometric Series o A series that is continuous o E.g 2 + 1 + ½ + ¼ o Use -1< r <1
How to use Sigma Notation • Sigma means “write the sum of” Eg. n= upper limit-lower limit+1 7 Upper limit i 1 5 2 Lower limit 3 i General formula t 3 + t 4 + t 5 + t 6 + t 7 1. n= 7-3+1 = 5 terms to add up 4. 2. t 1 = 5 t 1 = 20 t 2 = 5 t 2 = 40 3. r= = =2
1. n=12-1+1=12 2. t 1 =2(-3) 1-1 =2 t 2 =2(-3) 2-1 =-6 Use Sn formula 3. r= 𝑢 2 −6 𝑢 1 = 2 =-3 4. S 12 = -265720
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 the perimeter of the 5 th triangle?
Logarithms What is a log? Log is the inverse of functionsEx The inverse of: x 5 = 49 ------> Log x 49 = 5 Basic Rules:1) x 4 * x 5 = x 9 ------> Log(4)(5) = Log 4 + Log 5 2) x 4 ÷ x 5 = x -1 ------------> Log 4/5 = Log 4 - 5 3) (x 4 ) 5 = x 10 --------------> Log x -1 = -Log x 4) x 0 = 1 ------------------> Log 1 = 0
Logarithm Examples 1) Log 7 5 = Log5 = 0.827 Log7 2) Log 6 abc = Log 6 + Loga + Logb + Logc 3) 2Log3x - Log3y =Log3x2 - Log3y =Log3 (x2) ( y )
Trigonometry Stuff you should know already. :U 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 ° 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 SOH-CAH-TOA - S ome O ld H ippy C aught A nother an intial side and a terminal side. H ippy T ripping O n A cid
Degrees to Radians Arc Length: Radians = Degrees x π . s = θr 180° Radians to Degrees Degrees = Radians x 180° π s => arc length r => radius θ => central angle Make sure θ is measured in radians!
Give the exact value of sin π/3 . Special Angles √3 . 2 Give the exact value of cos5 π/4. -1 . √2
Identities Pythagorean Identities sin 2 θ + cos 2 θ = 1 1 + tan 2 θ = sec 2 θ 1 + cot 2 θ = csc 2 θ Reciprocal and Quotient Identities sec θ = 1/cosθ cscθ = 1/sinθ cotθ = 1/tanθ tan θ = sinθ/cosθ cotθ = cosθ/sinθ Sum and Difference Identities cos(α + β) = cosαcosβ - sinαsinβ sin(α + β) = sinαcosβ + cosαsinβ cos(α - β) = cosαcosβ + sinαsinβ sin(α - β) = sinαcosβ - cosαsinβ Double Angle Identities = cos 2 θ - sin 2 θ cos2θ sin2θ = 2sinθcosθ = 2cos 2 θ - 1 = 1 - 2sin 2 θ
Prove the following identities sin θ + cosθ x cotθ = secθ 1 + secθ = cscθ cos θ x cscθ sinθ + tanθ cos 2 x = 1 - 2sin 2 x cscx - sinx . = cotx 1 - tan 2 x 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 - n P r : permutation for n distinct objects taken r at a time E.X.- Arrangements of 5 books in a line 5 P 5 =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 9 C 3 =84 E.X.- How many hands of 5 cards with at least 4 hearts can be formed? 39 C 1 x 13 C 4 + 39 C 0 x 13 C 5 =29172
The Binomial Theorem - (a+b) n E.X.- Expand (a+b) 4 1a 4 +4a 3 b+6a 2 b 2 +4ab 3 +b 4 - t (K+1) = N C K (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? p(5)=1/6 in a deck of 52 cards, what is the probablility of getting a red card? p(red)=26/52 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'
formulas for combinatorics and probability
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