C Table of Contents

Transcription

1 Gen Page C Table of Contents Clear Sheet/Duotang/Notes/Homework C Formula Sheet C Bootcamp C TOC C Pictures C Methods C Remember C General Equation Graphs C Fraction/Exp/Algebra/Fact/Foil C Graph Transformations C Function/Point Review C Long/Synthetic Review C Polynomials Review C Trig Review C Trig Summary C Sine Definition C Trig Functions Review C Trig Identities Steps Review C Trig Identities Proofs C Trig Double Angle Review C Trig NPV C Exponentials Review C Logs Review C Rationals Review C Rationals Tree C Function Operation Review C Combinatorics Review C ncr npr Tables C Cards in Poker C -. - Point Translation C -. - Point Expansion Compression C -. - Point Reflection Inverse C Trans/Comp/Exp/Ref C Inverse Equation C Order Matters Point C -. - Radical Translations C -. - Radical Transformation C Radical Reflection C Radical Graphs C Long Division C Synthetic Division C Factor/Remainder Theorem C Factoring Trinomial's C Potential Factors C End Behaviour C Multiplicity C Graphing Polynomials C Open Top Rectangular Box C Degree/Radian Conversions C Arc Length/Sector area C Reference Angles and in Standard Position C ASTC/Special Triangles C Trig Ratios C Trigonometric Equations ASTC C Calculator C Algebra C Unit Circle C Factoring C Trigonometric Equations Period Change C Graphing Sin Cos and Tan C Amplitude and Period C Horizontal Centre-Line and Phase Shift C Graphing Equations C Finding Equations C Word Problems C Trig Identities C Trig Identities C Simplifying Exponentials C Exponential Graphs C Word Problems C Definition C Exponential Form/ Factoring C Change of Base C Adding and Subtracting Logs C Log Equations C Log Operations C Log Both Sides C Log Graphs/Inverse C Rationals Definition C Vertical Asymptote C Horizontal Asymptote C x,y Intercepts C Holes C Slant C Composite Functions C Composite Graphs C -. - Fundamental Counting Principle C -. - Factorial Notation C Outcomes Per Trial C ABC/3 Factorial/ Permutation/Combination C Cases C President/Committee C All Minus None C Identical Objects C Paths in Squares C ncr npr Algebra C Pascals Triangle C Binomial Expansion

2 Gen Page C Pictures,,

3 Gen Page 3 Graphing/Calc: General Formula TOV, 0 0, Domain/Range Word Problems C Methods Transformations Substitute the Opposite Operation for the Variable "Put ( ) in for ( )" Horizontal/Vertical Translations Expansions/Compressions Reflections Inverse: Switch x and y Invariant Points Factor Brackets Do functions stuff st. BEDMAS, Inside Out Steps Layout Mapping/Order Function Notation Radicals Trigonometry Radians <-> Degrees Arc Length/Sector Area ASTC/QI,II,III,IV/Unit Circle Special/Similar Triangles SOH-CAH-TOA,, STP/Reference Angles Solving Equations sin,,, 0, Linear/Angular Velocity Trig Functions Exponentials Exponent/Radical laws (Change of Base) Separating Exponents Find Intersection Same Base: Make exponents equal to each other Take both/sides to reciprocal exponent of variable Set 00 Logarithms Log/Exponential Form/Laws/Change of Base Bring Exponent down in front/separate log both sides/de-log both side Set Log arbitrarily Domain: Set the thing you are logging 0 Set the base of the log 0 VA: Set the thing you are logging 0 Graphing: Graph ->Switch x and y->trans Laws/Square both sides Domain: Set underneath the root 0 and solve Pick an value on Square root the y-value Draw the new point. Polynomials Definition Factoring Long Division Synthetic Division Factor Theorem Remainder Theorem Potential Factors (x-a) Solve by Inspection f(a) Store /Graph nd Calc Zero Graphing End behavior Multiplicity Box Model DACB Trig Identities Identities Fractions/LCD Factoring/FOIL Conjugates M8- Methods Bedmas/# Forms Substitution, let m = # Algebra/Fractions Exponents/Geometry Factoring/Quad Form Inequalities Rationals Steps: Factor, Holes, VA, HA, SA's VA: Set Denominator 0 LCD Fractions Long/Synthetic Division Function operations Combinatorics FCP Factorials Tree Diagram npr, ncr Cases All minus none Identical Objects Paths in Squares Pascal 'Triangle Binomial Theorem

4 Gen Page 4 C Remember Transformations 4 Logic! Do exactly what you see outside of the brackets on the right-hand side to the y-value "k" may be on the left hand side of the equation.. So add or subtract "k" to both sides. Do the opposite/reciprocal of what you see inside the brackets to x-value BEDMAS. Always do inverse first. Do Function Operations st. Inside Out. Factor the brackets so has a coefficient of (Cant factor/distribute out/into of a function) Order Matters How wide/tall is it/now? What happened? Any reflections? Inverse Check: Pick a point, not an intercept, do exp/comp/reflections, Has it moved? Radicals 4 Remember: Choose increments of x in your table of values that square root easily. Invariant Points:, 0,, Polynomials Long - Goes Into, Multiply, Subtract, Bring Down, Repeat Synthetic - Bring Down, Multiply, Add, Repeat 0, Missing Terms "Insert 0" Trigonometry Calculator must be in radian mode Rationalize sin Exponentials 3 6 Growth/Decay The exponent is the time or the number of time periods. Logarithms The thing you are Logging equals the Base to the other side. The base of the log is the base of the exponent. The exponent is the Answer. log 3 3 Logs and Exponentials are inverses of each other Trig Functions 0, 0,, 0,, 3 +Sin starts in the middle and goes up +Cos starts from the top and goes down Rational's Holes before VA's A graph can cross a horizontal asymptote Invariant Points:, Increments = "b" multiplies the # of original solutions between 0 Trig Identities sin sin sin cos, Function operations Combinatorics Repeats Replacement

5 Gen Page 5 C General Equation Graphs Real Life!, 0 0, Polynomial Linear Quadratic Cubic/Quartic/Quintic,,, Absolute Value Radical Rational 0, 0,,, Trigonometric a sin sin a cos cos a tan tan 0 Exponential, Logarithmic log, 0 log 0, 0,

6 Gen Page 6 C Fractions/Exp/Algebra/Fact/FOIL Theory 3 FOIL Conjugates: FL

7 Gen Page 7 C VHT VHCE VHR Graph Transformation Review Original VT Up VT Down HT Right HT Left Original Vertical Expansion Vertical Compression Horizontal Expansion Horizontal Compression Original Vertical Reflection (reflect over x-axis) Horizontal Reflection (reflect over y-axis) Horizontal and Vertical Reflection (reflect over y and x-axis)

8 Gen Page 8 C VHT VHCE VHR Function/Point Transformations Review : Vertical Translation (up) : Vertical Translation (down) : Horizontal Translation (right) ( is positive) : Horizontal Translation (left) ( is negative) 0: Vertical Reflection (VR) over the -axis : Vertical Expansion 0 : Vertical Compression 0: Horizontal Reflection (HR) over the y-axis : Horizontal Compression 0 : Horizontal Expansion Mapping, ; (,3) (,6) multiply y-value by two (,3) (4,3) multiply x-value by two ;, (3,4) (3,) multiply y-value by a half,,,,, (4,) (,) multiply x-value by a half, 0 0 (3,5) (3,5) Multiply y-value by (4,3) (4,3) Multiply x-value by,, 3 3 (,) (,4) add 3 to y-value, 3, (,4) (,) Subtract from y-value, (4,) (,) Subtract from x-value,, 4 4 (,3) (6,3) Add 4 to x-value 4, Inverse : switch x and y A reflection over the axis ( ) verse Check: Switch Domain and Range. Invariant Points: Horizontal Ref/Exp: y-intercepts (0,y) Vertical Ref/Exp: x-intercepts (x,0) Inverses: (a,a) (,) (any points on the line y=x) Roots: (x,0) (x,) Rationals, Inverse st Functions Operations st. DMAS, Inside Out

9 Gen Page 9 C VHT VHCE VHR Function Sub Review 3 Vertical Translation up Put ' 3 in for '' Substitute the Opposite Operation for the Variable 3 Horizontal Translation right Put ' 3 in for 'x' Substitute the Opposite Operation for the Variable 3 Horizontal Translation left Put ' 3 in for '' Substitute the Opposite Operation for the Variable Vertical Expansion of Put in for '' Substitute the Opposite Operation for the Variable Horizontal Expansion of 4 Put in for '' Substitute the Opposite Operation for the Variable Vertical Compression of Put in for '' Substitute the Opposite Operation for the Variable Horizontal Reflection Put -x in for x Substitute the Opposite Operation for the Variable Vertical Reflection Put -y in for y Substitute the Opposite Operation for the Variable Vertical Translation up 3 3 Horizontal Translation right

10 Gen Page 0 C Long/Synthetic Division ? 3 Review , Set denominator equal to zero and solve. 0,, List Coefficients Bring down the first coefficient , , Remainder ,0 Factor Theorem Remainder

11 Gen Page C Long/Synthetic Division ? 3 Review , Set denominator equal to zero and solve. 0,, List Coefficients Bring down the first coefficient , , Remainder ,3 Remainder Theorem Remainder

12 Gen Page C Polynomials Review Synthetic Steps: Bring Down Multiply Add Repeat last two steps Ascending Order 5; ; ; 5 ; ; ;, 0 0 The x intercept, the thing you put into synthetic division and is all the same. The factor is the only opposite Insert 0 if you are missing a degree term 0 4 If you are going to put the opposite sign of the x intercept into synthetic division, you must subtract Potential Factors Solve by Inspection x-intercepts End behavior Store x/graph Calc Zero 0 ; 0 :,0, Multiplicity 3 ; y-intercept 0, TOV Degree Max number of x-intercepts Degree Degree Min number of x-intercepts 0 Max number of turns Degree Degree Min number of turns 0* Long Division Steps: Goes Into Multiply Subtract Bring Down Repeat Coefficient Binomials/ Trinomials etc, Quadratic divisors - Use Long Division Min Degree = # turns + Calc: Store x TI84 up upenter Enter Graph and find x-intercepts TI83 nd or nd Entry Entry Enter

13 Gen Page 3 C Trigonometry Review Special Triangles. 80 Terminal arm S A , T Terminal Arm 3 Principal Axis C 0, Triangles hug the x-axis, Only inverse positives II nd Quadrant st Quadrant I sin III 3rd Quadrant 4th Quadrant IV 0, ± n : ±,,, (-,0) (,0) (0,-) Sine Graph Cosine Graph Negative Sine Graph Negative Cosine Graph GRAPH tan(x) Tan is Zero when sin is zero Tan is und when cos is zero

14 Gen Page 4 C Special Triangles 30,45,60 sin/cos/tan Notes Right Isosceles, with sides = Half an equilateral with sides Diagonal of a square with sides lengths of o 45 o 60 o 30 o must open up to the root 3. And Vice Versa 3 Check by Pythag 4 3

15 Gen Page 5 C Trig Summary θ θ, 0 θ, 0,, 0, 0,

16 Gen Page 6 C Sine Definition Notes 3, 3, ! 5! 7!! 4! 6! !!

17 Gen Page 7 C ASTC Notes Quadrant II Quadrant I S Only Sin positive. A All (sin, cos, tan) positive Only Tan positive. T Only Cos positive. C Quadrant III Quadrant IV Remember: the hypotenuse is always positive. Students All Take Calculus S A T C Hypotenuse is always +

18 Gen Page 8 C Advanced Unit Circle Table, 3, θ 90 0, 3, θ 6 60,, θ π 45 4 θ π 0 3 3, 3, θ ,,0 3, 3,, 3,,0 3,,

19 Gen Page 9 C Trigonometric Functions Review *(h,k)(c,d) a sin a cos : : : : Remember: Factor the brackets so has a coefficient of Sin starts on the y-axis on the centre line and goes up/down Cos starts on the y-axis on the top/(bottom) line and goes down/(up) a tan : Tan is Zero when sin is zero Tan is und when cos is zero -intercepts/domain Restrictions x-intercepts: Domain: : :, :, :, :, :, "b" multiplies the # of original solutions between 0 Rearranged Formula asin sin Transformations cos 90 cos90 90 sin 90 sin90 90 cos cot90 tan90 tan

20 Gen Page 0 C Trig Formula Sheet/Steps Review Pythagorean Identities sin cos tan sec IDs Fractions Factoring Conjugate cot csc Reciprocal and Quotient Identities Addition Identities sin sin cos cos Double Angle Identities cos sin cos sin Trig Identities: Steps Get into sin and cos Identities Conjugates, Foil Conjugate, Pythag, Simplify Distribution and FOIL where necessary Rearrange terms Fractions Simplify Adding and subtracting fractions LCD: Do to the top do to the bottom Multiplying by Flip and multiply Separate fractions (Complex Fractions) Add Fractions top and bottom, Flip and multiply OR Multiply top and bottom by LCD Factoring Check by distribution/foil GCF Trinomials Differences of Squares Chooses a to cross off the '' or combine with the #/ sin / cos.! tan tan tan tan tan

21 Gen Page C Trig Identities Proofs Pythagorean Identity sin cos Reciprocal Identities Double Angle Identities sin cos cos sin cos sin sin sin sin cos cos cos cos

22 C Trig Double Angle Review Gen Page

23 Gen Page 3 C Exponential Review Interest KEY ; ; with Compounding # : : ; ; ; 5 & ; Growth with "T" : / ; Continuous Growth :.78 : %.0 5%.5 40%.4 50%.5 60%.6 00% %. 5%.5 40%.4 50%.5 60%.6 95% : 00% 00 :. 0 0 ; Earthquakes, ph ; Sound KEY :,,,, log - Concentration of Hydrogen,,,,,

24 Gen Page 4 C Laws of Logarithms Review Math Alpha Math Laws of logarithms " " Must have same base to use laws. log What power must you raise "b" to, to equal "a"? 0 0, Slide "b" across.. The exponent of "a" can come down in front. Vice versa. 3. log Change of base. log log log :"c" is arbitrary log log log Positives go on top, negatives go on bottom and vice versa. Rule 3, 4 and 5 Must have coefficient of. "a" and "b" can both be taken to the same exponent, 0 Same base of exponent as logarithm, answer is "a" log 0 log Natural Logarithms: (same rules as logs) log ln ln Methods: "log" both sides "de-log" both sides "inverse": switch and Set Log arbitrarily Methods: "ln" both sides. "de-ln" both sides. log log Domain: Range: 0 0 : Domain: 0 Range: : 0 & 0,

25 Gen Page 5 C Rationals Review Holes: Factor the top, Factor the bottom. If a factor cancels, there is a hole when the factor equals zero : 3, 5 Domain Restriction: 3 Vertical Asymptote: 0 0. x y - : Domain Restriction: VA - und 0 Horizontal Asymptote: : : 3: :,,, : 0 : 3 3 : : 3 3?? Range Restrictions: : 0 : 0, 0 0, Slant Asymptote:,, t. # :,,,,,, Close to asymptote, through point/s, close to asymptote : 3 3

26 C Rationals Graphs Gen Page 6

27 Gen Page 7 C Function Operations Review Operations Composite Functions Pick an x-value to talk about. We aren't talking about another x-value till were done talking about that x-value. Inverse :,, x.

28 Gen Page 8 C Combinatorics Review Logic Guess and Check Fundamental Counting Principle Factorial Notation! Blanks,, # Repeats? #! Tree Diagrams :! Cases! Cases: Multiply within cases, add separate cases. Identical Objects #!!! Combinatorics Formulas Permutations: Order Matters Combinations: Doesn t Matter!!!!!! Binomial Theorem :!!! :!!!! ; ; Probability # of Options ;k is always one less than the term number.

29 Gen Page 9 C ncr, npr Tables! 3! 5! 3 5! 70! ! , We can only choose objects from a larger number of objects

30 Gen Page 30 C Table of Cards Hearts Diamonds Spades Clubs Ace Ace Ace Ace Jack Jack Jack Jack Queen Queen Queen Queen King King King King 5 card deck 4 suits 3 cards in each suit 4 of each rank Ace is both high and low 5 card poker hands # Hand # of Royal Flush Ace King Queen Jack 0 0-Ace same suit 4 4 Straight Flush card run same suit of a Kind same rank, other 64 Full House same rank, pair 3744 Flush 4 8 Jack 6 All same suit, no straight Straight card run, not all same suit of a kind kind, others not a pair 549 pair Queen different pairs, other 355 Pair King King 6 9 pair 3 others High Card Jack None of the above