C11 - Table of Contents

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1 C11 Txt Page 1 C11 - Table of Contents Duotang/Notes/Homework C11 - Methods C11 - Remember C11 - General Graphs C11 - General Equations C11-3//7 Calculator Buttons C Sequences and Series Review C Trigonometry Review C mbiguous Case of Sine C Rationalize C Graphing Quadratic's review C Solving quadratic's X int review C X intercepts Review C Quadform C Radicals Review C Rational's Review M Sequence and Series (Patterns) M rithmetic Sequence (List) M rithmetic Series (Sum) M Geometric Sequence M Geometric Series M Infinite Geometric Sequence M Infinite Geometric Series M Sigma Sequences and Series M Word Problems M Trigonometry M Reference ngle Theta M ngles in Standard Position M Co-terminal ngles M STC M Special Triangles M Trig Equations M Unit Circle M Graphs TOV M Factoring Trig Equations M Sine Law M Cosine Law M Trig Graphing C Quadratic Functions C Graphing Table of Values C Standard Form C Translations C Vertical Expansions/Compressions C Complete the Square C Vertex Form C11-3/ - Chapter 3/ Link C Word problems C Quadratic Equations C X intercepts C Factored Form C Square Root Method C Quadratic Formula C Discriminant C Word Problems C Radicals C Simplifying/Expanding Radicals C dding Subtracting Radicals C Multiplying Dividing Radicals C Rationalizing Denominator C Solving Radical Equations C Restrictions M Rationals (Fractions) M Simplifying M Restrictions M Multiplying/Dividing M dding/subtracting M Rational Equations M Word Problems M bsolute Value and Reciprocals M bsolute Value Equations M bsolute Value Graphs M Reciprocals M Systems M Number of Intersections M11-8.1/2 - Linear Quadratic M11-8.1/2 - Equation 1 Equals Equation 2 M11-8.1/2 - Equation 3 Equals Zero M Inequalities M Linear Inequalities in Two Variable M Linear/Quadratic Inequalities One Variables M Quadratic Inequalities in Two Variables

2 C11 Txt Page 2 C11 - Methods Radicals Laws Simplifying Expanding dding Subtracting Coefficients Multiplying Dividing Rationalizing/Conjugates Solving Isolate root/separate roots Square Both Sides/gain Restrictions: Set underneath root 0 and solve. Quadratics Graphing Table of Values TOV Transformations Complete the Square Calculator 2nd Calc bsolute Value Isolate bsolute Value +" case: distribute a + into absolute value " " case: distribute a into absolute value x = 3 Impossible, no solution. y = +case LHS, y = case LHS, y = RHS Piecewise function: { y = "+" case, Domain " " case, Domain Set absolute value 0 and solve Set absolute value < 0 and solve Reciprocals Restrictions: Set Denominator 0 and solve Domain: x Restrictions V: x = Restrictions Invariant points: set denominator = ±1 and solve Intersection of Original Graph-Line y = ±1 Solving x intercepts y = 0 (x, 0) Get = 0. Factor 0 = (x + 2)(2x + 3) Set brackets = 0 seperately and solve x + 2 = 0 x = 2 Square Root Method Quadratic Formula x = b ± b ac 2a (x + 2) = ± 5 Discriminant: b 2 ac Case 1: b ac > 0 Two x ints Case 2: b ac < 0 No x int Case 3: b ac = 0 One x int Systems y = y y ± y = 0 y = 0 2x + 3 = 0 x = Find Intersection OR Find x intercepts y ± y = y (a)(b) = 0 (a) = 0 (b) = 0 Rationals Do to top/do to bottom Multiply tops/bottoms Flip and Multiply dding Subtracting LCD Multiply LDC Do to one/do to all Solving State Restrictions Check answer. Trigonometry STC/Unit Circle Special Triangles θ = sin (+) θ, θ, θ Sine/Cos Law SS mbiguous θ Inequalities One Variable - Number Line Two Variables Shading Test Point(s) Linear Quadratic We only sin, cos or tan reference angles. We never sin, cos or tan angles in standard position. STC determines all positives and negatives.

3 Systems C11 Txt Page 3

4 C11 Txt Page C11 - Remember Radicals Laws General Check nswer Inequalities Laws Radicals x + x + a + b a + b Quadratics y = f(x) x = 9 x 9 = 0 (x 3)(x + 3) = 0 x = 3, x = 3 Solving = 7 When you FOIL a conjugate you only have to do F & L (because O & I cancel) 3 + = = 25 = 5 x = 9 x = ± 9 x = ±3 ±; Plus/Minus x 9 x ± 9 x +3 x 3 bsolute Values " + " Case " " Case Piecewise Sequences Series Formulas Blanks t, n =, d, r x 9 x ± 9 x +3 x 3 3 x 3 Reciprocals Vs NPVs IPs a b = (a + b) a + b = (a b) Systems (x, y), (x, y) Rationals Restrictions Invariant points (x, ±1) Be aware of negatives in front of brackets. Trigonometry STC Unit Circle SOHCHTO We only inverse (find angle) sin, cos or tan using Positive (+) reference angles. θ is always positive, between 0 and 90 Substitution Special Triangles Calculator Systems y = y or y ± y = 0 Common Mistakes = x + a 1 + a x Corrections + 3 = + 3 = = 1.75 x + a = x x x + a x = 1 + a x Separate Fractions Separate Fractions 1 + x 1 + x x 2 2 = 3 2 x = + 1 x 1 2 2

5 C11 Txt Page 5 C11 - General Graphs Linear y = mx + b y = x Quadratic y = a(x h) + k y = x Polynomial y = ax + bx + cx + d y = x bsolute Value y = a b(x h) + k y = x Reciprocals y = a + k x h y = 1 x

6 C11 Txt Page 6 C11 - General Equations General Equation y = af b(x h) + k Example Linear y = mx + b (y = x) y = 3x 5 Quadratic y = ax + bx + c (y = x ) y = 2x 6x + Polynomial (3rd degree) y = ax + bx + cx + d (y = x ) y = x + 3x 2

7 C11 Txt Page 7 C Sequences and Series Review rithmetic +d +d 2, 5, 8, 32, t t t? t t n = 1 n = 2 n = 3 n = 11 n = n t = first term (aka: "a") d = common difference t = term n n = Term #, or number of terms t = t + (n 1)d General Term Formula d = t t t = t + (n m)d *Difficult s = n 2 (t + t ) If "n" and t is known. If you substitute the general formula into this formula, you get the one below. s = n 2 (2t + (n 1)d) If n is known. Note: a = t Geometric r r 2, 6, 18, 86, t t t t? t r = common ratio t = t r General Term Formula t r = t t r = t r = t *Difficult t s = t (1 r ) 1 r If n is known. t = s s s = t rt 1 r If t is known. If you substitute the general formula into this formula, you get the one above. s = t 1 r 1 < r < 1 Convergent Has sum r > 1 Divergent No sum OR r > 1 r < 1

8 C11 Txt Page 8 1 C Trigonometry Review 5 o o Special Triangles o SOH - CH - TO o sinθ = O H cosθ = H tanθ = O θ = sin (+ O H ) tanθ = sinθ cosθ y = 0 x = r 180 Terminal arm S T Terminal rm θ θ 90 x = 0 y = r 270 Triangles hug the x-axis, θ C x = 0 y = r 360 Principal xis y = 0 x = r Only inverse positives = θ (0,1) 1 r = Hyp y (-1,0) x (1,0) y = Opp (0,-1) x = dj sinθ = y cosθ = x tanθ = y x b c 180 C a B Opposite Pairs Period II III Sine Law: 2nd Quadrant θ = 180 θ θ = θ 1st Quadrant I θ = θ θ = 360 θ 3rd Quadrant th Quadrant θ = θ ± 360 IV a = b = c sin sin sinb sinc or = sinb = sinc a b c (to find side) (to find angle) Put what you are looking for on top. Sine Law: Opposite pair and one other piece of information mbiguous Case: SS Four acute cases, two obtuse cases. Cosine Law: Find smallest ngle First. p = (sin, cos) p = (tan) c = a + b 2abcosC General Solution: θ = θ ± pn, n I Notice: This pattern should occur. Cosine Law: SSS (hard) and SS (easy) Sine Graph y = sinx Cosine Graph y = cosx y = sinx y = cosx

9 C11 Txt Page 9 C mbiguous Case of Sine (SS) Review cute Triangle 8 Steps 1. Draw the given angle as seen on left. 2. Draw the given side, going up (not opposite the angle) Calculate the height of the triangle 8 30 sin θ = Opp a = Hsinθ Hyp sin30 = Opp 8 8 sin30 = Opp Opp = a = a = Height one triangle a > 8 a > b one triangle a < a < H no triangle < a < 8 H < a < B Two triangles or Notice: Both triangles have an angle of 30, a side going up of 8, and a side opposite to 30 of 6. Notice the isosceles triangle. Obtuse Triangle a > b One triangle B a < b No triangle B In either the acute or obtuse triangles, we want to treat the opposite side to the angle like a door on a hinge that can swing accordingly.

10 C11 Txt Page 10 C Graphing Quadratics Review Complete the Square FOIL FOIL Factor Vertex Form Standard Form Factored Form General Equation: y = a(x p) + q y = ax + bx + c y = a(x z)(x r) Vertex (V): (p, q) b 2a, y z + r, y 2 p is opposite xis of Symmetry (OS): x = p z + r 2 x = b 2a x = Domain: x R x R x R X intercepts: set y = 0, and solve set y = 0, and solve Set brackets separately equal to zero, and solve (x z) = 0 (x r) = 0 x = z x = r Y intercepts: set x = 0, and solve y int: (0, c) set x = 0, and solve Remember: The "p" value is the opposite of the value inside the brackets with x. a a > 0: Minimum (a is positive) a < 0: Maximum (a is negative) Opens Up Opens down a > 1 a < 1 Vertical Expansion or 1 < a < 1 Vertical Compression Range y min y max a = 2 a = 0.5 Max/Min Value y = min y = max

11 C11 Txt Page 11 C11-3//8 - Calculator Buttons Use a graphing calculator to graph the following. Find the Vertex, Max/Min, intercepts, intersects. GRPH: y = y = x 2x 3 lways CLER If Y = is not empty ZOOM 6 If can't see parabola, or change window. VERTEX: 2ND 2ND CLC TRCE CLC TRCE Minimum 3 If opens upward Maximum If opens downward Left Bound? Right Bound? Move barely left of vertex Move barely right of vertex y- INTERCEPT: Guess? 2ND CLC TRCE Vertex: (1, ) Value 1 x- INTERCEPT: X =? 0 2ND CLC TRCE Zero 2 Y = 3 y int: (0, 3) Or y = 0 and find intersections Left Bound? Right Bound? Move barely left of x-int Move barely right of x-int Guess? x intercept: ( 1, 0) x intercept: (3, 0) INTERSECTION: Graph two equations: y = y = x 1 y = y = 2x + 3x 5 Find intersection: 2ND CLC TRCE Intersection 5 Using and go near an intersection First Curve? Second Curve? Guess? Intersection: (1,0) REPET Intersection: ( 2, 3)

12 C11 Txt Page 12 C Solving Quadratics for x-intercepts Review 1) Factoring: y = bx + c y = 12x + 8x y = x(3x + 2) 0 = (x)(3x + 2) x = 0 3x + 2 = 0 x = 0 x = 2) Factoring: y = ax + bx + c y = x + 5x + 6 y = (x +2)(x + 3) 0 = (x + 2)(x + 3) x + 3 = 0 x + 2 = 0 x = 3 x = 2 3) Factoring: y = ax + bx + c y = 2x + 7x + 6 y = 2x +3x + x + 6 y = (2x +3x) (+x + 6) y = x(2x + 3) + 2(2x + 3) y = (x + 2)(2x + 3) x + 2 = 0 x = 2 ) Factoring: y = a b y = x 9 y = (x + 3)(x 3) 0 = (x + 3)(x 3) x + 3 = 0 x = 3 2x + 3 = 0 x = x 3 = 0 x = 3 Remove Greatest Common Factor "GCF." Set y = 0 Set brackets equal to zero "0" Solve a = 1 Factor set y = 0 Set brackets equal to zero "0" Solve a 1 Decompose Group GCF Switch Set y = 0 Set brackets equal to zero "0" Solve Differences of Squares Factor Set y = 0 Set brackets equal to zero "0" Solve a = 0 2 X 3 = c = b 5 3 X = ac = b 7 (a)(b) = 0 b = 0 1,2,3,6 5) Factoring: y = ax + bx + c y = 2x + 6x + x = (6) ± (6) (2)() 2(2) x = x = x = x = 6 2 x = 1 x = 2 Quadratic Formula: x = ± One Posititve Equation, One Negative Equation Solve Discriminant: b 2 ac 6) Factoring: y = a(x p) + q y = 2(x 2) 2 0 = 2(x 2) 2 2 = 2(x 2) 1 = (x 2) ± 1 = (x 2) ±1 = (x 2) ±1 + 2 = x x = x = 3 x = x = 1 Solve using the Square Root Method set y = 0 Square root both sides of the equation One Posititve Equation, One Negative Equation Solve CakeMixMath Nicholas Cragg, BCOM Copyright 2012

13 C11 Txt Page 13 C x int, y = 0 (x, 0) Review Graphing TOV x y x int: -1 0 ( 1,0) (3,0) Graphing Calculator: 2ND CLC x int: Factoring y = x 2x 3 0 = (x 3)(x + 1) x 3 = 0 x = 3 (3,0) x + 1 = 0 x = 1 ( 1,0) ( 1,0) (3,0) Complete Square/Square Root Method Quad Formula x = b ± b ac 2a y = x 2x 3 y = (x 2x) 3 y = (x 2x + 1 1) 3 y = (x 1) 0 = (x 1) = (x 1) ± = (x 1) ±2 = x 1 +2 = x 1 x = 3 2 = x 1 x = 1 b 2 vertex: (1, ) y = x 2x 3 x = 2 ± ( 2) (1)( 3) 2(1) x = 2 ± 16 2 x = 2 + x = x = 3 x = 1 x int: (3,0) ( 1,0) x int: (3,0) ( 1,0)

14 C11 Txt Page 1 C Quadratic Formula TI-83/8 Program Notes Prgm New Enter 2nd lpha :if necessary Quadform Spell it out PROGRM:QUD :Prompt,B,C :Disp (-B- (B -C))/(2) :Disp (-B+ (B -C))/(2) Prgm I/O Prompt Enter lpha, B, C Prgm I/O Disp Enter (-B+ (B C))/(2) Negative 1st, minus' between Prgm I/O Disp Enter (-B (B C))/(2) Negative 1st, minus' between 2nd Quit Running the Program Prgm Quadform Enter Enter =# B=# C=# Enter values for, B, C Enter nswers are x-intercepts If I does not work Pgrm Edit If it does not look like the box above something is wrong Remember to try a question you know how to factor and solve, and graph a TOV and check your calculator

15 C11 Txt Page 15 C Radicals Review Simplifying Radicals Index x Radical Radicand x = x x = x x x = x x = x x 2 = 2 2 = = 2 2 = 2 2 x = x x = x x x = x x x = x 2 = 2 2 = = = 2 dding and Subtracting Radicals x a + y a = (x + y) a Can only add or subtract like radicals: same index, same radicand = (2 + 3) 3 = 5 3 Multiplying and Dividing Radicals Can only multiply and divide if it has the same root index. x a y b = xy a b x a y b = x y a b = = = 20 6 = = 2 Exponents, Roots and bsolute Values x = x 5 = 5 ( x ) = x 7 = 7 ( x ) = x 8 = 8 Rationalizing the Denominator a = b a b b = b a b b b = a b b 2 = = = a 1 b = a 1 b 1 + b 1 + b = a 1 + b = 1 b 1 + b a 1 + b = 1 b a 1 + b 1 b = = 1(3 + 5 ) = (3 5 )(3 + 5 ) = (9 ) Conjugate (a + b)(a b) = a ab + ab b = a b a + b a b = a a a b + a b b b = a b x + 3 x 3 = x x x 3 + x = x 9 = x 3

16 C11 Txt Page 16 C Rationals Review Simplify/Multiply/Divide NPVs/Restrictions Watch out for Factor Multiply/Divide: Flip and Multiply Simplify Common Mistakes Distribution GCF=-1 Bedmas dd/subtract Factor LCD Do to top, Do to bottom dd/subtract Sometimes factor the top to see if it simplifies first Equations Factor LCD Get an LCD/Multiply by LCD/Cross Multiply Simplify Solve Possibly refactor the top/simplify again at end

17 C11 Txt Page 17 C Reciprocal Steps So with reciprocals we always want to make sure we graph the original first If we forget how to graft the original we must use a table of values We want to remember the vertical asymptotes are the locations of the x intercepts on the original. (Factor!) Then we want to draw horizontal dotted lines where y = 1 equals one and y = -1 Then we want to put points on the intersection of the original and the two horizontal lines Then we are ready to graph Make sure you use the graphing calculator to check your answer

Core Mathematics 2 Trigonometry

Core Mathematics 2 Trigonometry Edited by: K V Kumaran Email: kvkumaran@gmail.com Core Mathematics 2 Trigonometry 2 1 Trigonometry Sine, cosine and tangent functions. Their graphs, symmetries and periodicity.