CBC MATHEMATICS MATH 2412-PreCalculus Exam Formula Sheets

CBC MATHEMATICSMATH 2412-PreCalculusExam Formula Sheets System of Equations and Matrices 3 Matrix Row Operations: Switch any two rows.Multiply any row by a nonzero constant.Add any constant-multiple row to another Even and Odd functions Even function: 𝑓( 𝑥) 𝑓(𝑥)Odd function: 𝑓( 𝑥) 𝑓(𝑥) Graph Symmetry 𝑥 -axis symmetry: if (𝑥, 𝑦) is on the graph, then (𝑥, 𝑦) is also on the graph 𝑦-axis symmetry: if (𝑥, 𝑦) is on the graph, then ( 𝑥, 𝑦) is also on the graph origin symmetry: if (𝑥, 𝑦)f is on the graph, then ( 𝑥, 𝑦) is also on the graph Function Transformations Stretch and Compress 𝑦 𝑎𝑓(𝑥), 𝑎 0vertical: stretch 𝑓(𝑥) if 𝑎 1 Reflections 𝑦 𝑓(𝑥)reflect 𝑓(𝑥) about 𝑥 -axis 𝑦 𝑓( 𝑥)reflect 𝑓(𝑥) about 𝑦-axis Stretch and Compress 𝑦 𝑎𝑓(𝑥), 𝑎 0vertical: stretch 𝑓(𝑥) if 𝑎 1: compress 𝑓(𝑥) if 0 𝑎 1 𝑦 𝑓(𝑎𝑥), 𝑎 0horizontal: stretch 𝑓(𝑥) if 0 𝑎 1: compress 𝑓(𝑥) if 𝑎 1 Shifts 𝑦 𝑓(𝑥) 𝑘, 𝑘 0𝑦 𝑓(𝑥) 𝑘, 𝑘 0vertical: shift 𝑓(𝑥) up: shift 𝑓(𝑥) down 𝑦 𝑓(𝑥 ℎ) ℎ 0𝑦 𝑓(𝑥 ℎ), ℎ 0horizontal: shift 𝑓(𝑥) left: shift 𝑓(𝑥) rightCBC Mathematics 2019Fall

CBC MATHEMATICSMATH 2412-PreCalculusExam Formula Sheets Formulas/Equations Slope Intercept: 𝑦 𝑚𝑥 𝑏𝑚 𝑦2 𝑦1Point-Slope: 𝑦 𝑦1 𝑚(𝑥 𝑥1 ); 𝑥2 𝑥1 0 Slope: Average Rate of Change: Circle: Triangle: 𝐴𝑟𝑒𝑎 𝑏ℎ Rectangle: 𝑃𝑒𝑟𝑖𝑚𝑒𝑡𝑒𝑟 2𝑙 2𝑤 , Rectangular Solid: 𝑉𝑜𝑙𝑢𝑚𝑒 𝑙𝑤ℎ, 𝑆𝑢𝑟𝑓𝑎𝑐𝑒 𝐴𝑟𝑒𝑎 2𝑙𝑤 2𝑙ℎ 2𝑤ℎ Sphere: 𝑉𝑜𝑙𝑢𝑚𝑒 𝜋𝑟 3 , Right Circular Cylinder: 𝑉𝑜𝑙𝑢𝑚𝑒 𝜋𝑟 2 ℎ ,𝑥2 𝑥1Δ𝑦Δ𝑥 𝑓(𝑏) 𝑓(𝑎)𝑏 𝑎, where 𝑎 𝑏𝐴𝑟𝑒𝑎 𝜋𝑟 ��� 2𝜋𝑟 𝜋𝑑,124𝐴𝑟𝑒𝑎 𝑙𝑤𝑆𝑢𝑟𝑓𝑎𝑐𝑒 𝐴𝑟𝑒𝑎 4𝜋𝑟 23𝑆𝑢𝑟𝑓𝑎𝑐𝑒 𝐴𝑟𝑒𝑎 2𝜋𝑟 2 2𝜋𝑟ℎ General Form of Quadratic Function: 𝑓(𝑥) 𝑎𝑥 2 𝑏𝑥 𝑐 , (𝑎 0) Vertex (ℎ, 𝑘):ℎ or 𝑥 Quadratic Formula:( 𝑏2𝑎𝑏2𝑎 𝑏 𝑏2 4𝑎𝑐2𝑎𝑘 𝑎(ℎ)2 𝑏(ℎ) 𝑐,, 𝑓 ( 𝑏2𝑎)),or( 𝑏2𝑎,4𝑎𝑐 𝑏24𝑎)Axis of symmetry: 𝑥 ℎ Vertex Form of Quadratic Function: 𝑓(𝑥) 𝑎(𝑥 ℎ)2 𝑘vertex (ℎ, 𝑘) Polynomial function: 𝑓(𝑥) 𝑎𝑛 𝑥 𝑛 𝑎𝑛 1 𝑥 𝑛 1 𝑎1 𝑥 1 𝑎0 Polynomial graph has at most 𝑛 1 turning points. Remainder Theorem If polynomial 𝑓(𝑥) (𝑥 𝑐), remainder is 𝑓(𝑐). Factor Theorem If 𝑓(𝑐) 0, then 𝑥 𝑐 is a linear factor of 𝑓(𝑥).If 𝑥 𝑐 is a linear factor of 𝑓(𝑥), then 𝑓(𝑐) 0.CBC Mathematics 2019Fall

CBC MATHEMATICSMATH 2412-PreCalculusExam Formula Sheets Rational Zeros Theorem: for polynomial function 𝑓(𝑥) 𝑎𝑛 𝑥 𝑛 𝑎𝑛 1 𝑥 𝑛 1 𝑎1 𝑥1 𝑎0having degree of at least 1 and integer coefficients with 𝑎𝑛 0, 𝑎0 0 If𝑝𝑞, in lowest terms, is a rational zero of 𝑓, then 𝑝 must be a factor of 𝑎0 , and 𝑞 mustbe a factor of 𝑎𝑛 . Intermediate Value Theorem(for continuous function 𝑓(𝑥)) If 𝑎 𝑏 and if 𝑓(𝑎) and 𝑓(𝑏) have opposite signs, then 𝑓(𝑥) has at least one realzero between 𝑥 𝑎 and 𝑥 𝑏. Conjugate Pairs Theorem For polynomial functions 𝑓(𝑥) with real coefficients: If 𝑥 𝑎 𝑏𝑖 is azero of 𝑓(𝑥), then 𝑥 𝑎 𝑏𝑖 is also. Rational function: 𝑓(𝑥) 𝑝(𝑥)𝑞(𝑥),𝑝(𝑥) and 𝑞(𝑥) polynomials, but 𝑞(𝑥) 0. Vertical Asymptote: 𝑥 zero of denominator in reduced 𝑓(𝑥) Horizontal Asymptote: 𝑦 0 if degree of 𝑝(𝑥) degree of 𝑞(𝑥) 𝑦 𝑙𝑒𝑎𝑑𝑖𝑛𝑔 𝑐𝑜𝑒𝑓𝑓𝑖𝑐𝑖𝑒𝑛𝑡 𝑜𝑓 𝑝(𝑥)𝑙𝑒𝑎𝑑𝑖𝑛𝑔 𝑐𝑜𝑒𝑓𝑓𝑖𝑐𝑖𝑒𝑛𝑡 𝑜𝑓 𝑞(𝑥)if degree of 𝑝(𝑥) degree of 𝑞(𝑥) Oblique Asymptote: 𝑦 𝑞𝑢𝑜𝑡𝑖𝑒𝑛𝑡 of𝑝(𝑥)𝑞(𝑥)if degree of 𝑝(𝑥) degree of 𝑞(𝑥) Composite Function (𝑓 𝑔)(𝑥) 𝑓( 𝑔(𝑥) ) Exponential Function: 𝑓(𝑥) 𝑎 𝑥 If 𝑎𝑢 𝑎𝑣 , then 𝑢 𝑣 Logarithmic Function: 𝑓(𝑥) log 𝑎 (𝑥) log 𝑎 (1) 0 ,log 𝑎 (𝑎) 1 ,𝑎log𝑎 (𝑀) 𝑀 ,log 𝑎 (𝑎 𝑝 ) 𝑝 log 𝑎 ( 𝑀 𝑁 ) log 𝑎 (𝑀) log 𝑎 (𝑁) log 𝑎 ( log 𝑎𝑀) log 𝑎 (𝑀) log 𝑎 (𝑁)𝑁(𝑀𝑝 ) 𝑝 log 𝑎 (𝑀) If log 𝑎 (𝑀) log 𝑎 (𝑁), then 𝑀 𝑁.CBC Mathematics 2019Fall

CBC MATHEMATICSMATH 2412-PreCalculusExam Formula Sheets If 𝑀 𝑁, then log 𝑎 (𝑀) log 𝑎 (𝑁).log(𝑀)log 𝑎 (𝑀) Change of Base formulalog(𝑎)orlog 𝑎 (𝑀) ln(𝑀)ln(𝑎) Exponential Models Formulas Simple Interest: 𝐼 𝑃𝑟𝑡𝑟 𝑛 𝑡 Compound Interest: 𝐴 𝑃 (1 )𝑛 Continuous Compounding: 𝐴 𝑃𝑒 𝑟 𝑡 Effective Rate of Interest:𝑟 𝑛Compounding 𝑛 times per yearCompounding continuously per year𝑟𝑒𝑓𝑓 (1 ) 1𝑛𝑟𝑒𝑓𝑓 𝑒 𝑟 1 Growth & Decay: 𝐴(𝑡) 𝐴0 𝑒 𝑘 𝑡 Newton’s Law of Cooling: 𝑢(𝑡) 𝑇 (𝑢0 𝑇)𝑒 𝑘 𝑡 Logistic Model: 𝑃(𝑡) 𝑐1 𝑎𝑒 𝑏 𝑡 Sequences and Series 𝑛! 𝑛(𝑛 1)(𝑛 2) (3)(2)(1)𝑛!𝑃(𝑛, 𝑟) Arithmetic Sequence:𝑛𝑡ℎ term𝑎𝑛 𝑎1 (𝑛 1)𝑑(𝑛 𝑟)!𝐶(𝑛, 𝑟) 𝑛! 𝑟!(𝑛 𝑟)!𝑛Sum of first 𝑛 terms 𝑆𝑛 𝑛𝑘 1(𝑎1 (𝑘 1)𝑑) 2 (𝑎1 𝑎𝑛 )𝑛or 𝑆𝑛 𝑛𝑘 1(𝑎1 (𝑘 1)𝑑) (2𝑎1 (𝑛 1)𝑑).2 Geometric Sequence:𝑛𝑡ℎ term𝑎𝑛 𝑎1 (𝑟)𝑛 11 𝑟 𝑛𝑘 1Sum of first 𝑛 terms 𝑆𝑛 𝑛 𝑎1 for 𝑟 0,1𝑘 1 𝑎1 𝑟1 𝑟 Geometric Series:𝑘 1 𝑘 1 𝑎1 𝑟𝑎11 𝑟if 𝑟 1CBC Mathematics 2019Fall

CBC MATHEMATICSMATH 2412-PreCalculusExam Formula Sheets Binomial Theorem:𝑛(𝑥 𝑎)𝑛 𝑛𝑗 0 (𝑛𝑗) 𝑥 𝑛 𝑗 𝑎 𝑗 (𝑛0)𝑥 𝑛 (𝑛1)𝑥 𝑛 1 𝑎 (𝑛 1)𝑥𝑎𝑛 1 (𝑛𝑛)𝑎𝑛 Trigonometry Circular Measure and Motion Formulas 𝑠 𝑟𝜃Arc Length𝑠Linear Speed 𝑣 , 𝑣 𝑟𝜔𝑡1Area of Sector𝐴 𝑟2𝜃Angular Speed𝜔 2𝜃𝑡 Acute �𝑖𝑡𝑒 sin(𝜃) csc(𝜃) cos(𝜃) sec(𝜃) 𝑎𝑐𝑐𝑎 𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡tan(𝜃) 𝑜𝑡𝑒𝑛𝑢𝑠𝑒cot(𝜃) 𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡𝑏𝑎𝑎𝑏 𝑝𝑝𝑜𝑠𝑖𝑡𝑒 General Angle𝑏 sin(𝜃) cos(𝜃) csc(𝜃) ,𝑏 0𝑟𝑟𝑏sec(𝜃) 𝑎𝑟𝑟𝑎tan(𝜃) cot(𝜃) ,𝑎 0𝑏𝑎𝑎𝑏,𝑏 0 Cofunctions𝜋𝜋𝜋 sin(𝜃) cos ( 𝜃) ,cos(𝜃) sin ( 𝜃) ,tan(𝜃 ) cot ( 𝜃) csc(𝜃) sec ( 𝜃) ,sec(𝜃) csc ( 𝜃) ,cot(𝜃 ) tan ( 𝜃)2𝜋22𝜋2𝜋22 Fundamental Identitiessin(𝜃) tan(𝜃) csc(𝜃) sin2 (𝜃) cos2 (𝜃) 1cos(𝜃)1sin(𝜃) Even-Odd Identities sin( 𝜃 ) sin(𝜃 ) csc( 𝜃 ) csc(𝜃 ) Inverse Functionscot(𝜃) sec(𝜃) cos(𝜃)sin(𝜃)1cos(𝜃)cot(𝜃) 1tan(𝜃)tan2 (𝜃) 1 sec 2 (𝜃)cot 2 (𝜃) 1 csc 2 (𝜃)cos( 𝜃) cos(𝜃)sec( 𝜃) sec(𝜃)tan( 𝜃) tan(𝜃)cot( 𝜃) cot(𝜃)𝜋𝜋22 𝑦 sin 1 (𝑥) means 𝑥 sin(𝑦) where 1 𝑥 1 and 𝑦 𝑦 cos 1 (𝑥) means 𝑥 cos(𝑦) where 1 𝑥 1 and 0 𝑦 𝜋 𝑦 tan 1 (𝑥) means 𝑥 tan(𝑦) where 𝑥 and 𝑦 𝜋𝜋22CBC Mathematics 2019Fall

CBC MATHEMATICSMATH 2412-PreCalculusExam Formula Sheets𝜋𝜋22 𝑦 csc 1 (𝑥) means 𝑥 csc(𝑦) where 𝑥 1 and 𝑦 , 𝑦 0 𝑦 sec 1 (𝑥) means 𝑥 sec(𝑦) where 𝑥 1 and 0 𝑦 𝜋, 𝑦 cot 1 (𝑥) means 𝑥 cot(𝑦) where 𝑥 and 0 𝑦 𝜋𝑦 𝜋2 Sum and Difference Formulas sin(𝛼 𝛽) sin(𝛼 ) cos(𝛽) cos(𝛼 ) sin(𝛽) sin(𝛼 𝛽) sin(𝛼 ) cos(𝛽) cos(𝛼) sin(𝛽) cos(𝛼 𝛽) cos(𝛼 ) cos(𝛽) sin(𝛼) sin(𝛽) cos(𝛼 𝛽) cos(𝛼 ) cos(𝛽) sin(𝛼) sin(𝛽) tan(𝛼 𝛽) tan(𝛼) tan(𝛽)tan(𝛼 𝛽) 1 tan(𝛼) tan(𝛽)tan(𝛼) tan(𝛽)1 tan(𝛼) tan(𝛽) Half-Angle Formulas𝛼1 cos(𝛼)22𝛼1 cos(𝛼)22𝛼1 cos(𝛼)21 cos(𝛼) sin ( ) cos ( ) tan ( ) 1 cos(𝛼)sin(𝛼) sin(𝛼)1 cos(𝛼) Double-Angle Formulas sin(2𝜃 ) 2 sin(𝜃 ) cos(𝜃 ) cos(2𝜃 ) cos2 (𝜃 ) sin2 (𝜃 ) 2cos 2 (𝜃 ) 1 1 2sin2 (𝜃 ) tan(2𝜃) 2 tan(𝜃)1 tan2 (𝜃) Product to Sum Formulas1 sin(𝛼) sin(𝛽) [cos(𝛼 𝛽) cos(𝛼 𝛽)] cos(𝛼) cos(𝛽) [cos(𝛼 𝛽) cos(𝛼 𝛽)] sin(𝛼) cos(𝛽) [sin(𝛼 𝛽) sin(𝛼 𝛽)]21212 Sum to Product Formulas sin(𝛼) sin(𝛽) 2 sin (𝛼 𝛽2) cos (𝛼 𝛽2)CBC Mathematics 2019Fall

CBC MATHEMATICSMATH 2412-PreCalculusExam Formula Sheets𝛼 𝛽 sin(𝛼) sin(𝛽) 2 sin ( cos(𝛼) cos(𝛽) 2 cos ( cos(𝛼) cos(𝛽) 2 sin () cos (2𝛼 𝛽𝛼 𝛽) cos (2𝛼 𝛽2)2𝛼 𝛽) sin ()2𝛼 𝛽2) Law of Sines sin(𝐴)𝑎 sin(𝐵)𝑏 sin(𝐶)𝑐 Law of Cosines 𝑎2 𝑏2 𝑐 2 2𝑏𝑐 cos(𝐴) 𝑏2 𝑎2 𝑐 2 2𝑎𝑐 cos(𝐵 ) 𝑐 2 𝑎2 𝑏2 2𝑎𝑏 cos(𝐶 ) Area of SSS Triangles (Heron’s Formula) 𝐾 𝑠(𝑠 𝑎)(𝑠 𝑏)(𝑠 𝑐) ,1where 𝑠 (𝑎 𝑏 𝑐)2 Area of SAS Triangles 1𝐾 𝑎𝑏 sin(𝐶) ,211𝐾 𝑏𝑐 sin(𝐴) ,𝐾 𝑎𝑐 sin(𝐵)22 For 𝑦 𝐴sin(𝜔𝑥 𝜑) or 𝑦 𝐴cos(𝜔𝑥 𝜑) , with 𝜔 0 Amplitude 𝐴 ,Period 𝑇 2𝜋𝜔,Phase shift 𝜑𝜔CBC Mathematics 2019Fall

CBC MATHEMATICS MATH 2412-PreCalculus Exam Formula Sheets CBC Mathematics 2019Fall Rational Zeros Theorem: for polynomial function ( T) T 1 T 1 1 T1 0 having degree of at least 1 and integer coefficients with 0, 0 0 If