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CBC MATHEMATICS MATH 2412-PreCalculus Exam Formula Sheets  System of Equations and Matrices  3 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 : 푓(−푥) = 푓(푥) Odd function: 푓(−푥) = −푓(푥)  Graph Symmetry • 푥-axis symmetry: if (푥, 푦) is on the graph, then (푥, −푦) is also on the graph • 푦-axis symmetry: if (푥, 푦) is on the graph, then (−푥, 푦) is also on the graph • origin symmetry: if (푥, 푦)f is on the graph, then (−푥, −푦) is also on the graph  Function Transformations  Stretch and Compress

• 푦 = 푎푓(푥), 푎 > 0 vertical: stretch 푓(푥) if 푎 > 1  Reflections

• 푦 = −푓(푥) reflect 푓(푥) about 푥-axis • 푦 = 푓(−푥) reflect 푓(푥) about 푦-axis  Stretch and Compress

• 푦 = 푎푓(푥), 푎 > 0 vertical: stretch 푓(푥) if 푎 > 1 : compress 푓(푥) if 0 < 푎 < 1 • 푦 = 푓(푎푥), 푎 > 0 horizontal: stretch 푓(푥) if 0 < 푎 < 1 : compress 푓(푥) if 푎 > 1  Shifts

• 푦 = 푓(푥) + 푘, 푘 > 0 vertical: shift 푓(푥) up 푦 = 푓(푥) − 푘, 푘 > 0 : shift 푓(푥) down

• 푦 = 푓(푥 + ℎ) ℎ > 0 horizontal: shift 푓(푥) left 푦 = 푓(푥 − ℎ), ℎ > 0 : shift 푓(푥) right

CBC Mathematics 2019Fall CBC MATHEMATICS MATH 2412-PreCalculus Exam Formula Sheets  Formulas/Equations

• Intercept: 푦 = 푚푥 + 푏 Point-Slope: 푦 − 푦1 = 푚(푥 − 푥1) 푦2−푦1 • Slope: 푚 = ; 푥2 − 푥1 ≠ 0 푥2−푥1 Δ푦 푓(푏)−푓(푎) • Average Rate of Change: = , where 푎 ≠ 푏 Δ푥 푏−푎

• Circle: 퐶𝑖푟푐푢푚푓푒푟푒푛푐푒 = 2휋푟 = 휋푑, 퐴푟푒푎 = 휋푟2 1 • Triangle: 퐴푟푒푎 = 푏ℎ 2 • Rectangle: 푃푒푟𝑖푚푒푡푒푟 = 2푙 + 2푤 , 퐴푟푒푎 = 푙푤 • Rectangular Solid: 푉표푙푢푚푒 = 푙푤ℎ, 푆푢푟푓푎푐푒 퐴푟푒푎 = 2푙푤 + 2푙ℎ + 2푤ℎ 4 • Sphere: 푉표푙푢푚푒 = 휋푟3 , 푆푢푟푓푎푐푒 퐴푟푒푎 = 4휋푟2 3 • Right Circular Cylinder: 푉표푙푢푚푒 = 휋푟2ℎ , 푆푢푟푓푎푐푒 퐴푟푒푎 = 2휋푟2 + 2휋푟ℎ

 General Form of Quadratic Function: 푓(푥) = 푎푥2 + 푏푥 + 푐 , (푎 ≠ 0)

−푏±√푏2−4푎푐 • Quadratic Formula: 푥 = 2푎 푏 • Vertex (ℎ, 푘): ℎ = − 푘 = 푎(ℎ)2 + 푏(ℎ) + 푐, 2푎 푏 푏 푏 4푎푐−푏2 or (− , 푓 (− )), or (− , ) 2푎 2푎 2푎 4푎 • Axis of symmetry: 푥 = ℎ  Vertex Form of Quadratic Function: 푓(푥) = 푎(푥 − ℎ)2 + 푘 vertex (ℎ, 푘)

푛 푛−1 1  function: 푓(푥) = 푎푛푥 + 푎푛−1푥 + ⋯ + 푎1푥 + 푎0  Polynomial graph has at most 푛 − 1 turning points.  Remainder Theorem • If polynomial 푓(푥) ÷ (푥 − 푐), remainder is 푓(푐).  Factor Theorem • If 푓(푐) = 0, then 푥 − 푐 is a linear factor of 푓(푥). • If 푥 − 푐 is a linear factor of 푓(푥), then 푓(푐) = 0.

CBC Mathematics 2019Fall CBC MATHEMATICS MATH 2412-PreCalculus Exam Formula Sheets

푛 푛−1 1  Rational Zeros Theorem: for polynomial function 푓(푥) = 푎푛푥 + 푎푛−1푥 + ⋯ + 푎1푥 + 푎0 having degree of at least 1 and integer coefficients with 푎푛 ≠ 0, 푎0 ≠ 0 푝 • If , in lowest terms, is a rational zero of 푓, then 푝 must be a factor of 푎 , and 푞 must 푞 0 be a factor of 푎푛.  Intermediate Value Theorem(for 푓(푥)) • If 푎 < 푏 and if 푓(푎) and 푓(푏) have opposite signs, then 푓(푥) has at least one real zero between 푥 = 푎 and 푥 = 푏.  Conjugate Pairs Theorem • For polynomial functions 푓(푥) with real coefficients: If 푥 = 푎 + 푏𝑖 is a zero of 푓(푥), then 푥 = 푎 − 푏𝑖 is also.

푝(푥)  : 푓(푥) = , 푝(푥) and 푞(푥) , but 푞(푥) ≠ 0. 푞(푥)  Vertical Asymptote: 푥 = zero of denominator in reduced 푓(푥)  Horizontal Asymptote: • 푦 = 0 if degree of 푝(푥) < degree of 푞(푥) 푙푒푎푑푖푛푔 푐표푒푓푓푖푐푖푒푛푡 표푓 푝(푥) • 푦 = if degree of 푝(푥) = degree of 푞(푥) 푙푒푎푑푖푛푔 푐표푒푓푓푖푐푖푒푛푡 표푓 푞(푥)  Oblique Asymptote: 푝(푥) • 푦 = 푞푢표푡𝑖푒푛푡 of if degree of 푝(푥) > degree of 푞(푥) 푞(푥)

 Composite Function (푓 ∘ 푔)(푥) = 푓( 푔(푥) )  : 푓(푥) = 푎푥 • If 푎푢 = 푎푣, then 푢 = 푣

 Logarithmic Function: 푓(푥) = log푎(푥)

log푎(푀) 푝 • log푎(1) = 0 , log푎(푎) = 1 , 푎 = 푀 , log푎(푎 ) = 푝

• log푎( 푀 ∙ 푁 ) = log푎(푀) + log푎(푁) 푀 • log ( ) = log (푀) − log (푁) 푎 푁 푎 푎 푝 • log푎(푀 ) = 푝 ∙ log푎(푀)

• If log푎(푀) = log푎(푁), then 푀 = 푁.

CBC Mathematics 2019Fall CBC MATHEMATICS MATH 2412-PreCalculus Exam Formula Sheets

• If 푀 = 푁, then log푎(푀) = log푎(푁). log(푀) ln(푀) • Change of Base formula log (푀) = or log (푀) = 푎 log(푎) 푎 ln(푎)

 Exponential Models Formulas

• Simple Interest: 퐼 = 푃푟푡

푟 푛∙푡 • : 퐴 = 푃 (1 + ) 푛 • Continuous Compounding: 퐴 = 푃푒푟∙푡 • Effective Rate of Interest: 푟 푛 Compounding 푛 times per year 푟 = (1 + ) − 1 푒푓푓 푛 푟 Compounding continuously per year 푟푒푓푓 = 푒 − 1

푘∙푡 • Growth & Decay: 퐴(푡) = 퐴0푒 푘∙푡 • Newton’s Law of Cooling: 푢(푡) = 푇 + (푢0 − 푇)푒 푐 • Logistic Model: 푃(푡) = 1+푎푒−푏∙푡

and • 푛! = 푛(푛 − 1)(푛 − 2) ∙ ⋯ ∙ (3)(2)(1) 푛! 푛! • 푃(푛, 푟) = 퐶(푛, 푟) = (푛−푟)! 푟!(푛−푟)! • Arithmetic : 푡ℎ 푛 term 푎푛 = 푎1 + (푛 − 1)푑 푛 Sum of first 푛 terms 푆 = ∑푛 (푎 + (푘 − 1)푑) = (푎 + 푎 ) 푛 푘=1 1 2 1 푛 푛 or 푆 = ∑푛 (푎 + (푘 − 1)푑) = (2푎 + (푛 − 1)푑). 푛 푘=1 1 2 1 • Geometric Sequence: 푡ℎ 푛−1 푛 term 푎푛 = 푎1(푟) 1−푟푛 Sum of first 푛 terms 푆 = ∑푛 푎 푟푘−1 = 푎 ∙ for 푟 ≠ 0,1 푛 푘=1 1 1 1−푟 푎 • : ∑∞ 푎 푟푘−1 = 1 if |푟| < 1 푘=1 1 1−푟 CBC Mathematics 2019Fall CBC MATHEMATICS MATH 2412-PreCalculus Exam Formula Sheets  : 푛 푛 푛 푛−푗 푗 푛 푛 푛 푛−1 푛 푛−1 푛 푛 (푥 + 푎) = ∑푗=0 (푗) 푥 푎 = (0)푥 + (1)푥 푎 + ⋯ + (푛−1)푥푎 + (푛)푎

 Circular Measure and Motion Formulas 1 • 푠 = 푟휃 Area of Sector 퐴 = 푟2휃 2 푠 휃 • Linear Speed 푣 = , 푣 = 푟휔 Angular Speed 휔 = 푡 푡  Acute Angle 푏 표푝푝표푠푖푡푒 푎 푎푑푗푎푐푒푛푡 푏 표푝푝표푠푖푡푒 • sin(휃) = = cos(휃) = = tan(휃) = = 푐 ℎ푦푝표푡푒푛푢푠푒 푐 ℎ푦푝표푡푒푛푢푠푒 푎 푎푑푗푎푐푒푛푡 푐 ℎ푦푝표푡푒푛푢푠푒 푐 ℎ푦푝표푡푒푛푢푠푒 푎 푎푑푗푎푐푒푛푡 • csc(휃) = = sec(휃) = = cot(휃) = = 푏 표푝푝표푠푖푡푒 푎 푎푑푗푎푐푒푛푡 푏 표푝푝표푠푖푡푒  General Angle 푏 푎 푏 • sin(휃) = cos(휃) = tan(휃) = 푟 푟 푎 푟 푟 푎 • csc(휃) = ,푏 ≠ 0 sec(휃) = ,푎 ≠ 0 cot(휃) = ,푏 ≠ 0 푏 푎 푏  Cofunctions 휋 휋 휋 • sin(휃) = cos ( − 휃) , cos(휃) = sin ( − 휃) , tan(휃) = cot ( − 휃) 2 2 2 휋 휋 휋 • csc(휃) = sec ( − 휃) , sec(휃) = csc ( − 휃) , cot(휃) = tan ( − 휃) 2 2 2  Fundamental Identities sin(휃) cos(휃) • tan(휃) = cot(휃) = cos(휃) sin(휃) 1 1 1 • csc(휃) = sec(휃) = cot(휃) = sin(휃) cos(휃) tan(휃) • sin2(휃) + cos2(휃) = 1 tan2(휃) + 1 = sec2(휃) cot2(휃) + 1 = csc2(휃)  Even-Odd Identities • sin(−휃) = −sin(휃) cos(−휃) = cos(휃) tan(−휃) = − tan(휃) • csc(−휃) = −csc(휃) sec(−휃) = sec(휃) cot(−휃) = − cot(휃)  Inverse Functions 휋 휋 • 푦 = sin −1(푥) means 푥 = sin (푦) where −1 ≤ 푥 ≤ 1 and − ≤ 푦 ≤ 2 2 • 푦 = cos −1(푥) means 푥 = cos (푦) where −1 ≤ 푥 ≤ 1 and 0 ≤ 푦 ≤ 휋 휋 휋 • 푦 = tan −1(푥) means 푥 = tan (푦) where −∞ ≤ 푥 ≤ ∞ and − < 푦 < 2 2

CBC Mathematics 2019Fall CBC MATHEMATICS MATH 2412-PreCalculus Exam Formula Sheets

휋 휋 • 푦 = csc −1(푥) means 푥 = csc (푦) where |푥| ≥ 1 and − ≤ 푦 ≤ , 푦 ≠ 0 2 2 휋 • 푦 = sec −1(푥) means 푥 = sec (푦) where |푥| ≥ 1 and 0 ≤ 푦 ≤ 휋, 푦 ≠ 2 • 푦 = cot −1(푥) means 푥 = cot (푦) where −∞ ≤ 푥 ≤ ∞ and 0 < 푦 < 휋

 Sum and Difference Formulas • sin(훼 + 훽) = sin(훼) cos(훽) + cos(훼) sin(훽) • sin(훼 − 훽) = sin(훼) cos(훽) − cos(훼) sin(훽) • cos(훼 + 훽) = cos(훼) cos(훽) − sin(훼) sin(훽) • cos(훼 − 훽) = cos(훼) cos(훽) + sin(훼) sin(훽)

tan(훼)+tan(훽) tan(훼)−tan(훽) • tan(훼 + 훽) = tan(훼 − 훽) = 1−tan(훼) tan(훽) 1+tan(훼) tan(훽)

 Half-Angle Formulas

훼 1−cos(훼) • sin ( ) = ±√ 2 2

훼 1+cos(훼) • cos ( ) = ±√ 2 2

훼 1−cos(훼) 1−cos(훼) sin(훼) • tan ( ) = ±√ = = 2 1+cos(훼) sin(훼) 1+cos(훼)

 Double-Angle Formulas • sin(2휃) = 2 sin(휃) cos(휃) • cos(2휃) = cos2(휃) − sin2(휃) = 2cos2(휃) − 1 = 1 − 2sin2(휃)

2 tan(휃) • tan(2휃) = 1−tan2(휃)  Product to Sum Formulas 1 • sin(훼) sin(훽) = [cos(훼 − 훽) − cos(훼 + 훽)] 2 1 • cos(훼) cos(훽) = [cos(훼 − 훽) + cos(훼 + 훽)] 2 1 • sin(훼) cos(훽) = [sin(훼 + 훽) + sin(훼 − 훽)] 2

 Sum to Product Formulas 훼+훽 훼−훽 • sin(훼) + sin(훽) = 2 sin ( ) cos ( ) 2 2

CBC Mathematics 2019Fall CBC MATHEMATICS MATH 2412-PreCalculus Exam Formula Sheets

훼−훽 훼+훽 • sin(훼) − sin(훽) = 2 sin ( ) cos ( ) 2 2 훼+훽 훼−훽 • cos(훼) + cos(훽) = 2 cos ( ) cos ( ) 2 2 훼+훽 훼−훽 • cos(훼) − cos(훽) = −2 sin ( ) sin ( ) 2 2  Law of Sines sin(퐴) sin(퐵) sin(퐶) • = = 푎 푏 푐

 Law of Cosines • 푎2 = 푏2 + 푐2 − 2푏푐 cos(퐴) • 푏2 = 푎2 + 푐2 − 2푎푐 cos(퐵) • 푐2 = 푎2 + 푏2 − 2푎푏 cos(퐶)

 Area of SSS Triangles (Heron’s Formula) 1 • 퐾 = √푠(푠 − 푎)(푠 − 푏)(푠 − 푐) , where 푠 = (푎 + 푏 + 푐) 2

 Area of SAS Triangles 1 1 1 • 퐾 = 푎푏 sin (퐶) , 퐾 = 푏푐 sin (퐴) , 퐾 = 푎푐 sin (퐵) 2 2 2

 For 푦 = 퐴sin (휔푥 − 휑) or 푦 = 퐴cos (휔푥 − 휑) , with 휔 > 0 2휋 휑 • Amplitude = |퐴| , Period= 푇 = , Phase shift = 휔 휔

CBC Mathematics 2019Fall