Idea Transcript
CBC MATHEMATICS MATH 2412-PreCalculus Exam 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 > 0
vertical: 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 > 0
vertical: stretch 𝑓(𝑥) if 𝑎 > 1 : compress 𝑓(𝑥) if 0 < 𝑎 < 1
•
𝑦 = 𝑓(𝑎𝑥), 𝑎 > 0
horizontal: stretch 𝑓(𝑥) if 0 < 𝑎 < 1 : compress 𝑓(𝑥) if 𝑎 > 1
Shifts •
𝑦 = 𝑓(𝑥) + 𝑘, 𝑘 > 0 𝑦 = 𝑓(𝑥) − 𝑘, 𝑘 > 0
vertical: shift 𝑓(𝑥) up : shift 𝑓(𝑥) down
•
𝑦 = 𝑓(𝑥 + ℎ) ℎ > 0 𝑦 = 𝑓(𝑥 − ℎ), ℎ > 0
horizontal: shift 𝑓(𝑥) left : shift 𝑓(𝑥) right
CBC Mathematics 2018Fall
CBC MATHEMATICS MATH 2412-PreCalculus Exam Formula Sheets Formulas/Equations • Slope Intercept: 𝑦 = 𝑚𝑥 + 𝑏
𝑚=
𝑦2 −𝑦1
Point-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 −𝑥1
Δ𝑦 Δ𝑥
=
𝑓(𝑏)−𝑓(𝑎) 𝑏−𝑎
, where 𝑎 ≠ 𝑏
𝐴𝑟𝑒𝑎 = 𝜋𝑟 2
𝐶𝑖𝑟𝑐𝑢𝑚𝑓𝑒𝑟𝑒𝑛𝑐𝑒 = 2𝜋𝑟 = 𝜋𝑑, 1 2
4
𝐴𝑟𝑒𝑎 = 𝑙𝑤
𝑆𝑢𝑟𝑓𝑎𝑐𝑒 𝐴𝑟𝑒𝑎 = 4𝜋𝑟 2
3
𝑆𝑢𝑟𝑓𝑎𝑐𝑒 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:
(−
𝑏
2𝑎
2𝑎
𝑘 = 𝑎(ℎ)2 + 𝑏(ℎ) + 𝑐,
2𝑎
𝑏
−𝑏±√𝑏2 −4𝑎𝑐
, 𝑓 (−
𝑏 2𝑎
)),
or
(−
𝑏 2𝑎
,
4𝑎𝑐−𝑏2 4𝑎
)
Axis of symmetry: 𝑥 = ℎ
Vertex Form of Quadratic Function: 𝑓(𝑥) = 𝑎(𝑥 − ℎ)2 + 𝑘
vertex (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.
Rational Zeros Theorem •
Possible rational zeros: ±
𝑝 𝑞
, where 𝑝 is a factor of 𝑎0 and 𝑞 is a factor of 𝑎𝑛 . CBC Mathematics 2018Fall
CBC MATHEMATICS MATH 2412-PreCalculus Exam Formula Sheets 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 real zero between 𝑥 = 𝑎 and 𝑥 = 𝑏.
Conjugate Pairs Theorem •
For polynomial functions 𝑓(𝑥) with real coefficients: If 𝑥 = 𝑎 + 𝑏𝑖 is a zero 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 of
𝑝(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) 𝑀
• log 𝑎 ( ) = log 𝑎 (𝑀) − log 𝑎 (𝑁) 𝑁 • log 𝑎 (𝑀𝑝 ) = 𝑝 ∙ log 𝑎 (𝑀) • If log 𝑎 (𝑀) = log 𝑎 (𝑁), then 𝑀 = 𝑁. • If 𝑀 = 𝑁, then log 𝑎 (𝑀) = log 𝑎 (𝑁). • Change of Base formula
log 𝑎 (𝑀) =
log(𝑀) log(𝑎)
or
log 𝑎 (𝑀) =
ln(𝑀) ln(𝑎)
CBC Mathematics 2018Fall
CBC MATHEMATICS MATH 2412-PreCalculus Exam Formula Sheets Exponential Models Formulas • Simple Interest: 𝐼 = 𝑃𝑟𝑡 𝑟 𝑛∙𝑡
• Compound Interest: 𝐴 = 𝑃 (1 + ) 𝑛
• Continuous Compounding: 𝐴 = 𝑃𝑒 𝑟∙𝑡 • Effective Rate of Interest:
𝑟 𝑛
Compounding 𝑛 times per year Compounding continuously per year
𝑟𝑒𝑓𝑓 = (1 + ) − 1 𝑛 𝑟 𝑟𝑒𝑓𝑓 = 𝑒 − 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) =
𝑐 1+𝑎𝑒 −𝑏∙𝑡
Sequences and Series
•
𝑛! = 𝑛(𝑛 − 1)(𝑛 − 2) ∙ ⋯ ∙ (3)(2)(1) 𝑛!
𝑃(𝑛, 𝑟) =
•
Arithmetic Sequence: 𝑛𝑡ℎ term 𝑎𝑛 = 𝑎1 + (𝑛 − 1)𝑑
(𝑛−𝑟)!
𝐶(𝑛, 𝑟) =
𝑛!
•
𝑟!(𝑛−𝑟)!
𝑛
Sum of first 𝑛 terms 𝑆𝑛 = ∑𝑛 𝑘=1(𝑎1 + (𝑘 − 1)𝑑) = 2 (𝑎1 + 𝑎𝑛 ) 𝑛 or 𝑆𝑛 = ∑𝑛 𝑘=1(𝑎1 + (𝑘 − 1)𝑑) = (2𝑎1 + (𝑛 − 1)𝑑). 2
•
Geometric Sequence: 𝑛𝑡ℎ term
𝑎𝑛 = 𝑎1 (𝑟)𝑛−1 1−𝑟 𝑛
𝑘−1 Sum of first 𝑛 terms 𝑆𝑛 = ∑𝑛 = 𝑎1 ∙ for 𝑟 ≠ 0,1 𝑘=1 𝑎1 𝑟 1−𝑟
•
Geometric Series:
𝑘−1 ∑∞ = 𝑘=1 𝑎1 𝑟
𝑎1 1−𝑟
if |𝑟| < 1
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
CBC Mathematics 2018Fall
CBC MATHEMATICS MATH 2412-PreCalculus Exam Formula Sheets Trigonometry Circular Measure and Motion Formulas • •
𝑠 = 𝑟𝜃
Arc Length
𝑠
Linear Speed 𝑣 = , 𝑣 = 𝑟𝜔 𝑡
1
Area of Sector
𝐴 = 𝑟2𝜃
Angular Speed
𝜔=
2 𝜃 𝑡
Acute Angle 𝑏
𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒
𝑐 𝑐
ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒 ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒
𝑏
𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒
•
sin(𝜃) = =
•
csc(𝜃) = =
cos(𝜃) = sec(𝜃) =
𝑎 𝑐 𝑐 𝑎
= =
𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡
tan(𝜃) =
ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒 ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒
cot(𝜃) =
𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡
𝑏 𝑎 𝑎 𝑏
= =
𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒 𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡 𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡 𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒
General Angle 𝑏
•
sin(𝜃) =
cos(𝜃) =
•
csc(𝜃) = ,𝑏 ≠ 0
𝑟 𝑟
𝑏
sec(𝜃) =
𝑎 𝑟 𝑟 𝑎
tan(𝜃) = cot(𝜃) =
,𝑎 ≠ 0
𝑏 𝑎 𝑎 𝑏
,𝑏 ≠ 0
Cofunctions 𝜋
𝜋
𝜋
•
sin(𝜃) = cos ( − 𝜃) ,
cos(𝜃) = sin ( − 𝜃) ,
tan(𝜃 ) = cot ( − 𝜃)
•
csc(𝜃) = sec ( − 𝜃) ,
sec(𝜃) = csc ( − 𝜃) ,
cot(𝜃 ) = tan ( − 𝜃)
2 𝜋 2
2 𝜋
2 𝜋
2
2
Fundamental Identities sin(𝜃)
•
tan(𝜃) =
•
csc(𝜃) =
•
sin2 (𝜃) + cos2 (𝜃) = 1
cos(𝜃) 1 sin(𝜃)
Even-Odd Identities • sin(−𝜃 ) = −sin(𝜃 ) • csc(−𝜃 ) = −csc(𝜃 )
cot(𝜃) = sec(𝜃) =
cos(𝜃) sin(𝜃) 1 cos(𝜃)
cot(𝜃) =
1 tan(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)
Inverse Functions
𝜋
𝜋
2
2
•
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 <
𝜋 2
𝜋
𝜋
2
2
𝜋 2
•
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 < 𝜋
𝑦≠
𝜋 2
CBC Mathematics 2018Fall
CBC MATHEMATICS MATH 2412-PreCalculus Exam Formula Sheets 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 Formulas 𝛼
1−cos(𝛼)
2
2
𝛼
1+cos(𝛼)
2
2
𝛼
1−cos(𝛼)
2
1+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) =
•
sin2 (𝜃) =
2 tan(𝜃) 1−tan2 (𝜃) 1−cos(2𝜃) 2
cos2 (𝜃) =
,
1+cos(2𝜃) 2
,
tan2 (𝜃) =
1−cos(2𝜃) 1−cos(2𝜃)
Product to Sum Formulas 1
•
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)]
2 1 2 1 2
Sum to Product Formulas 𝛼+𝛽
•
sin(𝛼) + sin(𝛽) = 2 sin (
•
sin(𝛼) − sin(𝛽) = 2 sin (
•
cos(𝛼) + cos(𝛽) = 2 cos (
•
cos(𝛼) − cos(𝛽) = −2 sin (
2 𝛼−𝛽
) cos ( ) cos (
2 𝛼+𝛽
𝛼−𝛽 2 𝛼+𝛽
) cos (
2 𝛼+𝛽 2
) )
2 𝛼−𝛽
) sin (
)
2 𝛼−𝛽 2
) CBC Mathematics 2018Fall
CBC MATHEMATICS MATH 2412-PreCalculus Exam Formula Sheets 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) ,
1
where 𝑠 = (𝑎 + 𝑏 + 𝑐) 2
Area of SAS Triangles •
1
𝐾 = 𝑎𝑏 sin(𝐶) , 2
1
1
𝐾 = 𝑏𝑐 sin(𝐴) ,
𝐾 = 𝑎𝑐 sin(𝐵)
2
2
For 𝑦 = 𝐴sin(𝜔𝑥 − 𝜑) or 𝑦 = 𝐴cos(𝜔𝑥 − 𝜑) , with 𝜔 > 0 •
Amplitude = |𝐴| ,
Period= 𝑇 =
2𝜋 𝜔
,
Phase shift =
𝜑 𝜔
CBC Mathematics 2018Fall