MATH 2412 Formula Sheets - Coastal Bend College [PDF]

CBC MATHEMATICS DIVISION. MATH 2412-PreCalculus. Exam Formula Sheets. Auth:C.Villarreal-Prof. CBC 2015Spring. ❑ 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.

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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

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