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CBC MATHEMATICS DIVISION MATH 1324-Exam Formula Sheets

Equations and Functions Linear Equations

• 𝑦 = 𝑚𝑥 + 𝑏 𝑦 − 𝑦1 = 𝑚(𝑥 − 𝑥1) 𝑦 = 𝑏 𝑥 = 𝑎

Linear Function: 𝑓(𝑥) = 𝑚𝑥 + 𝑏

• 𝑚 =𝑦2−𝑦1

𝑥2−𝑥1 ; 𝑥2 − 𝑥1 ≠ 0

General Form of Quadratic Function: 𝑓(𝑥) = 𝑎𝑥2 + 𝑏𝑥 + 𝑐 , (𝑎 ≠ 0)

• Quadratic Formula

𝑥 =−𝑏±√𝑏2−4𝑎𝑐

2𝑎

• Formulas to find Vertex (ℎ, 𝑘)

ℎ = −𝑏

2𝑎 𝑘 = 𝑎(ℎ)2 + 𝑏(ℎ) + 𝑐,

or (−𝑏

2𝑎, 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) = 𝑎𝑛𝑥𝑛 + 𝑎𝑛−1𝑥𝑛−1 + ⋯ + 𝑎1𝑥1 + 𝑎0

Rational function: 𝑓(𝑥) =𝑛(𝑥)

𝑑(𝑥) , 𝑛(𝑥) and 𝑑(𝑥) are polynomials, but 𝑑(𝑥) ≠ 0.

Vertical Asymptotes

• For 𝑓(𝑥) in simplified form, if 𝑑(𝑐) = 0, then 𝑥 = 𝑐 is a Vertical Asymptote.

Horizontal Asymptote

• 𝑦 = 0 is the Horizontal Asymptote if degree of 𝑛(𝑥) < degree of 𝑑(𝑥).

• 𝑦 =𝑎

𝑏 is the Horizontal Asymptote if degree of 𝑛(𝑥) = degree of 𝑑(𝑥),

where 𝑎 is leading coefficient of 𝑛(𝑥) and 𝑏 is leading coefficient of 𝑑(𝑥).

• If degree of 𝑛(𝑥) > degree of 𝑑(𝑥), then there is no Horizontal Asymptote.

Exponential Function: 𝑓(𝑥) = 𝑎𝑥, where 𝑎 > 0, 𝑎 ≠ 1.

Properties of Exponential Functions: 𝑎 > 0, 𝑏 > 0, 𝑎 ≠ 1, 𝑏 ≠ 1, and 𝑥, 𝑦 real.

• 𝑎𝑥𝑎𝑦 = 𝑎𝑥+𝑦 , (𝑎𝑥)𝑦 = 𝑎𝑥𝑦 , (𝑎𝑏)𝑥 = 𝑎𝑥𝑏𝑥

• . 𝑎𝑥

𝑎𝑦= 𝑎𝑥−𝑦 , (

𝑎

𝑏)

𝑥=

𝑎𝑥

𝑏𝑥

• 𝑎𝑥 = 𝑎𝑦, if and only if 𝑥 = 𝑦. • For 𝑥 ≠ 0, 𝑎𝑥 = 𝑏𝑥, if and only if 𝑎 = 𝑏.


Auth:C.Villarreal-Professor 2015Falll

Logarithmic Function: 𝑓(𝑥) = log𝑎(𝑥)

• log𝑎(1) = 0 , log𝑎(𝑎) = 1 , 𝑎log𝑎(𝑀) = 𝑀 , log𝑎(𝑎 𝑝) = 𝑝

• log𝑎( 𝑀 ∙ 𝑁 ) = log𝑎(𝑀) + log𝑎(𝑁)

• log𝑎 ( 𝑀

𝑁 ) = log𝑎(𝑀) − log𝑎(𝑁)

• log𝑎(𝑀𝑝) = 𝑝 ∙ log𝑎(𝑀)

• If log𝑎(𝑀) = log𝑎(𝑁), then 𝑀 = 𝑁.

• If 𝑀 = 𝑁, then log𝑎(𝑀) = log𝑎(𝑁).

• Change of Base formula log𝑎(𝑀) = log(𝑀)

log(𝑎) or log𝑎(𝑀) =

ln(𝑀)

ln(𝑎)

Function Transformations

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 vertical: shift 𝑓(𝑥) up

𝑦 = 𝑓(𝑥) − 𝑘, 𝑘 > 0 : shift 𝑓(𝑥) down

• 𝑦 = 𝑓(𝑥 + ℎ) ℎ > 0 horizontal: shift 𝑓(𝑥) left

𝑦 = 𝑓(𝑥 − ℎ), ℎ > 0 : shift 𝑓(𝑥) right

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.

Solve a system of equations(Gaussian Elimination) • Rewrite system of equations as augmented matrix. • Apply row operations to obtain Row Echelon form. • Write row equations and solve for solutions.(back-substitute if necessary)



Basic Properties of Matrices

• (𝐴 + 𝐵) + 𝐶 = 𝐴 + (𝐵 + 𝐶)

• 𝐴 + 𝐵 = 𝐵 + 𝐴

• 𝐴 + 0 = 0 + 𝐴 = 𝐴

• 𝐴 + (−𝐴) = (−𝐴) + 𝐴 = 0

• 𝐴(𝐵𝐶) = (𝐴𝐵)𝐶

• 𝐴𝐼 = 𝐼𝐴 = 𝐴

• If 𝐴 is square matrix and 𝐴−1 exists, then 𝐴𝐴−1 = 𝐴−1𝐴 = 𝐼.

• 𝐴(𝐵 + 𝐶) = 𝐴𝐵 + 𝐴𝐶 , (𝐵 + 𝐶)𝐴 = 𝐵𝐴 + 𝐶𝐴

• If 𝐴 = 𝐵, then 𝐴 + 𝐶 = 𝐵 + 𝐶

• If 𝐴 = 𝐵, then 𝐶𝐴 = 𝐶𝐵 and 𝐴𝐶 = 𝐵𝐶

Matrix Equation: 𝐴𝑋 = 𝐵 → 𝑋 = 𝐴−1𝐵, provided 𝐴 is square and 𝐴−1 exists.

Simplex Method: Summary of Problem Types

Problem Type

Constraints Constants Objective Function

Coefficients Method

Maximization ≤ Nonnegative Real Simplex

Minimization ≥ Real Nonnegative Dual

Maximization Mixed Nonnegative Real Big 𝑀

Minimization Mixed Nonnegative Real Big 𝑀

Exponential Models Formulas

• Simple Interest: 𝐼 = 𝑃𝑟𝑡

• Compound Interest: 𝐴 = 𝑃(1 + 𝑖)𝑛 → 𝐴 = 𝑃 (1 +𝑟

𝑚)

𝑚∙𝑡

• Continuous Compounding: 𝐴 = 𝑃𝑒𝑛 → 𝐴 = 𝑃𝑒𝑟∙𝑡

• Annual Percentage Yield(Effective Rate of Interest):

Compounding 𝑚 times per year 𝐴𝑃𝑌 = (1 +𝑟

𝑚)

𝑚− 1

Compounding continuously per year 𝐴𝑃𝑌 = 𝑒𝑟 − 1

• Annuity(Future Value): 𝐹𝑉 = 𝑃𝑀𝑇 ((1+

𝑟

𝑚)

𝑚∙𝑡 − 1

(𝑟

𝑚)

)

• Annuity(Present Value): 𝑃𝑉 = 𝑃𝑀𝑇 ( 1 − (1+

𝑟

𝑚)

−(𝑚∙𝑡)

(𝑟

𝑚)

)

• Logistic Model: 𝑃(𝑡) =𝑐

1+𝑎𝑒−(𝑏)∙𝑡



Counting Principles

• 𝑛(𝐴 ∪ 𝐵) = 𝑛(𝐴) + 𝑛(𝐵) − 𝑛(𝐴 ∩ 𝐵)

• 𝑛! = 𝑛(𝑛 − 1)(𝑛 − 2) ∙ ⋯ ∙ 2 ∙ 1

• 0! = 1 , 𝑛! = 𝑛(𝑛 − 1)!

• 𝑛𝑃𝑟 =𝑛!

(𝑛−𝑟)! for 0 ≤ 𝑟 ≤ 𝑛

• 𝑛𝐶𝑟 =𝑛!

𝑟!(𝑛−𝑟)! for 0 ≤ 𝑟 ≤ 𝑛

Probability

• 𝑃(𝐸) =𝑛(𝐸)

𝑛(𝑆)

• 𝑃(𝐴 ∪ 𝐵) = 𝑃(𝐴) + 𝑃(𝐵) − 𝑃(𝐴 ∩ 𝐵)

• Odds for 𝐸 =𝑃(𝐸)

𝑃(𝐸′)=

𝑃(𝐸)

1−𝑃(𝐸) where 𝑃(𝐸) ≠ 1.

• Odds against 𝐸 =𝑃(𝐸′)

𝑃(𝐸) where 𝑃(𝐸) ≠ 0.

• Conditional Probability: 𝑃(𝐴|𝐵) =𝑃(𝐴∩𝐵)

𝑃(𝐵) where 𝑃(𝐵) ≠ 0.

𝑃(𝐵|𝐴) =𝑃(𝐵∩𝐴)

𝑃(𝐴) where 𝑃(𝐴) ≠ 0.

• Product Rule: 𝑃(𝐴 ∩ 𝐵) = 𝑃(𝐴)𝑃(𝐵|𝐴) = 𝑃(𝐵)𝑃(𝐴|𝐵),

where 𝑃(𝐴) ≠ 0, 𝑃(𝐵) ≠ 0.

• If 𝑃(𝐴 ∩ 𝐵) = 𝑃(𝐴)𝑃(𝐵), then 𝐴 and 𝐵 independent.

• If 𝑃(𝐴) ≠ 0 and 𝑃(𝐵) ≠ 0, and either 𝑃(𝐴|𝐵) = 𝑃(𝐴) or 𝑃(𝐵|𝐴) = 𝑃(𝐵), then 𝐴 and 𝐵 are independent.

• If 𝐸1, 𝐸2, … , 𝐸𝑛 are independent, then

𝑃(𝐸1 ∩ 𝐸2 ∩ ⋯ ∩ 𝐸𝑛) = 𝑃(𝐸1) ∙ 𝑃(𝐸2) ∙ ⋯ ∙ 𝑃(𝐸𝑛)

• Bayes’ Formula: 𝑃(𝑈1|𝐸) =𝑃(𝑈1∩𝐸)

𝑃(𝐸)=

𝑃(𝑈1∩𝐸)

𝑃(𝑈1∩𝐸)+𝑃(𝑈2∩𝐸)+⋯+𝑃(𝑈𝑛∩𝐸)

=𝑃(𝐸|𝑈1)∙𝑃(𝑈1)

𝑃(𝐸|𝑈1)∙𝑃(𝑈1)+𝑃(𝐸|𝑈2)∙𝑃(𝑈2)+⋯+𝑃(𝐸|𝑈𝑛)∙𝑃(𝑈𝑛)