Algebra 2 (Grade 10) | Truth Carriers Academy

1Equations & Inequalities Review

RECEIVE - Foundational Skills

"A false balance is abomination to Yahuah: but a just weight is His delight." - Proverbs 11:1
Equations must balance - just as Yahuah's truth is always consistent!

Solving Multi-Step Equations

Simplify each side (distribute, combine like terms)
Move variables to one side, constants to the other
Isolate the variable
Check your solution

Example: Solve 3(2x - 4) + 5 = 2x + 9

6x - 12 + 5 = 2x + 9 (distribute)

6x - 7 = 2x + 9 (combine)

4x - 7 = 9 (subtract 2x)

4x = 16 (add 7)

x = 4

Absolute Value Equations

|ax + b| = c means ax + b = c OR ax + b = -c (if c ≥ 0)

Example: |2x - 3| = 7

2x - 3 = 7 → x = 5 OR 2x - 3 = -7 → x = -2

REFLECT - Practice

Solve:

1. 5(x - 2) = 3x + 8 → x =

2. |x + 4| = 10 → x = or x =

3. 2|3x - 1| = 16 → x = or x =

2Linear Functions & Systems

RECEIVE - Systems of Equations

Slope-Intercept Form

y = mx + b

m = slope, b = y-intercept

Methods for Solving Systems

Method	Best When
Graphing	Visual understanding, estimates
Substitution	One variable is already isolated
Elimination	Coefficients can easily cancel

Example: Elimination Method

2x + 3y = 12

4x - 3y = 6

Add equations: 6x = 18 → x = 3

Substitute: 2(3) + 3y = 12 → y = 2

Solution: (3, 2)

Systems of Three Variables

Use elimination or substitution to reduce to two variables, then solve.

REFLECT - Practice

Solve the system:

x + y = 7

2x - y = 5

x = , y =

3Quadratic Functions

RECEIVE - Parabolas and Quadratics

Standard Form: y = ax² + bx + c

Vertex Form: y = a(x - h)² + k where (h, k) is the vertex

Factored Form: y = a(x - r₁)(x - r₂) where r₁, r₂ are roots

Quadratic Formula

x = (-b ± √(b² - 4ac)) / 2a

The Discriminant: b² - 4ac

Positive: Two real solutions
Zero: One real solution (double root)
Negative: Two complex solutions

Example: Solve x² - 5x + 6 = 0

Method 1: Factoring

(x - 2)(x - 3) = 0

x = 2 or x = 3

Method 2: Quadratic Formula

x = (5 ± √(25 - 24)) / 2 = (5 ± 1) / 2

x = 3 or x = 2

REFLECT - Practice

Solve:

1. x² - 7x + 12 = 0 → x = or x =

2. 2x² + 5x - 3 = 0 → x = or x =

3. Find the discriminant of x² + 4x + 5 = 0:

How many real solutions?

4Polynomial Functions

RECEIVE - Working with Polynomials

Polynomial Terminology

Degree: Highest exponent
Leading coefficient: Coefficient of highest-degree term
End behavior: What happens as x → ±∞

End Behavior

Degree	Leading Coeff	Left End	Right End
Even	Positive	Up	Up
Even	Negative	Down	Down
Odd	Positive	Down	Up
Odd	Negative	Up	Down

Factoring Methods

Greatest Common Factor (GCF)
Grouping
Difference of Squares: a² - b² = (a + b)(a - b)
Sum/Difference of Cubes: a³ ± b³

Special Factoring Formulas

a³ + b³ = (a + b)(a² - ab + b²)

a³ - b³ = (a - b)(a² + ab + b²)

REFLECT - Practice

Factor completely:

1. x² - 16 =

2. x³ - 8 =

3. 2x³ + 6x² - 8x =

5Radical Functions

RECEIVE - Roots and Radicals

Radical Rules

√(ab) = √a · √b

√(a/b) = √a / √b

ⁿ√(aᵐ) = a^(m/n)

Simplifying Radicals

√50 = √(25 · 2) = 5√2

³√54 = ³√(27 · 2) = 3³√2

Rationalizing Denominators

1/√3 = (1 · √3)/(√3 · √3) = √3/3

1/(2 + √3) = (2 - √3)/((2 + √3)(2 - √3)) = (2 - √3)/(4 - 3) = 2 - √3

REFLECT - Practice

Simplify:

1. √72 =

2. ³√125 =

3. Rationalize: 4/√2 =

6Exponential Functions

RECEIVE - Growth and Decay

"Be fruitful, and multiply" - Genesis 1:28
Population growth follows exponential patterns - Yahuah's design in mathematics!

Exponential Form: y = abˣ

a = initial value, b = growth factor

If b > 1: exponential growth

If 0 < b < 1: exponential decay

Exponent Rules

Rule	Example
aᵐ · aⁿ = aᵐ⁺ⁿ	x² · x³ = x⁵
aᵐ / aⁿ = aᵐ⁻ⁿ	x⁵ / x² = x³
(aᵐ)ⁿ = aᵐⁿ	(x²)³ = x⁶
a⁰ = 1	5⁰ = 1
a⁻ⁿ = 1/aⁿ	x⁻² = 1/x²

Compound Interest

A = P(1 + r/n)^(nt)

P = principal, r = rate, n = compoundings/year, t = years

REFLECT - Practice

Simplify:

1. (3x²)(2x³) =

2. (2x³)² =

3. x⁵/x² =

7Logarithmic Functions

RECEIVE - The Inverse of Exponentials

Definition of Logarithm

log_b(x) = y means bʸ = x

"log base b of x equals y"

Logarithm Properties

Property	Formula
Product Rule	log(ab) = log(a) + log(b)
Quotient Rule	log(a/b) = log(a) - log(b)
Power Rule	log(aⁿ) = n·log(a)
Identity	log_b(b) = 1
Identity	log_b(1) = 0

Common Logarithms

log(x): Base 10 (common log)
ln(x): Base e ≈ 2.718 (natural log)

Example: Solve 2ˣ = 32

Take log of both sides: log(2ˣ) = log(32)

Power rule: x·log(2) = log(32)

x = log(32)/log(2) = 5

Check: 2⁵ = 32 ✓

REFLECT - Practice

Evaluate:

1. log₂(8) =

2. log₃(81) =

3. log(1000) =

4. ln(e⁵) =

8Rational Functions

RECEIVE - Fractions with Polynomials

Rational Function: f(x) = P(x)/Q(x)

Where P(x) and Q(x) are polynomials and Q(x) ≠ 0

Key Features

Domain: All real numbers except where denominator = 0
Vertical asymptote: Where denominator = 0 (simplified form)
Horizontal asymptote: Compare degrees of numerator and denominator
Holes: Where factors cancel

Horizontal Asymptote Rules

If degree of numerator < degree of denominator: y = 0

If degree of numerator = degree of denominator: y = (leading coefficients ratio)

If degree of numerator > degree of denominator: No horizontal asymptote

REFLECT - Practice

For f(x) = (x + 2)/(x - 3):

1. Find the vertical asymptote: x =

2. Find the horizontal asymptote: y =

3. Find the domain: All real numbers except x =

9Sequences & Series

RECEIVE - Patterns in Numbers

Arithmetic Sequences

Common difference d between terms

nth term: aₙ = a₁ + (n-1)d

Sum of n terms: Sₙ = n(a₁ + aₙ)/2

Geometric Sequences

Common ratio r between terms

nth term: aₙ = a₁ · r^(n-1)

Sum of n terms: Sₙ = a₁(1 - rⁿ)/(1 - r) when r ≠ 1

Example: Arithmetic

Sequence: 3, 7, 11, 15, ...

d = 4, a₁ = 3

10th term: a₁₀ = 3 + (10-1)(4) = 3 + 36 = 39

REFLECT - Practice

1. Arithmetic sequence: 5, 8, 11, 14, ...

Common difference d =

Find the 20th term:

2. Geometric sequence: 2, 6, 18, 54, ...

Common ratio r =

Find the 6th term:

10Probability & Statistics

RECEIVE - Data Analysis

Measures of Center

Mean: Average of all values
Median: Middle value when ordered
Mode: Most frequent value

Standard Deviation

Measures spread from the mean

σ = √(Σ(x - μ)²/n)

Probability Basics

P(event) = favorable outcomes / total outcomes

0 ≤ P(event) ≤ 1

P(A and B) = P(A) · P(B) [if independent]

P(A or B) = P(A) + P(B) - P(A and B)

REFLECT - Practice

Data set: 12, 15, 18, 15, 20

1. Mean =

2. Median =

3. Mode =

11Conic Sections

RECEIVE - Curves from Cones

Conic Section Equations

Conic	Standard Form
Circle	(x-h)² + (y-k)² = r²
Parabola	y = a(x-h)² + k or x = a(y-k)² + h
Ellipse	(x-h)²/a² + (y-k)²/b² = 1
Hyperbola	(x-h)²/a² - (y-k)²/b² = 1

Circle: (x-h)² + (y-k)² = r²

Center: (h, k)

Radius: r

REFLECT - Practice

For the circle (x-3)² + (y+2)² = 25:

1. Center: (, )

2. Radius =

12Trigonometry Introduction

RECEIVE - Beyond Right Triangles

"The heavens declare the glory of El; and the firmament shows His handiwork." - Psalm 19:1
Trigonometry helps us measure the heavens Yahuah created!

The Unit Circle

Circle with radius 1 centered at origin

For angle θ: (x, y) = (cos θ, sin θ)

Key Angles (Degrees → Radians)

Degrees	Radians	sin	cos
0°	0	0	1
30°	π/6	1/2	√3/2
45°	π/4	√2/2	√2/2
60°	π/3	√3/2	1/2
90°	π/2	1	0

Conversion

Degrees to Radians: multiply by π/180

Radians to Degrees: multiply by 180/π

REFLECT - Practice

Convert:

1. 180° = radians

2. π/3 radians = °

Evaluate:

3. sin(30°) =

4. cos(60°) =

Answer Key

Unit 1: 1) x=9; 2) x=6 or x=-14; 3) x=3 or x=-7/3

Unit 2: x=4, y=3

Unit 3: 1) x=3 or x=4; 2) x=1/2 or x=-3; 3) discriminant=-4, 0 real solutions

Unit 4: 1) (x+4)(x-4); 2) (x-2)(x²+2x+4); 3) 2x(x+4)(x-1)

Unit 5: 1) 6√2; 2) 5; 3) 2√2

Unit 6: 1) 6x⁵; 2) 4x⁶; 3) x³

Unit 7: 1) 3; 2) 4; 3) 3; 4) 5

Unit 8: 1) x=3; 2) y=1; 3) x=3

Unit 9: 1) d=3, a₂₀=62; 2) r=3, a₆=486

Unit 10: Mean=16, Median=15, Mode=15

Unit 11: Center (3,-2), r=5

Unit 12: 1) π; 2) 60°; 3) 1/2; 4) 1/2

Table of Contents

1Equations & Inequalities Review

RECEIVE - Foundational Skills

Solving Multi-Step Equations

Example: Solve 3(2x - 4) + 5 = 2x + 9

Absolute Value Equations

REFLECT - Practice

2Linear Functions & Systems

RECEIVE - Systems of Equations

Slope-Intercept Form

Methods for Solving Systems

Example: Elimination Method

Systems of Three Variables

REFLECT - Practice

3Quadratic Functions

RECEIVE - Parabolas and Quadratics

Standard Form: y = ax² + bx + c

Quadratic Formula

The Discriminant: b² - 4ac

Example: Solve x² - 5x + 6 = 0

REFLECT - Practice

4Polynomial Functions

RECEIVE - Working with Polynomials

Polynomial Terminology

End Behavior

Factoring Methods

Special Factoring Formulas

REFLECT - Practice

5Radical Functions

RECEIVE - Roots and Radicals

Radical Rules

Simplifying Radicals

Rationalizing Denominators

REFLECT - Practice

6Exponential Functions

RECEIVE - Growth and Decay

Exponential Form: y = abˣ

Exponent Rules

Compound Interest

REFLECT - Practice

7Logarithmic Functions

RECEIVE - The Inverse of Exponentials

Definition of Logarithm

Logarithm Properties

Common Logarithms

Example: Solve 2ˣ = 32

REFLECT - Practice

8Rational Functions

RECEIVE - Fractions with Polynomials

Rational Function: f(x) = P(x)/Q(x)

Key Features

Horizontal Asymptote Rules

REFLECT - Practice

9Sequences & Series

RECEIVE - Patterns in Numbers

Arithmetic Sequences

Geometric Sequences

Example: Arithmetic

REFLECT - Practice

10Probability & Statistics

RECEIVE - Data Analysis

Measures of Center

Standard Deviation

Probability Basics

REFLECT - Practice

11Conic Sections

RECEIVE - Curves from Cones

Conic Section Equations

Circle: (x-h)² + (y-k)² = r²

REFLECT - Practice

12Trigonometry Introduction

RECEIVE - Beyond Right Triangles

The Unit Circle

Key Angles (Degrees → Radians)

Conversion

REFLECT - Practice

Answer Key