Pre-Calculus & Trigonometry - Grade 11

Sacred Names Pronunciation Guide

Yahuah (yah-HOO-ah) - The Father's covenant name, meaning "I AM"

Yahusha (yah-HOO-sha) - The Son's name, meaning "Yah is salvation"

Elohim (el-oh-HEEM) - Hebrew word for God (plural majesty)

Ruach HaKodesh (ROO-akh hah-KOH-desh) - The Holy Spirit

Welcome to Pre-Calculus

Pre-Calculus bridges algebra and calculus, introducing you to the mathematical tools that describe our Creator's universe. From the curves of planetary orbits to the waves of sound and light, advanced mathematics reveals the elegant patterns Yahuah built into creation.

This course covers functions, trigonometry, and the foundations of calculus. These concepts are essential for engineering, physics, computer science, and understanding the mathematical language of creation.

Psalm 19:1

"The heavens declare the glory of Elohim; and the firmament sheweth his handywork."

Lesson 1: Functions Review

RECEIVE: What is a Function?

Function:

A relation where each input (x) produces exactly ONE output (y). Written as f(x) = expression.

Key Function Concepts:

Domain: All possible input values (x-values)
Range: All possible output values (y-values)
Vertical Line Test: If any vertical line crosses the graph more than once, it's NOT a function

Function Notation

f(x) = 2x + 3

Read as "f of x equals 2x plus 3"

If x = 4, then f(4) = 2(4) + 3 = 11

REFLECT: Functions in Creation

Yahuah's Consistent Laws

Functions represent consistent relationships—the same input always produces the same output. This mirrors Yahuah's unchanging nature. The laws of physics are functions: the same conditions always produce the same results. Malachi 3:6 says "For I am Yahuah, I change not." His creation reflects His consistency.

RECALL: Practice Problems

Given f(x) = 3x² - 2x + 1, evaluate:

1. f(0) =

2. f(2) =

3. f(-1) =

4. f(a) =

Find the domain of each function:

5. f(x) = √(x - 4)
Domain:

6. g(x) = 1/(x + 2)
Domain:

RESPOND: Application

Explain why finding the domain matters in real-world applications:

Lesson 2: Polynomial Functions

RECEIVE: Understanding Polynomials

Polynomial Function:

f(x) = aₙxⁿ + aₙ₋₁xⁿ⁻¹ + ... + a₁x + a₀, where n is a non-negative integer

Degree determines behavior:

Degree 0: Constant (horizontal line)
Degree 1: Linear (straight line)
Degree 2: Quadratic (parabola)
Degree 3: Cubic (S-curve)
Degree 4: Quartic (W or M shape)

End Behavior Rules

Even degree, positive leading coefficient: ↑ both ends up

Even degree, negative leading coefficient: ↓ both ends down

Odd degree, positive leading coefficient: ↙↗ down-left, up-right

Odd degree, negative leading coefficient: ↖↘ up-left, down-right

RECALL: End Behavior Practice

Describe the end behavior of each polynomial:

1. f(x) = 2x⁴ - 3x² + 1
As x → -∞, f(x) →
As x → +∞, f(x) →

2. g(x) = -x³ + 2x
As x → -∞, g(x) →
As x → +∞, g(x) →

3. h(x) = -2x⁶ + x⁴ - 3
As x → -∞, h(x) →
As x → +∞, h(x) →

Find all zeros (x-intercepts) by factoring:

4. f(x) = x³ - 4x
Zeros:

5. g(x) = x² - 5x + 6
Zeros:

Lesson 3: Rational Functions

RECEIVE: Ratios of Polynomials

Rational Function:

f(x) = P(x)/Q(x), where P and Q are polynomials and Q(x) ≠ 0

Key Features:

Vertical Asymptotes: Where denominator = 0 (and numerator ≠ 0)
Horizontal Asymptotes: Compare degrees of numerator and denominator
Holes: Where both numerator and denominator = 0
x-intercepts: Where numerator = 0

Horizontal Asymptote Rules

If degree(top) < degree(bottom): y = 0

If degree(top) = degree(bottom): y = leading coefficients ratio

If degree(top) > degree(bottom): No horizontal asymptote (slant possible)

Example: f(x) = (2x + 4)/(x - 1)

Vertical Asymptote: x = 1 (denominator = 0)

Horizontal Asymptote: y = 2/1 = 2 (equal degrees)

x-intercept: 2x + 4 = 0 → x = -2

RECALL: Analyze Rational Functions

Find the asymptotes and intercepts:

1. f(x) = 3/(x + 2)
Vertical Asymptote:
Horizontal Asymptote:
y-intercept:

2. g(x) = (x² - 9)/(x - 3)
Simplified form:
Hole at:

3. h(x) = (2x² + 1)/(x² - 4)
Vertical Asymptotes:
Horizontal Asymptote:

Lesson 4: Exponential Functions

RECEIVE: Powers of Growth

Exponential Function:

f(x) = a · bˣ, where a ≠ 0, b > 0, and b ≠ 1

Key Properties:

If b > 1: Exponential growth (increasing)
If 0 < b < 1: Exponential decay (decreasing)
Horizontal asymptote at y = 0
Domain: All real numbers
Range: (0, ∞) if a > 0

The Natural Base e ≈ 2.71828...

e = lim(n→∞) (1 + 1/n)ⁿ

Used in continuous growth: A = Pe^(rt)

Exponential Growth in Scripture

Yahuah told Adam and Eve to "be fruitful and multiply" (Genesis 1:28). Population growth is exponential. A grain of wheat produces many seeds, each producing more plants. This principle appears throughout creation—compound growth that shows Yahuah's abundant blessing.

RECALL: Evaluate and Apply

Evaluate each expression:

1. 2⁵ =

2. 3⁻² =

3. e⁰ =

4. (1/2)³ =

Application Problem:

A population of 1000 grows at 5% per year. Write the function and find the population after 10 years.

Function: P(t) =
P(10) =

Lesson 5: Logarithmic Functions

RECEIVE: The Inverse of Exponentials

Logarithm:

log_b(x) = y means bʸ = x

"The logarithm base b of x is the exponent you need to raise b to get x"

Logarithm Properties

log_b(MN) = log_b(M) + log_b(N) (Product Rule)

log_b(M/N) = log_b(M) - log_b(N) (Quotient Rule)

log_b(Mⁿ) = n · log_b(M) (Power Rule)

log_b(b) = 1

log_b(1) = 0

Converting Between Forms:

Exponential: 2³ = 8 ↔ Logarithmic: log₂(8) = 3

Exponential: 10² = 100 ↔ Logarithmic: log₁₀(100) = 2

Common Logarithms:

log(x) means log₁₀(x) — common logarithm

ln(x) means log_e(x) — natural logarithm

RECALL: Logarithm Practice

Evaluate without a calculator:

1. log₂(32) =

2. log₃(81) =

3. log₁₀(1000) =

4. ln(e⁵) =

5. log₅(1) =

Expand using logarithm properties:

6. log(x²y³) =

7. ln(x/y²) =

Lesson 6: Introduction to Trigonometry

RECEIVE: Angles and Measurement

Trigonometry studies relationships between angles and sides of triangles. It's essential for navigation, astronomy, engineering, and understanding wave phenomena.

Angle Measurements

Degrees: Full circle = 360°

Radians: Full circle = 2π radians

π radians = 180°

To convert: degrees × (π/180) = radians

To convert: radians × (180/π) = degrees

Circles in Creation

The circle is fundamental to creation—the sun, moon, and stars appear circular. The path of the sun (as we observe it) traces an arc. Yahuah's design uses circular patterns throughout nature. Understanding angles helps us appreciate the precision of His handiwork.

RECALL: Angle Conversions

Convert degrees to radians:

1. 90° = radians

2. 45° = radians

3. 60° = radians

4. 180° = radians

Convert radians to degrees:

5. π/6 = °

6. π/3 = °

7. 2π/3 = °

8. 3π/2 = °

Lesson 7: Right Triangle Trigonometry

RECEIVE: SOH-CAH-TOA

The Three Main Trig Ratios

sin(θ) = Opposite / Hypotenuse (SOH)

cos(θ) = Adjacent / Hypotenuse (CAH)

tan(θ) = Opposite / Adjacent (TOA)

The Reciprocal Functions:

csc(θ) = 1/sin(θ) = Hypotenuse/Opposite
sec(θ) = 1/cos(θ) = Hypotenuse/Adjacent
cot(θ) = 1/tan(θ) = Adjacent/Opposite

Example: Right triangle with sides 3, 4, 5

If angle θ is opposite the side of length 3:

sin(θ) = 3/5 = 0.6

cos(θ) = 4/5 = 0.8

tan(θ) = 3/4 = 0.75

RECALL: Calculate Trig Ratios

For a right triangle with legs 5 and 12, and hypotenuse 13:

Let θ be the angle opposite the side of length 5.

1. sin(θ) =

2. cos(θ) =

3. tan(θ) =

4. csc(θ) =

Solve for x (round to nearest tenth):

5. In a right triangle, angle = 30°, hypotenuse = 10. Find the opposite side.
x =

6. In a right triangle, angle = 45°, adjacent side = 8. Find the opposite side.
x =

Lesson 8: The Unit Circle

RECEIVE: Circle of Radius 1

Unit Circle:

A circle centered at the origin with radius 1. Any point on the circle is (cos θ, sin θ) where θ is the angle from the positive x-axis.

Key Unit Circle Values to Memorize:

Angle (°)	Angle (rad)	cos θ	sin θ
0°	0	1	0
30°	π/6	√3/2	1/2
45°	π/4	√2/2	√2/2
60°	π/3	1/2	√3/2
90°	π/2	0	1
180°	π	-1	0
270°	3π/2	0	-1

Memory Trick:

For first quadrant angles (0°, 30°, 45°, 60°, 90°):

sin values: √0/2, √1/2, √2/2, √3/2, √4/2 = 0, 1/2, √2/2, √3/2, 1

cos values are the reverse!

RECALL: Unit Circle Practice

Find the exact value without a calculator:

1. sin(π/3) =

2. cos(π/4) =

3. tan(π/6) = sin(π/6)/cos(π/6) =

4. sin(π) =

5. cos(3π/2) =

What quadrant is each angle in? What are the signs of sin and cos?

6. 150° — Quadrant: sin: cos:

7. 225° — Quadrant: sin: cos:

8. 315° — Quadrant: sin: cos:

Lesson 9: Graphing Sine and Cosine

RECEIVE: Wave Functions

General Form

y = A sin(B(x - C)) + D

A = Amplitude (height from center to peak)

B = Affects period (Period = 2π/B)

C = Phase shift (horizontal shift)

D = Vertical shift (midline)

Basic sine wave y = sin(x):

Amplitude = 1
Period = 2π
Starts at origin, goes up first

Basic cosine wave y = cos(x):

Amplitude = 1
Period = 2π
Starts at maximum (0, 1)

Waves in Creation

Sound travels in waves. Light behaves as waves. The tides rise and fall in wave patterns. Seasons cycle. Yahuah built rhythmic, wave-like patterns throughout creation. Sine and cosine functions describe these patterns mathematically—another example of the mathematical order underlying our world.

RECALL: Identify Transformations

For each function, identify the amplitude, period, phase shift, and vertical shift:

1. y = 3sin(x)
Amplitude: Period: Phase Shift: Vertical Shift:

2. y = cos(2x)
Amplitude: Period: Phase Shift: Vertical Shift:

3. y = 2sin(x - π/2) + 1
Amplitude: Period: Phase Shift: Vertical Shift:

RESPOND: Sketch the Graph

Sketch y = 2sin(x) on the coordinate plane below:

[Graph paper for student work]

Lesson 10: Trigonometric Identities

RECEIVE: Fundamental Identities

Pythagorean Identities

sin²θ + cos²θ = 1

1 + tan²θ = sec²θ

1 + cot²θ = csc²θ

Quotient Identities

tan θ = sin θ / cos θ

cot θ = cos θ / sin θ

Reciprocal Identities

csc θ = 1/sin θ

sec θ = 1/cos θ

cot θ = 1/tan θ

Simplify: sin²θ + cos²θ + tan²θ

sin²θ + cos²θ = 1 (Pythagorean identity)

1 + tan²θ = sec²θ (Pythagorean identity)

Answer: sec²θ

RECALL: Use Identities

Simplify each expression:

1. (1 - cos²θ)/sin θ =

2. sin θ · csc θ =

3. cos²θ · tan²θ =

4. sec²θ - 1 =

If sin θ = 3/5 and θ is in Quadrant I, find:

5. cos θ =

6. tan θ =

Lesson 11: Sum and Difference Formulas

RECEIVE: Combining Angles

Sum and Difference Formulas

Sine:

sin(A + B) = sin A cos B + cos A sin B

sin(A - B) = sin A cos B - cos A sin B

Cosine:

cos(A + B) = cos A cos B - sin A sin B

cos(A - B) = cos A cos B + sin A sin B

Tangent:

tan(A + B) = (tan A + tan B)/(1 - tan A tan B)

tan(A - B) = (tan A - tan B)/(1 + tan A tan B)

Find sin(75°) exactly:

75° = 45° + 30°

sin(75°) = sin(45° + 30°)

= sin 45° cos 30° + cos 45° sin 30°

= (√2/2)(√3/2) + (√2/2)(1/2)

= √6/4 + √2/4 = (√6 + √2)/4

RECALL: Apply the Formulas

Find the exact value:

1. cos(75°) = cos(45° + 30°) =

2. sin(15°) = sin(45° - 30°) =

3. cos(π/12) = cos(π/3 - π/4) =

Lesson 12: Sequences and Series

RECEIVE: Patterns in Numbers

Sequence:

An ordered list of numbers following a pattern.

Series:

The sum of the terms of a sequence.

Arithmetic Sequence

Each term differs by a constant d (common difference)

aₙ = a₁ + (n-1)d

Sum of first n terms:

Sₙ = n(a₁ + aₙ)/2

Geometric Sequence

Each term multiplied by a constant r (common ratio)

aₙ = a₁ · r^(n-1)

Sum of first n terms:

Sₙ = a₁(1 - rⁿ)/(1 - r), r ≠ 1

Arithmetic: 2, 5, 8, 11, 14, ...

a₁ = 2, d = 3

a₁₀ = 2 + (10-1)(3) = 2 + 27 = 29

Geometric: 3, 6, 12, 24, 48, ...

a₁ = 3, r = 2

a₆ = 3 · 2⁵ = 3 · 32 = 96

RECALL: Sequence Practice

Identify as arithmetic or geometric, then find the indicated term:

1. 4, 7, 10, 13, ... Type: a₂₀ =

2. 2, 6, 18, 54, ... Type: a₇ =

3. 100, 90, 80, 70, ... Type: a₁₅ =

Find the sum:

4. Sum of first 10 terms of: 1, 3, 5, 7, 9, ...
S₁₀ =

5. Sum of first 5 terms of: 1, 2, 4, 8, 16, ...
S₅ =

Lesson 13: Introduction to Limits

RECEIVE: Approaching a Value

Limit:

The value that a function approaches as the input approaches some value.

Written as: lim(x→c) f(x) = L

"The limit of f(x) as x approaches c equals L"

The limit exists if:

The function approaches the same value from both left and right
lim(x→c⁻) f(x) = lim(x→c⁺) f(x)

Example: lim(x→2) (x² - 4)/(x - 2)

Direct substitution gives 0/0 (indeterminate)

Factor: (x² - 4)/(x - 2) = (x+2)(x-2)/(x-2)

Cancel: = x + 2 (for x ≠ 2)

Now evaluate: lim(x→2) (x + 2) = 4

Infinity and Eternity

Limits often involve approaching infinity—something that keeps going forever. This gives us a mathematical window into concepts like eternity. While we cannot fully grasp the infinite, mathematics helps us reason about it. Yahuah is eternal, without beginning or end—beyond even the mathematical concept of infinity.

RECALL: Evaluate Limits

Find each limit:

1. lim(x→3) (2x + 1) =

2. lim(x→0) (x² + 3x)/x =

3. lim(x→4) (x² - 16)/(x - 4) =

4. lim(x→∞) (3x² + 2)/(x² - 1) =

Lesson 14: Introduction to Vectors

RECEIVE: Magnitude and Direction

Vector:

A quantity with both magnitude (size) and direction. Represented by an arrow or ordered pair ⟨a, b⟩.

Vector Operations

Magnitude: |v| = √(a² + b²)

Addition: ⟨a, b⟩ + ⟨c, d⟩ = ⟨a+c, b+d⟩

Scalar Multiplication: k⟨a, b⟩ = ⟨ka, kb⟩

Unit Vector: v/|v| (vector with magnitude 1)

Vector v = ⟨3, 4⟩

Magnitude: |v| = √(9 + 16) = √25 = 5

Unit vector: ⟨3/5, 4/5⟩ = ⟨0.6, 0.8⟩

Real-World Applications:

Force (pushes/pulls in specific directions)
Velocity (speed with direction)
Wind (speed and direction)
Navigation (course plotting)

RECALL: Vector Operations

Given u = ⟨2, 5⟩ and v = ⟨-1, 3⟩, find:

1. u + v =

2. u - v =

3. 3u =

4. |u| =

5. |v| =

RESPOND: Reflection

How does studying advanced mathematics deepen your appreciation for Yahuah's creation? Write your thoughts:

Answer Key for Parents/Teachers

Lesson 1: Functions Review

1. f(0) = 1 | 2. f(2) = 9 | 3. f(-1) = 6 | 4. f(a) = 3a² - 2a + 1

5. Domain: x ≥ 4 or [4, ∞) | 6. Domain: x ≠ -2 or (-∞, -2) ∪ (-2, ∞)

Lesson 2: Polynomials

1. Both ends up (+∞) | 2. Left up (+∞), Right down (-∞) | 3. Both ends down (-∞)

4. x = 0, 2, -2 | 5. x = 2, 3

Lesson 3: Rational Functions

1. VA: x = -2, HA: y = 0, y-int: (0, 3/2)

2. Simplified: x + 3, Hole at x = 3

3. VA: x = 2, x = -2, HA: y = 2

Lesson 4: Exponentials

1. 32 | 2. 1/9 | 3. 1 | 4. 1/8

P(t) = 1000(1.05)ᵗ; P(10) ≈ 1628.89

Lesson 5: Logarithms

1. 5 | 2. 4 | 3. 3 | 4. 5 | 5. 0

6. 2log(x) + 3log(y) | 7. ln(x) - 2ln(y)

Lesson 6: Angle Conversions

Degrees to Radians: 1. π/2 | 2. π/4 | 3. π/3 | 4. π

Radians to Degrees: 5. 30° | 6. 60° | 7. 120° | 8. 270°

Lesson 7: Right Triangle Trig

1. sin(θ) = 5/13 | 2. cos(θ) = 12/13 | 3. tan(θ) = 5/12 | 4. csc(θ) = 13/5

5. x = 10 sin(30°) = 5 | 6. x = 8 tan(45°) = 8

Lesson 8: Unit Circle

1. √3/2 | 2. √2/2 | 3. √3/3 or 1/√3 | 4. 0 | 5. 0

6. Q II, sin +, cos - | 7. Q III, sin -, cos - | 8. Q IV, sin -, cos +

Lesson 9: Graphing

1. A = 3, P = 2π, PS = 0, VS = 0

2. A = 1, P = π, PS = 0, VS = 0

3. A = 2, P = 2π, PS = π/2 right, VS = 1 up

Lesson 10: Identities

1. sin θ | 2. 1 | 3. sin²θ | 4. tan²θ

5. cos θ = 4/5 | 6. tan θ = 3/4

Lesson 11: Sum/Difference

1. (√6 - √2)/4 | 2. (√6 - √2)/4 | 3. (√6 + √2)/4

Lesson 12: Sequences

1. Arithmetic, a₂₀ = 61 | 2. Geometric, a₇ = 1458 | 3. Arithmetic, a₁₅ = -40

4. S₁₀ = 100 | 5. S₅ = 31

Lesson 13: Limits

1. 7 | 2. 3 | 3. 8 | 4. 3

Lesson 14: Vectors

1. ⟨1, 8⟩ | 2. ⟨3, 2⟩ | 3. ⟨6, 15⟩ | 4. √29 ≈ 5.39 | 5. √10 ≈ 3.16

Romans 1:20

"For the invisible things of him from the creation of the world are clearly seen, being understood by the things that are made, even his eternal power and Godhead."