Chapter 01
Functions & Their Properties
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Memory Key
DOMAIN = INPUT = x | You can't take √ of negative, can't divide by zero
Quick Example
f(x) = √(x − 3) → Need x − 3 ≥ 0 → Domain: x ≥ 3 or [3, ∞)
What is the domain of \( f(x) = \dfrac{1}{\sqrt{x-2}} \)?
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Memory Key
EVEN = f(−x) = f(x) (y-axis symmetric) | ODD = f(−x) = −f(x) (origin symmetric)
Quick Example
f(x) = x² → f(−x) = (−x)² = x² = f(x) → EVEN ✓
Which function is odd?
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Memory Key
(f∘g)(x) = f(g(x)) → plug g into f, NOT f into g! Order matters!
Quick Example
f(x) = x+1, g(x) = 2x → (f∘g)(3) = f(g(3)) = f(6) = 7
Let \( f(x) = x^2 + 1 \) and \( g(x) = 2x - 3 \). Find \( (f \circ g)(2) \).
Chapter 02
Polynomials & Factoring
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Memory Key
SUM of cubes: a³+b³ = (a+b)(a²−ab+b²)
DIFF of cubes: a³−b³ = (a−b)(a²+ab+b²)
Tip: "Same, Opposite, Positive" (SOAP)
Factor completely: \( x^3 - 8 \)
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Memory Key
REMAINDER THEOREM: f(c) = remainder when f(x) ÷ (x−c)
Factor Theorem: (x−c) is a factor ↔ f(c) = 0
Quick Example
f(x) = x²−5x+6, remainder when ÷(x−3): f(3) = 9−15+6 = 0 → (x−3) is a factor!
What is the remainder when \( f(x) = 2x^3 - 3x + 5 \) is divided by \( (x - 2) \)?
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Memory Key
RATIONAL ROOT THEOREM: possible roots = ±(factors of constant) / (factors of leading coeff)
Find all real zeros of \( f(x) = x^3 - 6x^2 + 11x - 6 \).
Chapter 03
Exponential & Logarithmic Functions
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Memory Key · LOG RULES
log(AB) = logA + logB | log(A/B) = logA − logB
log(Aⁿ) = n·logA | log_b(b) = 1 | log_b(1) = 0
Simplify: \( \log_2 8 + \log_2 4 - \log_2 16 \)
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Memory Key
Same base → set exponents equal: bˣ = bʸ → x = y
Different base → take log of both sides
Quick Example
4ˣ = 8 → (2²)ˣ = 2³ → 2²ˣ = 2³ → 2x = 3 → x = 3/2
Solve for \( x \): \( 3^{2x-1} = 27 \)
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Memory Key
CHANGE OF BASE: log_b(a) = log(a) / log(b) = ln(a) / ln(b)
Solve: \( \log_3(x+1) = 2 \)
Chapter 04
Trigonometry
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Memory Key · UNIT CIRCLE
"All Students Take Calculus" → Q1: All +, Q2: Sin +, Q3: Tan +, Q4: Cos +
30°→(√3/2, 1/2) | 45°→(√2/2, √2/2) | 60°→(1/2, √3/2)
Find the exact value of \( \sin\!\left(\dfrac{5\pi}{6}\right) \).
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Memory Key · PYTHAGOREAN IDENTITIES
sin²θ + cos²θ = 1 (THE BIG ONE)
1 + tan²θ = sec²θ | 1 + cot²θ = csc²θ
If \( \sin\theta = \dfrac{3}{5} \) and \( \theta \) is in the first quadrant, what is \( \cos\theta \)?
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Memory Key · y = A·sin(Bx + C) + D
A = Amplitude | Period = 2π/B | D = vertical shift
Phase shift = −C/B (left if C > 0, right if C < 0)
What is the period of \( f(x) = 3\sin(2x - \pi) + 1 \)?
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Memory Key · DOUBLE ANGLE FORMULAS
sin(2θ) = 2·sinθ·cosθ
cos(2θ) = cos²θ − sin²θ = 2cos²θ−1 = 1−2sin²θ
If \( \sin\theta = \dfrac{1}{2} \) and \( \theta \) is in Quadrant I, find \( \sin(2\theta) \).
Chapter 05
Sequences, Series & Binomial Theorem
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Memory Key · ARITHMETIC SEQUENCE
aₙ = a₁ + (n−1)d (nth term)
Sₙ = n/2 · (a₁ + aₙ) (sum of n terms)
The first term of an arithmetic sequence is \(3\) and the common difference is \(5\). What is the 10th term?
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Memory Key · GEOMETRIC SERIES
aₙ = a₁ · rⁿ⁻¹ (nth term)
Infinite sum = a₁/(1−r) when |r| < 1
Find the sum of the infinite geometric series: \( 1 + \dfrac{1}{3} + \dfrac{1}{9} + \dfrac{1}{27} + \cdots \)
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Memory Key · BINOMIAL THEOREM
(a+b)ⁿ = Σ C(n,k) · aⁿ⁻ᵏ · bᵏ
kth term (from index 0): C(n,k) · aⁿ⁻ᵏ · bᵏ
What is the coefficient of \( x^2 \) in the expansion of \( (x + 2)^5 \)?
Chapter 06
Conic Sections & Analytic Geometry
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Memory Key · CIRCLE EQUATION
(x−h)² + (y−k)² = r²
Center = (h, k) | Radius = r (NOT r²!)
What is the center and radius of the circle \( (x-3)^2 + (y+2)^2 = 25 \)?
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Memory Key · PARABOLA
Vertex form: y = a(x−h)² + k → Vertex = (h, k)
a > 0: opens UP | a < 0: opens DOWN
Find the vertex of the parabola \( y = 2x^2 - 8x + 5 \) by completing the square.
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Memory Key · ELLIPSE
x²/a² + y²/b² = 1
If a > b: horizontal major axis. Foci: (±c, 0) where c² = a²−b²
For the ellipse \( \dfrac{x^2}{25} + \dfrac{y^2}{9} = 1 \), what is the length of the major axis?
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Memory Key · DOT PRODUCT
a⃗ · b⃗ = |a||b|cosθ
Perpendicular (⊥) ↔ dot product = 0 | Parallel ↔ cross = 0
Quick Example
⟨1, 0⟩ · ⟨0, 1⟩ = 1·0 + 0·1 = 0 → perpendicular ✓ (x and y axis)
Vectors \( \vec{u} = \langle 3, -4 \rangle \) and \( \vec{v} = \langle a, 6 \rangle \) are perpendicular. Find \( a \).