📖 Concept Review & Key Formulas
U1–U2
Absolute Value & Linear Systems
Absolute value equations: |ax + b| = c splits into two cases:
ax + b = c OR ax + b = −cAbsolute value inequalities:
|x| < a → −a < x < a | |x| > a → x < −a or x > a3-variable systems: Use elimination / substitution; back-substitute.
Worked Example
Solve |3x − 9| = 12.
Case 1: 3x − 9 = 12 → x = 7 | Case 2: 3x − 9 = −12 → x = −1
Answer: x ∈ {−1, 7}
Case 1: 3x − 9 = 12 → x = 7 | Case 2: 3x − 9 = −12 → x = −1
Answer: x ∈ {−1, 7}
U3–U4
Quadratics & Complex Numbers
Quadratic Formula: x = (−b ± √(b²−4ac)) / 2a
Discriminant Δ = b²−4ac: Δ>0 two real | Δ=0 one real | Δ<0 two complex
Vertex form: f(x)=a(x−h)²+k → vertex (h,k); opens up if a>0, down if a<0
Complex numbers: i² = −1 | (a+bi)(c+di) = (ac−bd) + (ad+bc)i
Worked Example
f(x) = −3(x+2)²+7 → vertex = (−2, 7), opens downward (a=−3<0)
U5–U6
Polynomials: Roots & Theorems
Remainder Theorem: f(x) ÷ (x−c) has remainder = f(c)
Factor Theorem: (x−c) is a factor ⟺ f(c) = 0
Rational Root Theorem: Possible rational roots = ±(factors of constant) / (factors of leading coeff)
Complex Conjugate Roots: If a+bi is a root, a−bi is also a root (real coefficients).
Worked Example
f(x)=x³−6x²+11x−6: test f(1)=1−6+11−6=0 → (x−1) is factor. Fully factors as (x−1)(x−2)(x−3).
U7–U8
Rational & Radical Functions
Domain of √f(x): f(x) ≥ 0 | Domain of 1/f(x): f(x) ≠ 0
HA rules: deg(num) < deg(den) → y=0 | equal → y=ratio of leading coeffs | num>den → oblique VA: set denominator = 0 (only if it does NOT cancel with numerator)Worked Example
f(x) = (3x²−1)/(x²−4): same degree → HA: y=3. Denom: x²−4=0 → VA: x=±2.
U9–U10
Exponential & Logarithmic Functions
Product: log_b(xy) = log_b(x)+log_b(y) | Quotient: log_b(x/y) = log_b(x)−log_b(y)
Power: log_b(xⁿ) = n·log_b(x) | Change of base: log_b(x) = ln(x)/ln(b)
Conversion: b^x = y ↔ log_b(y) = x
Worked Example
Solve 4^(x−1) = 8^(x/3): write as powers of 2. 2^(2x−2) = 2^x → 2x−2 = x → x = 2. ✓
U11–U12
Sequences, Series & Binomial Theorem
Arithmetic: a_n = a₁+(n−1)d | S_n = n/2·(a₁+a_n)
Geometric: a_n = a₁·r^(n−1) | S_n = a₁(1−rⁿ)/(1−r) | S∞ = a₁/(1−r), |r|<1
Binomial: (a+b)ⁿ = Σ C(n,k)·a^(n−k)·b^k, k=0 to n
Worked Example
Infinite geometric sum: a₁=18, r=12/18=2/3 → S∞ = 18/(1−2/3) = 18/(1/3) = 54.
U13–U14
Conic Sections
Circle: (x−h)²+(y−k)²=r²
Ellipse: x²/a²+y²/b²=1 (a>b). c²=a²−b². Foci: (±c,0)
Hyperbola: x²/a²−y²/b²=1. Asymptotes: y=±(b/a)x
Parabola: x²=4py → vertex (0,0), focus (0,p), directrix y=−p
Worked Example
x²=12y → 4p=12 → p=3 → focus: (0,3), directrix: y=−3.
U15
Counting, Probability & Inverse Functions
P(n,r)=n!/(n−r)! (order matters) | C(n,r)=n!/[r!(n−r)!] (order doesn't matter)
P(A∪B)=P(A)+P(B)−P(A∩B) | P(A|B)=P(A∩B)/P(B)
Inverse function: swap x and y in y=f(x), then solve for y → gives f⁻¹(x).
Worked Example
C(8,3)=8!/(3!·5!)=(8×7×6)/(3×2×1)=336/6=56.
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