Solve linear, rational, and absolute value equations and inequalities. Graph solution sets on number lines and coordinate planes.
β Must Memorize
|ax + b| = c β ax + b = c OR ax + b = βc
|ax + b| < c β βc < ax + b < c
|ax + b| > c β ax + b < βc OR ax + b > c
π Example
Solve |2x β 3| = 7
β 2xβ3=7 β x=5 OR 2xβ3=β7 β x=β2 Answer: x=5 or x=β2
Solve quadratics by factoring, completing the square, and the quadratic formula. Analyze vertex, axis of symmetry, and discriminant.
β Must Memorize
Quadratic Formula: x = (βb Β± β(bΒ²β4ac)) / 2a
Vertex form: y = a(x β h)Β² + k where vertex = (h, k)
Discriminant: D = bΒ²β4ac
D > 0 β two real roots
D = 0 β one real root (repeated)
D < 0 β two complex roots
π Example
Find vertex of y = 2xΒ² β 8x + 5
β h = β(β8)/(2Β·2) = 2, k = 2(4)β16+5 = β3 Vertex: (2, β3)
Factor higher-degree polynomials, find zeros using the Rational Root Theorem, apply Remainder and Factor Theorems.
β Must Memorize
Remainder Theorem: f(a) = remainder when f(x) Γ· (xβa)
Factor Theorem: (xβa) is a factor βΊ f(a) = 0
Rational Root Theorem: possible rational roots = Β±(factors of constant) / (factors of leading coeff)
Sum of roots = βb/a, Product of roots = c/a (for axΒ²+bx+c)
π Example
Is (xβ2) a factor of f(x) = xΒ³β3xΒ²+x+2?
β f(2) = 8β12+2+2 = 0 β Yes, (xβ2) is a factor.
Find domain restrictions, vertical/horizontal/oblique asymptotes, holes, and graph rational functions.
β Must Memorize
Vertical asymptote: denominator = 0 (after canceling holes)
Horizontal asymptote:
deg(num) < deg(den) β y = 0
deg(num) = deg(den) β y = lead coeff ratio
deg(num) > deg(den) β oblique asymptote (polynomial division)
Hole: common factor in num & den (canceled factor)
π Example
Find asymptotes of f(x) = (xΒ²β1)/(xΒ²β4)
β VA: x=Β±2 HA: y=1 (same degree, ratio of leading coeffs = 1/1)
Work with exponential growth/decay, convert between exponential and logarithmic form, apply log properties, and solve log equations.
β Must Memorize
log_b(x) = y βΊ b^y = x
log(xy) = log x + log y
log(x/y) = log x β log y
log(x^n) = nΒ·log x
Change of base: log_b(x) = ln(x)/ln(b)
Continuous growth: A = Pe^(rt)
π Example
Solve logβ(x) + logβ(xβ2) = 3
β logβ(x(xβ2))=3 β xΒ²β2x=8 β xΒ²β2xβ8=0 β (xβ4)(x+2)=0 β x=4 β (x must be >0 and x>2)
Simplify radicals, perform operations with complex numbers, solve radical equations, and rationalize denominators.
β Must Memorize
i = β(β1), iΒ² = β1, iΒ³ = βi, iβ΄ = 1 (cycle of 4)
(a+bi)(c+di) = (acβbd) + (ad+bc)i
Conjugate of (a+bi) = (aβbi)
|a+bi| = β(aΒ²+bΒ²)
β(a)Β·β(b) = β(ab) [valid for a,b β₯ 0]
π Example
Simplify (3+2i)(1βi)
β 3β3i+2iβ2iΒ² = 3βi+2 = 5βi
Solve 2Γ2 and 3Γ3 systems algebraically and with matrices. Use Cramer's Rule and row reduction (Gaussian elimination).
β Must Memorize
For 2Γ2 system: ax+by=e, cx+dy=f
det(A) = adβbc
x = (edβbf)/(adβbc), y = (afβec)/(adβbc)
Matrix multiplication: (AB)α΅’β±Ό = Ξ£ Aα΅’βBββ±Ό
Inverse: Aβ»ΒΉ = (1/det) Β· adjugate
π Example
Solve: 2x+y=7 and xβy=2
β Add: 3x=9 β x=3, then y=7β6=1 Answer: (3, 1)
Work with arithmetic and geometric sequences and series. Find partial sums and infinite geometric series sums.
β Must Memorize
Arithmetic: aβ = aβ + (nβ1)d, S_n = n(aβ+aβ)/2
Geometric: aβ = aβΒ·r^(nβ1), S_n = aβ(1βrβΏ)/(1βr)
Infinite geometric (|r|<1): S_β = aβ/(1βr)
Sigma notation: Ξ£(k=1 to n) k = n(n+1)/2
π Example
Find S_β of: 12, 6, 3, 1.5, β¦
β aβ=12, r=0.5, S_β = 12/(1β0.5) = 12/0.5 = 24
Identify and graph circles, ellipses, hyperbolas, and parabolas. Write equations in standard form.
β Must Memorize
Circle: (xβh)Β²+(yβk)Β² = rΒ²
Ellipse: (xβh)Β²/aΒ² + (yβk)Β²/bΒ² = 1 (a>b, major axis horizontal)
Hyperbola: (xβh)Β²/aΒ² β (yβk)Β²/bΒ² = 1 (opens L/R)
Parabola: (xβh)Β² = 4p(yβk) or (yβk)Β² = 4p(xβh)
focus is p units from vertex
π Example
Identify: 4xΒ²+9yΒ²=36
β Divide by 36: xΒ²/9+yΒ²/4=1 β Ellipse with a=3, b=2, center (0,0)
Convert degrees/radians, use unit circle values, apply basic trig identities, and solve trig equations.
β Must Memorize
Radians β Degrees: ΞΈ(rad) = ΞΈ(deg)Β·Ο/180
sinΒ²ΞΈ + cosΒ²ΞΈ = 1
tan ΞΈ = sin ΞΈ / cos ΞΈ
Unit circle key values:
sin(30Β°)=1/2, sin(45Β°)=β2/2, sin(60Β°)=β3/2
cos(30Β°)=β3/2, cos(45Β°)=β2/2, cos(60Β°)=1/2
π Example
Convert 210Β° to radians and find sin(210Β°)
β 210Β·(Ο/180) = 7Ο/6, sin(210Β°) = β1/2 (3rd quadrant, ref angle 30Β°)