Concept Review · Timed Practice · Detailed Explanations
20
Questions
60
Minutes
10
Units
01Algebra
02Systems
03Quadratics
04Polynomials
05Rationals
06Radicals
07Exponentials
08Trigonometry
09Geometry
10Statistics
Concept Summary
Key formulas & strategies — memorize before you practice
Unit 01 · Algebra
Linear Equations & Inequalities
Standard: ax + b = c → x = (c − b) / a
Slope-intercept: y = mx + b
Point-slope: y − y₁ = m(x − x₁)
Slope: m = (y₂ − y₁) / (x₂ − x₁)
Key: When multiplying/dividing an inequality by a negative number, flip the sign. Parallel lines share the same slope. Perpendicular lines have slopes that multiply to −1.
Quick Example
If 3(x − 2) = 15, find x.
3x − 6 = 15 → 3x = 21 → x = 7
Unit 02 · Systems
Systems of Equations
Substitution: isolate one variable, substitute
Elimination: multiply & add equations to cancel one variable
Number of solutions:
• One: lines intersect (different slopes)
• None: parallel (same slope, different intercept)
• Infinite: same line (identical equations)
Key: If the system has NO solution, the equations produce a contradiction (e.g., 0 = 5). If INFINITE solutions, the equations are multiples of each other.
Quick Example
Solve: 2x + y = 7, x − y = 2
Add: 3x = 9 → x = 3, y = 1
Unit 03 · Quadratics
Quadratic Functions
Standard: f(x) = ax² + bx + c
Vertex: x = −b/(2a), y = f(−b/(2a))
Quadratic Formula: x = [−b ± √(b²−4ac)] / (2a)
Discriminant: b²−4ac
> 0: two real roots = 0: one root < 0: no real roots
Factored: f(x) = a(x − r₁)(x − r₂)
Key: The vertex is always the axis of symmetry. If a > 0, parabola opens up (minimum). If a < 0, opens down (maximum). Sum of roots = −b/a; Product = c/a.
Quick Example
Find the vertex of f(x) = 2x² − 8x + 5
x = 8/(2·2) = 2, y = 2(4)−16+5 = −3 → Vertex: (2, −3)
Unit 04 · Polynomials & Rationals
Polynomial Operations & Rational Expressions
FOIL: (a+b)(c+d) = ac + ad + bc + bd
Difference of squares: a² − b² = (a+b)(a−b)
Perfect square: (a+b)² = a² + 2ab + b²
Rational: multiply/divide → factor & cancel common terms
LCD for addition/subtraction of rational expressions
Key: Always factor completely before canceling. A rational expression is undefined when the denominator = 0. Excluded values must be stated.
Quick Example
Simplify: (x²−4)/(x+2)
(x+2)(x−2)/(x+2) = x−2, where x ≠ −2
Unit 05 · Radicals & Exponents
Radical Equations & Exponential Functions
Radical rules: √(ab) = √a · √b, √(a/b) = √a/√b
Exponent rules: aᵐ · aⁿ = aᵐ⁺ⁿ, (aᵐ)ⁿ = aᵐⁿ, a⁻ⁿ = 1/aⁿ
Exponential growth: f(t) = a · bᵗ (b > 1)
Exponential decay: f(t) = a · bᵗ (0 < b < 1)
Logarithm: logₐb = c ↔ aᶜ = b
log(xy) = log x + log y, log(x/y) = log x − log y
Key: When solving radical equations, square both sides — always check for extraneous solutions! For exponential equations, take log of both sides.
Quick Example
Solve: √(2x + 1) = 5
2x + 1 = 25 → x = 12 ✓ (check: √25 = 5)
Unit 06 · Trigonometry
Trig Ratios, Identities & the Unit Circle
SOH-CAH-TOA:
sin θ = opp/hyp, cos θ = adj/hyp, tan θ = opp/adj
Pythagorean Identity: sin²θ + cos²θ = 1
→ 1 + tan²θ = sec²θ, 1 + cot²θ = csc²θ
Special angles:
sin 30°=1/2, cos 30°=√3/2, tan 30°=√3/3
sin 45°=√2/2, cos 45°=√2/2, tan 45°=1
sin 60°=√3/2, cos 60°=1/2, tan 60°=√3
Law of Sines: a/sin A = b/sin B = c/sin C
Law of Cosines: c² = a² + b² − 2ab·cos C
Key: All Students Take Calculus (ASTC) — signs by quadrant: Q1 all +, Q2 sin +, Q3 tan +, Q4 cos +.
Quick Example
If sin θ = 3/5 and θ is in Q1, find cos θ.
cos θ = 4/5 (3-4-5 right triangle)
Unit 07 · Geometry
Circles, Triangles & Coordinate Geometry
Circle: Area = πr², Circumference = 2πr
Equation: (x−h)² + (y−k)² = r²
Arc length = (θ/360°) · 2πr
Sector area = (θ/360°) · πr²
Triangle area = (1/2)bh
Special right: 30-60-90 → sides 1 : √3 : 2
Special right: 45-45-90 → sides 1 : 1 : √2
Distance: d = √[(x₂−x₁)² + (y₂−y₁)²]
Midpoint: M = ((x₁+x₂)/2, (y₁+y₂)/2)
Key: A tangent line to a circle is always perpendicular to the radius at the point of tangency. Interior angles of a triangle sum to 180°.
Quick Example
Find the radius of (x−3)² + (y+1)² = 25
r² = 25, so r = 5
Unit 08 · Functions
Function Composition, Inverse & Matrices
Composition: (f∘g)(x) = f(g(x))
Inverse: swap x and y, then solve for y
f(f⁻¹(x)) = x
Matrix multiplication: [a b][e f] = [ae+bg af+bh]
[c d][g h] [ce+dg cf+dh]
Determinant of 2×2: |a b| = ad − bc
|c d|
Key: Domain of f∘g: start with domain of g, then restrict to values where g(x) is in the domain of f. A function has an inverse only if it is one-to-one.
Quick Example
If f(x) = 2x+1 and g(x) = x², find f(g(3)).
g(3) = 9, f(9) = 2(9)+1 = 19
Unit 09 · Statistics & Probability
Mean, Median, Probability & Counting
Mean = (sum of values) / (number of values)
Median = middle value when sorted
Mode = most frequent value
Weighted mean = Σ(weight × value) / Σ(weights)
Probability: P(A) = favorable/total
P(A and B) = P(A) · P(B) [independent]
P(A or B) = P(A) + P(B) − P(A and B)
Permutation: P(n,r) = n!/(n−r)!
Combination: C(n,r) = n!/[r!(n−r)!]
Key: Adding a constant to all values shifts mean and median by that constant, but does NOT change standard deviation. Multiplying by k scales standard deviation by |k|.
Key: Powers of i cycle with period 4. For iⁿ: divide n by 4 and use the remainder (0→1, 1→i, 2→−1, 3→−i). To divide complex numbers, multiply numerator and denominator by the conjugate of the denominator.