ACT Math Master — 20 Essential Questions

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⏱ Suggested Time: 35 minutes

Answer Key & Full Explanations

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ACT Math Master
20 Core Questions · All Major Topics · Explained
ACT Official Style · 2025 Edition

Master the ACT
Math Section

20 carefully crafted questions across every major topic. Study the concepts, then prove your mastery — with instant explanations.

📐
Algebra
Q 1–5 · Linear, Quadratic, Systems
△
Geometry
Q 6–10 · Angles, Circles, Area
〜
Trig & Functions
Q 11–15 · SOH-CAH-TOA, Graphs
∑
Stats & Numbers
Q 16–20 · Probability, Data
Study Guide

Concepts & Key Formulas

Review each topic carefully. Memorize the formulas — they appear directly in the questions.

C1 Linear Equations & Inequalities
Slope-Intercept & Standard Form

A linear equation represents a straight line. The slope m tells how steep the line is. The y-intercept b is where the line crosses the y-axis.

y = mx + b (slope-intercept form) Ax + By = C (standard form) Slope: m = (y₂ - y₁) / (x₂ - x₁) Parallel lines: same slope (m₁ = m₂) Perpendicular lines: m₁ × m₂ = -1
🔖 Must Memorize
★ Slope = rise ÷ run ★ y-int: set x=0 ★ x-int: set y=0 ★ Perp: negative reciprocal
Quick Example

Line passes through (2, 5) and (6, 13). What is the slope?

m = (13 − 5) / (6 − 2) = 8/4 = 2
C2 Quadratic Equations
Solving & Graphing Parabolas

Quadratic equations have the form ax² + bx + c = 0. They can be solved by factoring, completing the square, or using the quadratic formula.

ax² + bx + c = 0 Quadratic Formula: x = (−b ± √(b² − 4ac)) / 2a Discriminant: Δ = b² − 4ac Δ > 0: two real roots Δ = 0: one real root (repeated) Δ < 0: no real roots (complex) Vertex: x = −b / 2a Sum of roots = −b/a, Product = c/a
🔖 Must Memorize
★ Quad formula: x = (−b ± √Δ) / 2a ★ Δ = b²−4ac ★ Vertex x = −b/2a
Quick Example

Solve: x² − 5x + 6 = 0

Factor: (x − 2)(x − 3) = 0 → x = 2 or 3
C3 Systems of Equations
Substitution & Elimination

A system of equations has multiple equations with multiple unknowns. Find values that satisfy all equations simultaneously.

Substitution: solve one equation for a variable, substitute into the other. Elimination: add/subtract equations to cancel one variable. No solution: parallel lines (same slope, different y-int) Infinite solutions: same line (identical equations)
🔖 Must Memorize
★ Subst: isolate one variable first ★ No soln: slopes equal, y-int different ★ ∞ soln: identical lines
Quick Example

Solve: x + y = 7, x − y = 3

Add: 2x = 10 → x = 5, y = 2. Solution: (5, 2)
C4 Functions & Transformations
Evaluating, Composing & Shifting

A function maps each input to exactly one output. Function notation f(x) means "plug x into the rule." Transformations shift, stretch, or reflect the graph.

f(x): plug x into the function rule f(g(x)): composition — compute g(x) first, then f Vertical shift up k: y = f(x) + k Vertical shift down k: y = f(x) − k Horizontal shift right h: y = f(x − h) Horizontal shift left h: y = f(x + h) Reflection over x-axis: y = −f(x) Vertical stretch by a: y = a·f(x)
🔖 Must Memorize
★ f(x−h) shifts RIGHT by h ★ f(x+h) shifts LEFT by h ★ Compose: inside out
Quick Example

If f(x) = 2x + 1 and g(x) = x², find f(g(3)).

g(3) = 9 → f(9) = 2(9)+1 = 19
C5 Exponents & Radicals
Laws of Exponents
aᵐ · aⁿ = aᵐ⁺ⁿ aᵐ / aⁿ = aᵐ⁻ⁿ (aᵐ)ⁿ = aᵐⁿ a⁻ⁿ = 1/aⁿ a⁰ = 1 a^(1/n) = ⁿ√a a^(m/n) = ⁿ√(aᵐ) √(a·b) = √a · √b
🔖 Must Memorize
★ Multiply: add exponents ★ Divide: subtract exponents ★ Negative exp = reciprocal ★ a^(1/2) = √a
Quick Example

Simplify: (x³)² / x⁴

x⁶ / x⁴ = x²
C6 Angles & Triangles
Triangle Properties & Special Triangles

Angles in a triangle always sum to 180°. The Pythagorean theorem applies to right triangles.

Triangle angle sum: A + B + C = 180° Pythagorean Theorem: a² + b² = c² (c = hypotenuse) Special right triangles: 30-60-90: sides → 1 : √3 : 2 45-45-90: sides → 1 : 1 : √2 Exterior angle = sum of two non-adjacent interior angles Isoceles triangle: base angles are equal
🔖 Must Memorize
★ 30-60-90: x, x√3, 2x ★ 45-45-90: x, x, x√2 ★ a²+b²=c²
Quick Example

Right triangle: legs = 5 and 12. Hypotenuse = ?

c = √(25 + 144) = √169 = 13
C7 Circles
Area, Circumference & Arc
Area = π r² Circumference = 2π r = π d Arc length = (θ/360°) × 2πr Sector area = (θ/360°) × πr² Circle equation: (x − h)² + (y − k)² = r² Center: (h, k), Radius: r
🔖 Must Memorize
★ A = πr² ★ C = 2πr ★ Arc: (θ/360) × 2πr ★ (x−h)²+(y−k)²=r²
Quick Example

Circle with radius 6. Area of a 90° sector?

(90/360) × π(36) = (1/4)(36π) = 9π
C8 Area, Perimeter & Volume
2D and 3D Shape Formulas
Rectangle: Area = l×w, Perimeter = 2(l+w) Triangle: Area = ½ b×h Trapezoid: Area = ½(b₁+b₂)×h Parallelogram: Area = b×h Cube: Volume = s³, Surface Area = 6s² Rectangular Prism: V = l×w×h Cylinder: V = πr²h Cone: V = ⅓πr²h Sphere: V = (4/3)πr³
🔖 Must Memorize
★ Trap: ½(b₁+b₂)h ★ Cyl: πr²h ★ Cone: ⅓πr²h
Quick Example

Trapezoid: bases 4 and 10, height 5. Area?

A = ½(4+10)(5) = ½(14)(5) = 35
C9 Coordinate Geometry
Distance, Midpoint & Translations
Distance formula: d = √[(x₂−x₁)² + (y₂−y₁)²] Midpoint formula: M = ((x₁+x₂)/2, (y₁+y₂)/2) Translation: (x, y) → (x+a, y+b) Reflection over x-axis: (x, y) → (x, −y) Reflection over y-axis: (x, y) → (−x, y) Reflection over y=x: (x, y) → (y, x)
🔖 Must Memorize
★ Distance: √(Δx²+Δy²) ★ Midpoint: average coords ★ Reflect x-axis: flip y sign
Quick Example

Midpoint of (1, 3) and (7, 9)?

M = ((1+7)/2, (3+9)/2) = (4, 6)
C10 Similar & Congruent Triangles
Ratios of Corresponding Parts
Similar triangles: same shape, different size Corresponding sides are proportional Corresponding angles are equal If scale factor = k: Side lengths ratio = k Area ratio = k² Volume ratio = k³ Congruent triangles: SSS, SAS, ASA, AAS, HL (right triangles)
🔖 Must Memorize
★ Similar: sides proportional ★ Area scales as k² ★ Volume scales as k³
Quick Example

Two similar triangles have sides in ratio 3:5. Ratio of their areas?

Area ratio = 3²:5² = 9:25
C11 Trigonometry
SOH-CAH-TOA & Key Values
sin θ = Opposite / Hypotenuse (SOH) cos θ = Adjacent / Hypotenuse (CAH) tan θ = Opposite / Adjacent (TOA) csc θ = 1/sin θ, sec θ = 1/cos θ, cot θ = 1/tan θ Pythagorean Identity: sin²θ + cos²θ = 1 Law of Sines: a/sin A = b/sin B = c/sin C Law of Cosines: c² = a² + b² − 2ab·cos C Key values: sin 30° = ½, cos 30° = √3/2, tan 30° = 1/√3 sin 45° = √2/2, cos 45° = √2/2, tan 45° = 1 sin 60° = √3/2, cos 60° = ½, tan 60° = √3
🔖 Must Memorize
★ SOH-CAH-TOA ★ sin²+cos²=1 ★ sin 30°=½, cos 60°=½ ★ tan 45°=1
Quick Example

Right triangle: hypotenuse = 10, angle = 30°. Find the opposite side.

opp = 10 × sin 30° = 10 × ½ = 5
C12 Trigonometric Graphs
Amplitude, Period & Phase Shift
y = A sin(Bx + C) + D Amplitude = |A| Period = 2π / B Phase shift = −C / B (left if positive, right if negative) Vertical shift = D y = A cos(Bx + C) + D (same rules) sin has midline crossings at 0, π, 2π cos has maximum at 0, minimum at π
🔖 Must Memorize
★ Amplitude = |A| ★ Period = 2π/B ★ Phase shift = −C/B
Quick Example

y = 3 sin(2x). Amplitude and period?

Amplitude = 3, Period = 2π/2 = π
C13 Polynomials & Rational Expressions
Factoring, FOIL & Rational Simplification
FOIL: (a+b)(c+d) = ac + ad + bc + bd Difference of squares: a² − b² = (a+b)(a−b) Perfect square trinomial: (a+b)² = a² + 2ab + b² Sum/Diff of cubes: a³ + b³ = (a+b)(a²−ab+b²) a³ − b³ = (a−b)(a²+ab+b²) Rational expression: factor top & bottom, cancel common factors Undefined when denominator = 0
🔖 Must Memorize
★ Diff of squares: (a+b)(a−b) ★ (a+b)²=a²+2ab+b² ★ Cancel common factors
Quick Example

Simplify: (x² − 9) / (x − 3)

= (x+3)(x−3) / (x−3) = x + 3 (x ≠ 3)
C14 Sequences & Series
Arithmetic & Geometric Sequences
Arithmetic Sequence (common difference d): aₙ = a₁ + (n−1)d Sum: Sₙ = n/2 × (a₁ + aₙ) Geometric Sequence (common ratio r): aₙ = a₁ × r^(n−1) Sum: Sₙ = a₁(1 − rⁿ) / (1 − r), r ≠ 1
🔖 Must Memorize
★ Arith: aₙ = a₁ + (n−1)d ★ Geo: aₙ = a₁·rⁿ⁻¹
Quick Example

Arithmetic sequence: a₁ = 3, d = 4. Find a₁₀.

a₁₀ = 3 + (10−1)(4) = 3 + 36 = 39
C15 Probability & Counting
Basic Probability & Combinations
P(event) = favorable outcomes / total outcomes P(A and B) = P(A) × P(B) [independent events] P(A or B) = P(A) + P(B) − P(A and B) P(not A) = 1 − P(A) Combinations (order doesn't matter): C(n,r) = n! / [r!(n−r)!] Permutations (order matters): P(n,r) = n! / (n−r)!
🔖 Must Memorize
★ P = favorable/total ★ P(A or B): add, subtract overlap ★ nCr = n!/(r!(n−r)!)
Quick Example

Choosing 2 from 5 people. How many ways? (order doesn't matter)

C(5,2) = 5!/(2!×3!) = 10
C16 Statistics & Data Analysis
Mean, Median, Mode & Range
Mean = sum of values / count Median = middle value (sorted data) Mode = most frequent value Range = max − min Standard deviation: measures spread from mean (lower SD = data clustered tightly) Weighted average: Σ(weight × value) / Σ(weights)
🔖 Must Memorize
★ Mean: sum/count ★ Median: sort, pick middle ★ Outlier raises/lowers mean
Quick Example

Data: {3, 7, 7, 9, 14}. Find mean, median, mode.

Mean = 40/5 = 8, Median = 7, Mode = 7
C17 Number Properties & Ratios
Factors, Multiples, Percentages & Ratios
LCM: smallest number divisible by both a and b GCF: largest factor common to both a and b Percent: part/whole × 100 Percent change: (new − old) / old × 100 Ratio a:b means a/(a+b) and b/(a+b) of total Proportion: a/b = c/d → ad = bc (cross multiply)
🔖 Must Memorize
★ % change = Δ/original × 100 ★ Cross-multiply proportions ★ LCM: list multiples
Quick Example

Price increased from $80 to $100. Percent increase?

(100−80)/80 × 100 = 20/80 × 100 = 25%