ACT Math – Hard Practice Test

01 Algebra — Equations, Systems & Functions

Memorize these identities cold — they appear in disguised form on hard ACT questions.

(a + b)² = a² + 2ab + b² (a − b)² = a² − 2ab + b² (a + b)(a − b) = a² − b² Quadratic Formula: x = [−b ± √(b²−4ac)] / 2a Discriminant: Δ = b²−4ac (Δ>0: 2 real roots; Δ=0: 1; Δ<0: none)

Quick Example

If 2x² − 5x + 2 = 0, find x.

→ x = (5 ± √(25−16))/4 = (5 ± 3)/4 → x = 2 or x = ½

02 Functions — Composition, Inverse & Transformations

Composite functions: apply inner first, then outer.

(f ∘ g)(x) = f(g(x)) Inverse: swap x and y, solve for y → f⁻¹(x) Vertex form: f(x) = a(x − h)² + k → vertex at (h, k) Reflection over y = x gives the inverse function

Quick Example

f(x) = 2x+1, g(x) = x². Find f(g(3)).

→ g(3) = 9 → f(9) = 2(9)+1 = 19

03 Geometry — Angles, Circles & Coordinate Geometry

High-value formulas the ACT tests repeatedly:

Circle: (x−h)² + (y−k)² = r² → center (h,k), radius r Arc length = (θ/360°) · 2πr Sector area = (θ/360°) · πr² Midpoint: ((x₁+x₂)/2, (y₁+y₂)/2) Distance: √[(x₂−x₁)² + (y₂−y₁)²] Slope: m = (y₂−y₁)/(x₂−x₁) Perpendicular slopes: m₁ · m₂ = −1

Quick Example

What is the radius of (x+3)² + (y−2)² = 25?

→ r² = 25 → r = 5; center = (−3, 2)

04 Trigonometry — SOH-CAH-TOA & Identities

Know these exactly — the ACT tests trig identities heavily in its hard questions.

sin²θ + cos²θ = 1 tan θ = sin θ / cos θ sin(2θ) = 2 sin θ cos θ cos(2θ) = cos²θ − sin²θ = 1 − 2sin²θ = 2cos²θ − 1 Special angles (degrees → radians → sin, cos, tan): 30° → π/6 → ½, √3/2, 1/√3 45° → π/4 → √2/2, √2/2, 1 60° → π/3 → √3/2, ½, √3 Law of Sines: a/sin A = b/sin B = c/sin C Law of Cosines: c² = a² + b² − 2ab·cos C

Quick Example

If sin θ = 3/5 and θ is in Quadrant I, find cos θ.

→ cos θ = √(1 − 9/25) = √(16/25) = 4/5

05 Statistics, Probability & Number Theory

Frequently tested on ACT hard questions in data interpretation contexts.

Mean = (sum of values) / n Median = middle value when sorted Combinations: C(n,r) = n! / [r!(n−r)!] Permutations: P(n,r) = n! / (n−r)! P(A and B) = P(A) · P(B) [if independent] P(A or B) = P(A) + P(B) − P(A and B) Arithmetic sequence: aₙ = a₁ + (n−1)d Geometric sequence: aₙ = a₁ · rⁿ⁻¹ Sum of geometric (|r|<1): S = a₁/(1−r)

Quick Example

How many ways can 3 people be chosen from a group of 7?

→ C(7,3) = 7!/(3!·4!) = 35

Master the
Hard Questions

Key Concepts & Formulas

Master theHard Questions

Key Concepts & Formulas

Master the
Hard Questions