ACT Formulas

Percent

Percent = (part ÷ whole) × 100. Use to find what portion of a total something represents.

Percent Change

Percent change = [(new − original) ÷ original] × 100. Use to find the percentage increase or decrease.

Mean

Mean = sum of values ÷ number of values. Use to find the average of a data set.

Weighted Average

Weighted average = Σ(value × weight) ÷ Σweights. Use when different values contribute different amounts to an overall average.

Ratio

A ratio compares two quantities. If a/b = c/d, then ad = bc. Use to compare quantities or solve proportions.

Proportion

a/b = c/d. Use when two ratios are equivalent; cross-multiply to solve.

Product Rule of Exponents

x^a × x^b = x^(a+b). Use when multiplying powers with the same base.

Quotient Rule of Exponents

x^a ÷ x^b = x^(a−b). Use when dividing powers with the same base.

Power Rule of Exponents

(x^a)^b = x^(ab). Use when raising a power to another power.

Power of a Product

(xy)^a = x^a y^a. Use when raising a product to a power.

Negative Exponent

x^(−a) = 1/x^a. Use to rewrite negative exponents as positive exponents.

Zero Exponent

x^0 = 1 for x ≠ 0. Use to simplify expressions with a zero exponent.

Fractional Exponent

x^(1/n) = ⁿ√x and x^(m/n) = ⁿ√(x^m). Use to convert between exponents and radicals.

Product Rule for Radicals

√(ab) = √a × √b. Use to multiply or simplify radicals.

Quotient Rule for Radicals

√(a/b) = √a ÷ √b. Use to simplify radicals involving fractions.

Distributive Property

a(b + c) = ab + ac. Use to remove parentheses by multiplying a factor by every term inside.

Difference of Squares

a² − b² = (a − b)(a + b). Use to factor expressions that are the difference of two perfect squares.

Perfect Square Trinomial

a² + 2ab + b² = (a + b)² and a² − 2ab + b² = (a − b)². Use to factor certain quadratic trinomials.

Slope

m = (y₂ − y₁)/(x₂ − x₁). Use to find the steepness and direction of a line.

Slope-Intercept Form

y = mx + b, where m is slope and b is y-intercept. Use to write or graph a line.

Point-Slope Form

y − y₁ = m(x − x₁). Use to write a line when given a point and slope.

Standard Form of a Line

Ax + By = C. Use as another form for writing linear equations.

Parallel Lines

Parallel lines have equal slopes: m₁ = m₂. Use to determine whether two lines are parallel.

Perpendicular Lines

Perpendicular lines have slopes that are negative reciprocals: m₁m₂ = −1. Use to determine whether two lines meet at 90°.

System of Equations

A set of equations solved simultaneously. Use substitution, elimination, or graphing to find values satisfying all equations.

Quadratic Formula

x = [−b ± √(b² − 4ac)] ÷ 2a. Use to solve ax² + bx + c = 0.

Quadratic Vertex Form

y = a(x − h)² + k. Vertex is (h,k). Use to identify the vertex and graph a parabola.

Axis of Symmetry

x = −b/(2a). Use to find the vertical line through the vertex of a quadratic.

Discriminant

b² − 4ac. Use to determine the number of real solutions of a quadratic: positive = 2, zero = 1, negative = 0.

Arithmetic Sequence

aₙ = a₁ + (n − 1)d. Use to find a term in a sequence with a constant difference.

Geometric Sequence

aₙ = a₁r^(n−1). Use to find a term in a sequence with a constant ratio.

Arithmetic Series

Sₙ = n/2(a₁ + aₙ). Use to find the sum of the terms in an arithmetic sequence.

Function Notation

f(x) represents the output of a function for input x. Use to evaluate functions and represent relationships.

Composition of Functions

(f ∘ g)(x) = f(g(x)). Use when one function is applied to the output of another.

Average Rate of Change

[f(b) − f(a)]/(b − a). Use to find how quickly a function changes over an interval.

Inverse Functions

f(f⁻¹(x)) = x. Use to undo a function or find its inverse.

Distance Formula

d = √[(x₂ − x₁)² + (y₂ − y₁)²]. Use to find the distance between two points.

Midpoint Formula

((x₁ + x₂)/2, (y₁ + y₂)/2). Use to find the point exactly halfway between two points.

Circle Standard Form

(x − h)² + (y − k)² = r². Center = (h,k), radius = r. Use to identify or graph a circle.

Circle Area

A = πr². Use to find the area inside a circle.

Circle Circumference

C = 2πr or C = πd. Use to find the distance around a circle.

Polygon Interior Angle Sum

(n − 2)180°. Use to find the sum of all interior angles of an n-sided polygon.

Regular Polygon Interior Angle

[(n − 2)180°]/n. Use to find each interior angle of a regular polygon.

Exterior Angle Sum

360°. Use to find the total of the exterior angles of any polygon.

Regular Polygon Exterior Angle

360°/n. Use to find each exterior angle of a regular polygon.

Triangle Area

A = 1/2bh. Use to find the area of a triangle.

Pythagorean Theorem

a² + b² = c², where c is the hypotenuse. Use for right triangles.

Triangle Angle Sum

A + B + C = 180°. Use to find a missing angle in a triangle.

45-45-90 Triangle

Side ratio = x : x : x√2. Use to find missing sides in a 45-45-90 right triangle.

30-60-90 Triangle

Side ratio = x : x√3 : 2x. Use to find missing sides in a 30-60-90 right triangle.

Similar Triangles

Corresponding sides are proportional. If the scale factor is k, lengths scale by k, areas by k², and volumes by k³. Use to solve for missing dimensions in similar figures.

Sine

sin θ = opposite/hypotenuse. Use to find a missing side or angle in a right triangle.

Cosine

cos θ = adjacent/hypotenuse. Use to find a missing side or angle in a right triangle.

Tangent

tan θ = opposite/adjacent. Use to find a missing side or angle in a right triangle.

Pythagorean Trigonometric Identity

sin²θ + cos²θ = 1. Use to relate sine and cosine values.

Law of Sines

a/sin A = b/sin B = c/sin C. Use to solve non-right triangles when you know an angle and its opposite side.

Law of Cosines

c² = a² + b² − 2ab cos C. Use to solve non-right triangles when given two sides and the included angle or all three sides.

Rectangle Area

A = lw. Use to find the area of a rectangle.

Rectangle Perimeter

P = 2l + 2w. Use to find the distance around a rectangle.

Square Area

A = s². Use to find the area of a square.

Square Perimeter

P = 4s. Use to find the perimeter of a square.

Square Diagonal

d = s√2. Use to find the diagonal of a square.

Parallelogram Area

A = bh. Use to find the area of a parallelogram.

Trapezoid Area

A = 1/2(b₁ + b₂)h. Use to find the area of a trapezoid.

Rhombus Area

A = 1/2d₁d₂. Use to find the area of a rhombus when its diagonals are known.

Rectangular Prism Volume

V = lwh. Use to find the volume of a rectangular prism.

Cube Volume

V = s³. Use to find the volume of a cube.

Cylinder Volume

V = πr²h. Use to find the volume of a cylinder.

Pyramid Volume

V = 1/3Bh. Use to find the volume of a pyramid, where B is the base area.

Cone Volume

V = 1/3πr²h. Use to find the volume of a cone.

Sphere Volume

V = 4/3πr³. Use to find the volume of a sphere.

Rectangular Prism Surface Area

SA = 2lw + 2lh + 2wh. Use to find the total surface area of a rectangular prism.

Cylinder Surface Area

SA = 2πr² + 2πrh. Use to find the total surface area of a cylinder.

Sphere Surface Area

SA = 4πr². Use to find the surface area of a sphere.

Basic Probability

P(A) = favorable outcomes ÷ total outcomes. Use to find the probability of an event.

Complement Rule

P(A') = 1 − P(A). Use to find the probability that an event does not happen.

Addition Rule of Probability

P(A or B) = P(A) + P(B) − P(A and B). Use to find the probability that at least one of two events occurs.

Independent Events

P(A and B) = P(A)P(B). Use when two events do not affect each other.

Conditional Probability

P(A|B) = P(A and B) ÷ P(B). Use to find the probability of A when B has already occurred.

Factorial

n! = n(n−1)(n−2)…1. Use in counting problems involving arrangements.

Permutations

nPr = n!/(n−r)!. Use when selecting and arranging objects where order matters.

Combinations

nCr = n!/[r!(n−r)!]. Use when selecting objects where order does not matter.

Exponential Growth

A = P(1 + r)^t. Use for quantities increasing by a constant percentage.

Exponential Decay

A = P(1 − r)^t. Use for quantities decreasing by a constant percentage.

Compound Interest

A = P(1 + r/n)^(nt). Use to calculate interest compounded n times per year.

Logarithm Definition

log_b(x) = y means b^y = x. Use to rewrite between logarithmic and exponential forms.

Product Rule for Logs

log_b(xy) = log_b(x) + log_b(y). Use to expand logarithms involving multiplication.

Quotient Rule for Logs

log_b(x/y) = log_b(x) − log_b(y). Use to expand logarithms involving division.

Power Rule for Logs

log_b(x^n) = n log_b(x). Use to move an exponent in a logarithm to the front.

Change of Base Formula

log_b(x) = log(x)/log(b). Use to calculate a logarithm with a different base.

Ellipse Standard Form

(x−h)²/a² + (y−k)²/b² = 1 for a horizontal ellipse, or (x−h)²/b² + (y−k)²/a² = 1 for a vertical ellipse. Use to identify the center, axes, and vertices of an ellipse.

Ellipse Foci

c² = a² − b². Use to find the distance c from the center to each focus of an ellipse.

Hyperbola Horizontal Form

(x−h)²/a² − (y−k)²/b² = 1. Use to identify a horizontal hyperbola that opens left and right.

Hyperbola Vertical Form

(y−k)²/a² − (x−h)²/b² = 1. Use to identify a vertical hyperbola that opens up and down.

Hyperbola Foci

c² = a² + b². Use to find the distance c from the center to each focus of a hyperbola.

Horizontal Hyperbola Asymptotes

y − k = ±(b/a)(x − h). Use to find the two diagonal lines a horizontal hyperbola approaches.

Vertical Hyperbola Asymptotes

y − k = ±(a/b)(x − h). Use to find the two diagonal lines a vertical hyperbola approaches.

Parabola Vertical Form

(x − h)² = 4p(y − k). Use for a parabola opening up or down.

Parabola Horizontal Form

(y − k)² = 4p(x − h). Use for a parabola opening left or right.

Parabola Vertex

(h,k). Use to identify the turning point of a parabola.