Complete Guide to Acing the ACT Mathematics Section

ACT Math Section

Consists of 60 questions to be answered in 60 minutes without a formula sheet.

Real Numbers

Include all rational and irrational numbers found on the number line.

Complex Numbers

Involve the imaginary unit i, defined as i = √(-1).

Imaginary Unit Rules

i^1 = i, i^2 = -1, i^3 = -i, i^4 = 1; the cycle repeats every 4 powers.

Complex Form

Expressed as a + bi, where a is the real part and b is the imaginary part.

Conjugates

Used to divide complex numbers by multiplying the numerator and denominator by the conjugate of the denominator.

Matrix

A rectangular array of numbers used in various mathematical operations.

Matrix Addition/Subtraction

Can only add or subtract matrices with the same dimensions, by adding corresponding elements.

Matrix Multiplication

To multiply Matrix A (m x n) by Matrix B (n x p), the inner dimensions must match.

Vectors

Objects that have both magnitude and direction, often represented as ⟨a, b⟩ or ai + bj.

Component Form of Vectors

Given by ⟨x2-x1, y2-y1⟩.

Magnitude of Vectors

Calculated as ||v|| = √(a² + b²), similar to the Pythagorean theorem.

Vector Addition

Performed by adding the x-components and y-components separately.

Common Mistake: Imaginary Unit

Forgetting that i² = -1 during simplifications.

Common Mistake: Matrix Dimensions

Attempting to multiply matrices where the columns of the first don't match the rows of the second.

Linear Equations

Often represented in different forms, such as slope-intercept, standard, and point-slope forms.

Slope-Intercept Form

y = mx + b, where m is the slope and b is the y-intercept.

Point-Slope Form

y - y1 = m(x - x1), used to represent a line given a specific point.

Inequalities

Flipping the inequality sign when multiplying or dividing by a negative number.

Systems of Equations

Finding the intersection points of two lines through methods like substitution and elimination.

Quadratics

Typically represented as parabolas in quadratic equations.

Quadratic Formula

x = (-b ± √(b² - 4ac)) / (2a), used for solving quadratics.

Discriminant

Δ = b² - 4ac; indicates the nature of the roots of a quadratic.

Logarithms

The inverse of exponents, represented as log_b(y) = x if b^x = y.

Product Rule of Logarithms

logb(xy) = logb(x) + log_b(y).

Common Mistake: Negative Distribution

Incorrectly distributing a negative sign in expressions.

Common Mistake: Undefined Slopes

Confusing undefined slopes of vertical lines with zero slopes of horizontal lines.

Plane Geometry

Focuses on shapes where angles sum to 180º and provides area formulas.

Pythagorean Theorem

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

Special Right Triangle 30-60-90

Sides are in the ratio of x : x√3 : 2x.

Distance Formula

d = √((x2 - x1)² + (y2 - y1)²).

Midpoint Formula

M = ((x1 + x2)/2, (y1 + y2)/2).

Volume of Rectangular Prism

V = l * w * h.

Volume of Cylinder

V = πr²h.

Volume of Sphere

V = (4/3)πr³.

Common Mistake: Radii vs. Diameter

Confusing diameter with radius in calculations.

SOH CAH TOA

Sine = opposite/hypotenuse, Cosine = adjacent/hypotenuse, Tangent = opposite/adjacent.

Unit Circle

Displays angles and coordinates, crucial for trigonometric functions.

Law of Sines

Describes the relationship between angles and sides in non-right triangles.

Basic Probability Definition

P(A) = Desired Outcomes / Total Possible Outcomes.

Independent Events Probability

P(A and B) = P(A) * P(B).

Arithmetic Sequence Formula

an = a1 + (n-1)d.

Geometric Sequence Formula

an = a1 * r^(n-1).

Final Exam Tips

Plug in numbers, back-solving, draw diagrams, and be calculator literate.