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ACT (American College Testing) Mathematics Syllabus
The Mathematics syllabus for ACT (American College Testing) in full: 5 chapters, 23 topics and 53 sub-topics, every one listed below, plus 50 Mathematics flashcards with answers.
Mathematics syllabus: all 5 chapters
Expand any chapter to see its topics and sub-topics.
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Number and Quantity
4 topics- Real Number System
- Integers, rationals, and irrationals
- Absolute value and number line operations
- Order of operations
- Exponents and Radicals
- Integer and rational exponents
- Simplifying radical expressions
- Scientific notation
- Complex Numbers
- Imaginary unit i and powers of i
- Operations with complex numbers
- Vectors and Matrices
- Matrix addition, subtraction, and scalar multiplication
- Basic vector operations
- Real Number System
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Algebra
5 topics- Linear Equations and Inequalities
- Solving one-variable linear equations
- Solving and graphing linear inequalities
- Absolute value equations and inequalities
- Systems of Equations
- Substitution and elimination methods
- Graphical interpretation of solutions
- Polynomials and Factoring
- Adding, subtracting, and multiplying polynomials
- Factoring quadratics and special products
- Quadratic Equations
- Solving by factoring, completing the square, and the quadratic formula
- The discriminant and nature of roots
- Expressions and Word Problems
- Translating verbal statements into expressions
- Rates, ratios, and proportions
- Linear Equations and Inequalities
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Functions
5 topics- Function Notation and Evaluation
- Evaluating f(x) for given inputs
- Domain and range
- Composition of functions
- Linear and Quadratic Functions
- Slope-intercept and point-slope forms
- Parabolas, vertex, and axis of symmetry
- Polynomial, Radical, and Rational Functions
- Identifying graphs and key features
- Asymptotes of rational functions
- Exponential and Logarithmic Functions
- Growth and decay models
- Logarithm rules and conversions
- Transformations of Graphs
- Shifts, reflections, and stretches
- Function Notation and Evaluation
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Geometry
5 topics- Plane Geometry
- Angles, lines, and triangles
- Polygons and their properties
- Congruence and similarity
- Circles
- Arc length, sector area, and central angles
- Equation of a circle in the coordinate plane
- Perimeter, Area, and Volume
- Composite two-dimensional figures
- Surface area and volume of solids
- Coordinate Geometry
- Distance and midpoint formulas
- Slopes of parallel and perpendicular lines
- Right Triangles and the Pythagorean Theorem
- Special right triangles
- Applications in measurement problems
- Plane Geometry
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Statistics and Probability
4 topics- Data Interpretation
- Reading tables, charts, and graphs
- Two-way frequency tables
- Measures of Center and Spread
- Mean, median, mode
- Range and standard deviation concepts
- Probability
- Simple and compound events
- Counting principles, permutations, and combinations
- Expected value
- Trigonometry
- SOH-CAH-TOA in right triangles
- Trigonometric values and the unit circle
- Law of Sines and Law of Cosines
- Graphs of trigonometric functions
- Data Interpretation
ACT (American College Testing) Mathematics flashcards — 50 questions with answers
What are the subsets of the real number system, from most to least restrictive?
Natural numbers (counting), whole numbers (naturals plus 0), integers (whole numbers plus negatives), rationals (ratios of integers), and irrationals (non-repeating, non-terminating decimals). Rationals and irrationals together make up all real numbers.
What distinguishes a rational number from an irrational number?
A rational number can be written as a fraction a/b of two integers (its decimal terminates or repeats); an irrational number cannot (its decimal is non-terminating and non-repeating), e.g., pi, e, sqrt(2).
State the product and quotient rules for exponents.
Product: x^a * x^b = x^(a+b). Quotient: x^a / x^b = x^(a-b). Same base, add exponents to multiply and subtract to divide.
What do a zero exponent, a negative exponent, and a fractional exponent each mean?
x^0 = 1 (x not 0); x^(-n) = 1/x^n; x^(m/n) = the n-th root of x^m = (n-th root of x)^m.
How do you simplify a radical like sqrt(72)?
Factor out the largest perfect square: 72 = 36*2, so sqrt(72) = sqrt(36)*sqrt(2) = 6*sqrt(2).
How do you rationalize a denominator such as 1/sqrt(3)?
Multiply numerator and denominator by the radical: 1/sqrt(3) * sqrt(3)/sqrt(3) = sqrt(3)/3.
What is the imaginary unit i, and what is i^2?
i is defined as sqrt(-1), so i^2 = -1.
What is the cycle of powers of i?
i^1 = i, i^2 = -1, i^3 = -i, i^4 = 1, then it repeats every 4. To evaluate i^n, find n mod 4.
How do you multiply complex numbers, e.g., (2 + 3i)(1 - 4i)?
Use FOIL then replace i^2 with -1: 2 - 8i + 3i - 12i^2 = 2 - 5i + 12 = 14 - 5i.
How do you add two matrices, and what is required?
Add corresponding entries. The matrices must have the same dimensions (same number of rows and columns).
What is the condition for multiplying matrix A by matrix B, and what are the result's dimensions?
The number of columns of A must equal the number of rows of B. If A is m x n and B is n x p, the product is m x p.
How do you add vectors and multiply a vector by a scalar (in component form)?
Add vectors component-wise: <a,b> + <c,d> = <a+c, b+d>. Scalar multiply each component: k<a,b> = <ka, kb>.
How do you solve a linear equation like 3x + 7 = 22?
Isolate the variable using inverse operations: subtract 7 (3x = 15), then divide by 3 (x = 5).
What happens to an inequality's sign when you multiply or divide both sides by a negative number?
The inequality sign reverses direction (e.g., < becomes >).
What are the two main algebraic methods for solving a system of linear equations?
Substitution (solve one equation for a variable and plug into the other) and elimination (add/subtract equations to cancel a variable).
How can you tell whether a system of two linear equations has no solution or infinitely many?
No solution if the lines are parallel (same slope, different intercepts); infinitely many if the lines are identical (same slope and intercept). One solution if slopes differ.
What does the FOIL method expand, and what is (x + a)(x + b)?
FOIL multiplies two binomials (First, Outer, Inner, Last): (x + a)(x + b) = x^2 + (a+b)x + ab.
State the difference of squares and the perfect-square trinomial factoring patterns.
Difference of squares: a^2 - b^2 = (a + b)(a - b). Perfect square: a^2 + 2ab + b^2 = (a + b)^2.
What is the quadratic formula?
For ax^2 + bx + c = 0, x = (-b plus or minus sqrt(b^2 - 4ac)) / (2a).
What does the discriminant b^2 - 4ac tell you about a quadratic's roots?
Positive: two distinct real roots. Zero: one repeated real root. Negative: two complex (non-real) roots.
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What are the sum and product of the roots of ax^2 + bx + c = 0?
Sum of roots = -b/a; product of roots = c/a.
How do you translate 'a number increased by 5 is twice the number' into an equation?
Let the number be x: x + 5 = 2x, which gives x = 5.
What is the distance = rate x time relationship, and how do you find time?
d = r * t. Therefore time t = d/r and rate r = d/t.
In function notation, what does f(3) mean?
It means evaluate the function f by substituting 3 for the input variable x and computing the output.
If f(x) = 2x^2 - x + 1, what is f(-2)?
Substitute -2: 2(-2)^2 - (-2) + 1 = 2(4) + 2 + 1 = 8 + 2 + 1 = 11.
What are the slope-intercept and point-slope forms of a line?
Slope-intercept: y = mx + b (m = slope, b = y-intercept). Point-slope: y - y1 = m(x - x1).
For a quadratic in form y = ax^2 + bx + c, what is the x-coordinate of the vertex?
x = -b/(2a). Substitute back to find the y-coordinate of the vertex.
What does the leading coefficient's sign tell you about a parabola y = ax^2 + bx + c?
If a > 0 the parabola opens upward (vertex is a minimum); if a < 0 it opens downward (vertex is a maximum).
What causes a vertical asymptote and a hole in a rational function?
A vertical asymptote occurs where the denominator equals zero (and the factor does not cancel); a hole occurs where a factor cancels from both numerator and denominator.
What is the domain restriction for a square-root function sqrt(g(x))?
The radicand must be non-negative: g(x) >= 0. The domain is all x satisfying that inequality.
What does the exponential function y = a*b^x model, and what determines growth vs decay?
It models exponential growth/decay with initial value a. If b > 1 it grows; if 0 < b < 1 it decays.
How are logarithms and exponents related (definition of log)?
log_b(x) = y means b^y = x. A logarithm answers 'to what power must the base be raised to get x?'
State the three main logarithm rules (product, quotient, power).
log(mn) = log m + log n; log(m/n) = log m - log n; log(m^p) = p * log m.
How does y = f(x) + k versus y = f(x - h) transform a graph?
f(x) + k shifts the graph vertically (up if k > 0, down if k < 0); f(x - h) shifts horizontally (right if h > 0, left if h < 0).
What transformation does y = -f(x) versus y = f(-x) produce?
y = -f(x) reflects the graph across the x-axis; y = f(-x) reflects it across the y-axis.
What is the sum of the interior angles of a triangle, and of an n-sided polygon?
A triangle's interior angles sum to 180 degrees; an n-sided polygon's interior angles sum to (n - 2) * 180 degrees.
What are complementary and supplementary angles?
Complementary angles sum to 90 degrees; supplementary angles sum to 180 degrees.
What does it mean for two triangles to be similar, and what follows about their sides?
Similar triangles have equal corresponding angles; their corresponding sides are proportional (same ratio).
What are the circumference and area formulas for a circle of radius r?
Circumference C = 2*pi*r (or pi*d); Area A = pi*r^2.
How is an inscribed angle related to the central angle subtending the same arc?
An inscribed angle is half the measure of the central angle (and half the intercepted arc) that subtends the same arc.
What is the equation of a circle with center (h, k) and radius r?
(x - h)^2 + (y - k)^2 = r^2.
State the area formulas for a triangle, rectangle, and trapezoid.
Triangle: (1/2)*base*height. Rectangle: length*width. Trapezoid: (1/2)*(b1 + b2)*height.
What are the volume formulas for a rectangular box, a cylinder, and a sphere?
Box: l*w*h. Cylinder: pi*r^2*h. Sphere: (4/3)*pi*r^3.
What is the distance formula between points (x1, y1) and (x2, y2)?
d = sqrt((x2 - x1)^2 + (y2 - y1)^2).
What is the midpoint formula for the segment from (x1, y1) to (x2, y2)?
Midpoint = ((x1 + x2)/2, (y1 + y2)/2) — average the x-coordinates and the y-coordinates.
How are the slopes of parallel lines and perpendicular lines related?
Parallel lines have equal slopes; perpendicular lines have slopes that are negative reciprocals (their product is -1).
State the Pythagorean theorem and the side ratios of 30-60-90 and 45-45-90 triangles.
a^2 + b^2 = c^2 (c = hypotenuse). 45-45-90 sides: 1 : 1 : sqrt(2). 30-60-90 sides: 1 : sqrt(3) : 2.
How do you find the mean, median, and mode of a data set?
Mean = sum of values divided by count. Median = middle value when ordered (average of two middle values if even count). Mode = most frequently occurring value.
What is the range of a data set, and what does standard deviation measure?
Range = maximum value minus minimum value. Standard deviation measures the spread/dispersion of values about the mean (larger means more spread out).
What is the basic probability formula, and how do you find the probability of independent events both occurring?
P(event) = (favorable outcomes)/(total outcomes). For independent events A and B both occurring, multiply: P(A and B) = P(A)*P(B). The SOH-CAH-TOA ratios are sin = opp/hyp, cos = adj/hyp, tan = opp/adj.
ACT (American College Testing) Mathematics FAQ
What is in the ACT (American College Testing) Mathematics syllabus?
ACT (American College Testing) Mathematics is split into 5 chapters: Number and Quantity, Algebra, Functions, Geometry and Statistics and Probability. Together they contain 23 topics and 53 sub-topics.
How many chapters are in ACT (American College Testing) Mathematics?
5 chapters, containing 23 topics and 53 sub-topics. The largest chapter is Algebra, with 5 topics.
Are there flashcards for ACT (American College Testing) Mathematics?
Yes — a 50-card Mathematics deck, printed in full in the flashcards section of this page and available in the Examius app with spaced repetition. The free plan covers one exam of your choice, including its flashcards; Examius Pro, a one-time purchase, unlocks every other exam and deck.