ACT (American College Testing) Mathematics Syllabus

Mathematics syllabus — full chapter and topic list

Expand any chapter to see its topics and sub-topics. This is the whole examinable outline for Mathematics in ACT (American College Testing), not a summary of it.

1. 1

   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
2. 2

   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
3. 3

   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
4. 4

   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
5. 5

   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

Mathematics flashcards for ACT (American College Testing)

21 of 50 cards from the Mathematics deck — real questions with worked answers.

1. 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.

2. 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).

3. 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.

4. 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.

5. 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).

6. 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.

7. What is the imaginary unit i, and what is i^2?

   i is defined as sqrt(-1), so i^2 = -1.

8. 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.

9. 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.

10. 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).

11. 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.

12. 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>.

13. 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).

14. 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 >).

15. 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).

16. 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.

17. 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.

18. 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.

19. What is the quadratic formula?

    For ax^2 + bx + c = 0, x = (-b plus or minus sqrt(b^2 - 4ac)) / (2a).

20. 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.

21. 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.

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