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High School Diploma Mathematics Syllabus
The Mathematics syllabus for High School Diploma in full: 5 chapters, 24 topics and 54 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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Algebra I
5 topics- Expressions, Equations, and Inequalities
- Simplifying and evaluating expressions
- Solving linear equations and inequalities
- Literal equations and formulas
- Linear Functions
- Slope and rate of change
- Slope-intercept and point-slope forms
- Graphing and interpreting lines
- Systems of Equations
- Solving by graphing, substitution, and elimination
- Systems of inequalities
- Exponents and Polynomials
- Laws of exponents
- Adding, subtracting, and multiplying polynomials
- Factoring techniques
- Quadratic Functions
- Graphing parabolas
- Solving by factoring and the quadratic formula
- Expressions, Equations, and Inequalities
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Geometry
6 topics- Foundations and Logic
- Points, lines, planes, and angles
- Deductive reasoning and proofs
- Triangles and Congruence
- Triangle properties and classification
- Congruence postulates and theorems
- Similarity and Right Triangles
- Similar polygons and ratios
- Pythagorean theorem
- Right triangle trigonometry
- Polygons and Circles
- Properties of quadrilaterals
- Arcs, chords, and tangents
- Area, Surface Area, and Volume
- Two-dimensional area formulas
- Solids: surface area and volume
- Transformations and Coordinate Geometry
- Translations, reflections, rotations, dilations
- Distance and midpoint formulas
- Foundations and Logic
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Algebra II
5 topics- Functions and Their Graphs
- Function notation and transformations
- Domain, range, and inverse functions
- Polynomial and Rational Functions
- Higher-degree polynomials and roots
- Rational expressions and equations
- Exponential and Logarithmic Functions
- Exponential growth and decay
- Properties of logarithms
- Solving exponential and log equations
- Complex Numbers and Quadratics
- Imaginary and complex numbers
- Completing the square and the discriminant
- Sequences and Series
- Arithmetic and geometric sequences
- Summation notation
- Functions and Their Graphs
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Statistics and Probability
4 topics- Descriptive Statistics
- Measures of center and spread
- Data displays and distributions
- Probability
- Sample spaces and events
- Conditional probability and independence
- Permutations and combinations
- Data Collection and Inference
- Sampling methods and bias
- Experiments and observational studies
- Bivariate Data
- Scatterplots and correlation
- Linear regression and prediction
- Descriptive Statistics
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Precalculus and Trigonometry
4 topics- Trigonometric Functions
- Unit circle and radian measure
- Graphs of sine, cosine, and tangent
- Trigonometric Identities and Equations
- Pythagorean and angle identities
- Solving trigonometric equations
- Analytic Geometry
- Conic sections
- Polar coordinates and vectors
- Introduction to Limits
- Concept of a limit
- Continuity and end behavior
- Trigonometric Functions
High School Diploma Mathematics flashcards — 50 questions with answers
What is the distributive property, and give an example expanding 3(x + 4)?
a(b + c) = ab + ac. So 3(x + 4) = 3x + 12.
What does it mean to solve an inequality, and what operation reverses the inequality sign?
To find all values that make it true. Multiplying or dividing both sides by a negative number reverses the inequality sign.
What is the slope-intercept form of a linear function, and what does each symbol represent?
y = mx + b, where m is the slope (rate of change) and b is the y-intercept (value of y when x = 0).
What is the slope formula given two points (x1, y1) and (x2, y2)?
m = (y2 − y1) / (x2 − x1).
How do the slopes of parallel lines and perpendicular lines compare?
Parallel lines have equal slopes; perpendicular lines have slopes that are negative reciprocals (their product is −1).
What are the three methods for solving a system of linear equations, and what does the number of solutions indicate geometrically?
Graphing, substitution, and elimination. One solution = lines intersect; no solution = parallel lines; infinitely many = same line.
State the product rule and the quotient rule for exponents.
Product: a^m · a^n = a^(m+n). Quotient: a^m / a^n = a^(m−n).
What do a negative exponent and a zero exponent equal?
a^(−n) = 1/a^n (for a ≠ 0), and a^0 = 1 (for a ≠ 0).
What is the degree of a polynomial, and what is a binomial?
The degree is the highest exponent on the variable. A binomial is a polynomial with exactly two terms.
State the quadratic formula for ax² + bx + c = 0.
x = [−b ± √(b² − 4ac)] / (2a).
What does the discriminant b² − 4ac tell you about a quadratic's roots?
Positive = two real roots; zero = one repeated real root; negative = two complex (non-real) roots.
What is the vertex form of a quadratic, and what is the vertex?
y = a(x − h)² + k, where the vertex is (h, k) and x = h is the axis of symmetry.
In logic, what is the converse, inverse, and contrapositive of 'If p then q'?
Converse: if q then p. Inverse: if not p then not q. Contrapositive: if not q then not p (logically equivalent to the original).
What is the difference between an axiom (postulate) and a theorem?
An axiom/postulate is accepted as true without proof; a theorem is a statement proven true using axioms, definitions, and logic.
State the Triangle Sum Theorem and the Exterior Angle Theorem.
The interior angles of a triangle sum to 180°. An exterior angle equals the sum of the two non-adjacent interior angles.
Name the triangle congruence shortcuts (postulates/theorems).
SSS, SAS, ASA, AAS, and HL (Hypotenuse-Leg, for right triangles). Note: SSA and AAA do not prove congruence.
What is the difference between congruent and similar figures?
Congruent figures have the same shape and size (corresponding parts equal). Similar figures have the same shape but proportional sides and equal corresponding angles.
State the Pythagorean Theorem and what type of triangle it applies to.
a² + b² = c², where c is the hypotenuse, applying only to right triangles.
Give the side ratios for 45-45-90 and 30-60-90 special right triangles.
45-45-90: legs in ratio 1 : 1 : √2 (hypotenuse). 30-60-90: sides in ratio 1 : √3 : 2 (opposite 30°, 60°, 90°).
What is the sum of the interior angles of a convex polygon with n sides?
(n − 2) × 180°.
Show the other 30 Mathematics flashcards
State the relationship between an inscribed angle and its intercepted arc in a circle.
An inscribed angle equals half the measure of its intercepted arc (central angle).
Give the formulas for the circumference and area of a circle of radius r.
Circumference C = 2πr; Area A = πr².
Give the area formulas for a triangle, a trapezoid, and a parallelogram.
Triangle: ½bh. Trapezoid: ½(b1 + b2)h. Parallelogram: bh.
Give the volume formulas for a cylinder, a cone, and a sphere.
Cylinder: πr²h. Cone: ⅓πr²h. Sphere: (4/3)πr³.
What are the four basic rigid transformations (isometries), and which one changes orientation by flipping?
Translation, rotation, reflection, and glide reflection. A reflection flips orientation across a line.
State the distance formula and the midpoint formula for points (x1, y1) and (x2, y2).
Distance: √[(x2−x1)² + (y2−y1)²]. Midpoint: ((x1+x2)/2, (y1+y2)/2).
What is a function, and what does the vertical line test determine?
A relation where each input (x) has exactly one output (y). The vertical line test confirms a graph is a function if no vertical line crosses it more than once.
What are the domain and range of a function?
Domain is the set of all valid inputs (x-values); range is the set of all resulting outputs (y-values).
What is a vertical asymptote of a rational function, and where does it occur?
A vertical line the graph approaches but never touches; it occurs where the denominator equals zero (and the numerator does not).
State the Remainder Theorem and the Factor Theorem.
Remainder Theorem: dividing a polynomial f(x) by (x − a) gives remainder f(a). Factor Theorem: (x − a) is a factor of f(x) if and only if f(a) = 0.
What is the general form of an exponential function, and what distinguishes growth from decay?
f(x) = a·b^x (a ≠ 0, b > 0, b ≠ 1). Growth when b > 1; decay when 0 < b < 1.
How are logarithms and exponents related (definition of log)?
log_b(x) = y means b^y = x. A logarithm is the inverse of exponentiation.
State the three main logarithm laws (product, quotient, power).
log(xy) = log x + log y; log(x/y) = log x − log y; log(x^n) = n·log x.
Define the imaginary unit i and state i².
i = √(−1), so i² = −1.
How do you add complex numbers, and what is the complex conjugate of a + bi?
Add real and imaginary parts separately: (a+bi)+(c+di) = (a+c)+(b+d)i. The conjugate of a + bi is a − bi.
What is the explicit formula for the nth term of an arithmetic sequence?
a_n = a_1 + (n − 1)d, where d is the common difference.
What is the explicit formula for the nth term of a geometric sequence?
a_n = a_1 · r^(n−1), where r is the common ratio.
What is the sum of a finite arithmetic series of n terms?
S_n = n(a_1 + a_n)/2, or equivalently n/2 · [2a_1 + (n−1)d].
What is the sum of an infinite geometric series, and when does it converge?
S = a_1 / (1 − r), valid only when |r| < 1.
Define mean, median, and mode.
Mean: the arithmetic average. Median: the middle value when data is ordered. Mode: the most frequently occurring value.
What does standard deviation measure, and how does it relate to variance?
It measures the spread of data around the mean. Standard deviation is the square root of the variance.
State the probability formula for the union of two events, P(A or B).
P(A or B) = P(A) + P(B) − P(A and B). For mutually exclusive events, P(A and B) = 0.
What is the formula for conditional probability P(A | B)?
P(A | B) = P(A and B) / P(B), for P(B) ≠ 0.
What is the difference between a population and a sample, and what is random sampling?
A population is the entire group of interest; a sample is a subset studied. Random sampling gives every member an equal chance of selection to reduce bias.
In experimental design, what is the difference between an observational study and a controlled experiment?
An observational study records data without intervention; a controlled experiment imposes a treatment to establish cause-and-effect, using control and treatment groups.
What is the correlation coefficient r, and what range of values can it take?
A measure of the strength and direction of a linear relationship between two variables, ranging from −1 (perfect negative) to +1 (perfect positive); 0 means no linear correlation.
What does the slope of a least-squares regression line represent in bivariate data?
The predicted change in the response variable (y) for each one-unit increase in the explanatory variable (x).
Define the six trigonometric ratios using SOH-CAH-TOA and reciprocals.
sin = opp/hyp, cos = adj/hyp, tan = opp/adj; csc = 1/sin, sec = 1/cos, cot = 1/tan.
State the Pythagorean trig identity and the definition of one radian.
sin²θ + cos²θ = 1. One radian is the angle subtending an arc equal in length to the radius; 180° = π radians.
What is the intuitive meaning of the limit of f(x) as x approaches a?
The value f(x) gets arbitrarily close to as x approaches a (from both sides), regardless of whether f(a) itself is defined.
High School Diploma Mathematics FAQ
What is in the High School Diploma Mathematics syllabus?
High School Diploma Mathematics is split into 5 chapters: Algebra I, Geometry, Algebra II, Statistics and Probability and Precalculus and Trigonometry. Together they contain 24 topics and 54 sub-topics.
How many chapters are in High School Diploma Mathematics?
5 chapters, containing 24 topics and 54 sub-topics. The largest chapter is Geometry, with 6 topics.
Are there flashcards for High School Diploma 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.