Glossary

Distance from zero on the number line — always non-negative. Symbol |x|.

Two sides and a non-included angle — the one setup where the Law of Sines can yield zero, one, or two triangles.

Half the distance between a wave's highest and lowest points.

The angle you tilt your eyes up (elevation) or down (depression) from level to spot something.

The rate an angle changes over time, in radians per unit time.

The distance along a circle's curve, s = r\theta for \theta in radians.

Which way a complex number points from the origin — the angle in its polar form.

A list of numbers where each term is the last one plus a fixed common difference.

Grouping doesn't matter — (a+b)+c=a+(b+c).

A line a graph approaches but (usually) never crosses near a bad point or at infinity.

A system's coefficient matrix with an extra column of constants tacked on, separated by a bar.

Total change in output divided by total change in input over an interval.

A formula for reversing which event a conditional probability is conditioned on.

A compass direction written like N30°E — face north or south, then rotate toward east or west.

Symbol \iff — "if and only if"; the implication holds in both directions at once.

A polynomial with exactly two terms, like x+7.

The probability of exactly k successes across n independent, equally-likely trials.

The maximum sustainable value a logistic-growth quantity approaches.

The number multiplying a variable term, like the 3 in 3x^2.

A trig function paired with another via complementary angles, e.g. sine and cosine.

Three or more points that all lie on one straight line.

A selection of items where order doesn't matter — counted by C(n,r).

The fixed amount added to each term of an arithmetic sequence to get the next one.

Logarithm base 10, written \log x with no base shown.

The fixed factor each term of a geometric sequence is multiplied by to get the next one.

Order doesn't matter — a+b=b+a and ab=ba.

A fraction with a fraction living inside its own numerator or denominator.

The 2D picture of complex numbers — real part along the horizontal axis, imaginary part along the vertical.

A new function built by feeding one function's output into another — (f\circ g)(x)=f(g(x)).

Two inequality conditions combined with "and" or "or".

Interest earned on both your original money and the interest it already earned.

The same two-term expression with the middle sign flipped — a+b and a-b.

The perpendicular axis of a hyperbola that sets the height of its sketching box, without touching the curve.

Nonreal zeros of a real-coefficient polynomial always come in conjugate pairs a \pm bi.

The term with no variable at all — just a number.

The two numbers, written (x,y), that pin down one exact point on a graph.

A number from -1 to 1 measuring how tightly two variables move together in a straight-line pattern.

The reciprocal of sine — hypotenuse over opposite side.

The reciprocal of tangent — adjacent side over opposite side.

Angles that share the same terminal side despite different rotations.

A zero of an expression (and, for a rational expression, also a zero of its denominator) — the only places its sign can change.

The highest power of the variable appearing in a polynomial.

The bottom number in a fraction — how many equal pieces the whole is cut into.

The output variable — its value depends on the input.

Counts the possible positive and negative real zeros from a polynomial's sign changes.

A single number computed from a square matrix that reveals whether the system it represents has a unique solution.

The expression \frac{f(x+h)-f(x)}{h} — the slope between two nearby points on a curve.

One quantity is a constant multiple of another (direct), or a constant divided by it (inverse).

The fixed line a parabola's points stay exactly as far from as they are from the focus.

b^2-4ac — tells you how many real roots a quadratic has, before solving.

Two sets that share no elements — their intersection is empty.

A factor outside parentheses multiplies every term inside — a(b+c)=ab+ac.

The complete set of allowed inputs to a function.

Multiply matching components and add — turns two vectors into one number measuring how aligned they are.

The time for a growing quantity to double.

A single number, e, measuring how squashed or stretched a conic is.

Symbol \in — "is one of the members of" a set.

Add or subtract two equations, scaled so one variable's coefficients cancel out.

The set with no elements at all, written \varnothing.

A claim that two expressions are equal, joined by =.

A function whose graph mirrors across the y-axis — f(-x)=f(x).

Symbol \exists — "there exists"; at least one member of a set satisfies the statement.

The small raised number telling you how many times to multiply a number by itself.

A function of the form f(x) = a \cdot b^x, with the variable in the exponent.

A combination of numbers/variables/operations with no equals sign — a recipe, not a claim.

A candidate answer produced by valid algebra that fails to solve the original equation.

A piece multiplied with others to build an expression — or the act of finding those pieces.

x - c is a factor of f(x) exactly when f(c) = 0.

n! — multiply every whole number from n down to 1.

Splitting a four-term expression into pairs, each with its own factorable GCF.

A fixed reference point built into a conic's defining distance rule.

How many full cycles happen per unit of time — period's flip side.

A rule assigning exactly one output to each input.

A list of numbers where each term is the last one times a fixed common ratio.

The largest monomial that divides every term of an expression.

The time for a decaying quantity to drop to half its amount.

A two-branch curve where the difference of distances to two fixed points stays constant.

The longest side of a right triangle — the one opposite the right angle.

An equation true for EVERY value of the variable, not just some.

Adding 0 or multiplying by 1 changes nothing.

Symbol \implies — a one-way "if the left side is true, the right side must be true."

Two events where knowing one happened tells you nothing about the other's probability.

The input variable — the one you choose freely.

The small number in the crook of a radical sign telling you which root to take.

The counter variable in Σ notation, and the start/stop values that bound it.

The ray an angle starts from, before any rotation.

Whole numbers, including zero and negatives — symbol \mathbb{Z}.

If a continuous function changes sign between two points, it has a zero between them.

Symbol \cap — only what's shared by both sets.

Shorthand for a stretch of the number line, e.g. [2, 5).

The function that undoes another — swaps its inputs and outputs.

Every number has a partner that undoes it — a negative, or a reciprocal.

Real numbers that are NOT a ratio of two integers.

An irrational zero of the form a+\sqrt b in a rational-coefficient polynomial always comes paired with its conjugate a-\sqrt b.

A polynomial's highest-power coefficient.

The smallest expression every denominator in a problem divides into evenly.

Terms with identical variable parts — only their coefficients may differ.

The actual rate a point travels along its circular path, arc length per time.

An equation with more than one letter, solved for one specific variable.

The exponent needed on a base b to produce x, \log_b x = y \iff b^y = x.

Growth that starts exponential but levels off at a carrying capacity.

Dividing one polynomial by another the same way you divide numbers by hand.

How big a vector is — its length, with no direction attached.

The longer axis of an ellipse, running through both foci; length 2a.

Proving a statement true for every whole number by knocking over an infinite line of dominoes.

A rectangular table of numbers, used to organize and solve a system of equations.

The horizontal line a sinusoid oscillates around.

The shorter axis of an ellipse, perpendicular to the major axis; length 2b.

How far a complex number sits from the origin — its distance, written |z|.

A polynomial with exactly one term, like 5x^3.

How many times a zero repeats — controls whether a graph crosses or bounces there.

Two events that share no outcomes — they can never both happen at once.

Logarithm base e, written \ln x.

The counting numbers: 1, 2, 3, … — symbol \mathbb{N}.

The top number in a fraction — how many pieces you have.

A triangle with no 90° angle — SOH-CAH-TOA doesn't apply, but the Law of Sines and Law of Cosines do.

A function whose graph has pinwheel symmetry through the origin — f(-x)=-f(x).

A likelihood expressed as favorable-to-unfavorable, not as a fraction of the whole.

A function where distinct inputs always give distinct outputs.

The agreed sequence for evaluating an expression — parentheses, exponents, multiply/divide, add/subtract.

The U-shaped (or upside-down-U) curve every quadratic graphs into.

Lines that never meet — same steepness, forever the same distance apart.

A third variable, often t, that both x and y are written in terms of to trace a curve.

Splitting one complicated rational expression back into the simple fractions that were combined to make it.

The smallest positive interval after which a function's pattern repeats exactly.

An arrangement of items where order matters — counted by P(n,r).

Crossing at a perfect right (90°) angle.

The horizontal shift of a sinusoid, the C in A sin(B(x−C))+D.

A function defined by different formulas on different zones of its domain.

The leading entry in a row that row operations use as an anchor to clear the column around it.

The fixed reference ray a polar angle θ is measured from — plays the role of the positive x-axis.

The fixed center point polar coordinates are measured from — plays the role of the origin.

A sum of terms like a_n x^n + \dots + a_1 x + a_0, whole-number exponents only.

The nonnegative square root — the one the \sqrt{\ } symbol always means.

A complex number with no real part — just bi for a nonzero real b.

In a right triangle, the two short sides squared and added equal the long side squared.

An angle whose terminal side lies exactly on an axis, not inside any quadrant.

Involving a squared variable and nothing higher — degree 2.

The angle subtended by an arc equal in length to the circle's radius.

The expression sitting underneath a radical sign.

The complete set of outputs a function can actually produce.

Any number expressible as a ratio of two integers — symbol \mathbb{Q}.

Lists every possible rational zero of a polynomial, as a fraction of two factors.

Rewriting a fraction so no radical remains in the denominator (or numerator).

The full number line — symbol \mathbb{R} — rationals plus irrationals.

The number you multiply by to get 1 — flip the fraction.

Cosecant, secant, and cotangent — 1 divided by sine, cosine, and tangent.

A rule that defines each term of a sequence using the term(s) before it, plus a starting value.

The acute angle between an angle's terminal side and the x-axis.

Flipping a graph across a line, like a mirror image.

Dividing f(x) by x - c leaves a remainder of exactly f(c).

How far off a model's prediction was at one data point — actual minus predicted.

The single vector you get by adding two or more vectors together.

Tilting the x- and y-axes themselves so a slanted conic lines up straight.

A matrix shape where each row's leading entry steps right of the one above — the target of Gaussian elimination.

The complete set of everything that could possibly happen in a probability experiment.

A quantity described by a single number alone — no direction attached.

Writing a number as a \times 10^n with 1 \le |a| < 10.

The pie-slice region between two radii and their connecting arc.

What you get when you add up the terms of a sequence instead of just listing them.

A well-defined collection of distinct objects (its elements).

Describing a set by a rule instead of listing its members.

A table testing one point in each interval between critical values to track where an expression is positive or negative.

A square matrix with determinant zero — it has no inverse.

Any curve of the form y = A\sin(B(x-C)) + D, or the cosine equivalent.

Every number that makes an equation or inequality true.

An angle drawn with its vertex at the origin and initial side on the positive x-axis.

Symbol \subset — one set sits entirely inside another.

Solve one equation for a variable, then plug that expression into the other equation.

Symbol \Sigma — shorthand for adding up a pattern of terms.

A shortcut for long division by x - c, using only the coefficients.

The ray where an angle's rotation ends up.

The axis of a hyperbola running through both vertices and both foci.

Sine, cosine, tangent, and their reciprocals, defined from a right triangle's sides.

A polynomial with exactly three terms, like x^2-4x+3.

Symbol \cup — everything in either set, combined.

The circle of radius 1 centered at the origin, x^2+y^2=1.

A vector with magnitude exactly 1 — pure direction, no size.

Symbol \forall — "for all" / "for every"; a statement holds with no exceptions across a set.

Synthetic division checks that certify no real zero lies beyond a given number.

A symbol standing in for a number that is unknown or can vary.

The parabola's turning point — its lowest point (opens up) or highest point (opens down).

Where a graph crosses the x-axis — found by setting y=0.

Where a graph crosses the y-axis — found by setting x=0.

The ratio of a circle's circumference to its diameter, ≈ 3.14159.