Math Glossary

Amplitude

The height of a wave measured from the middle line to its peak. For $y = a \sin(x)$ or $y = a \cos(x)$, the amplitude is $|a|$.

$y = 3\sin(x)$ has amplitude $3$.

Arithmetic sequence

A list of numbers where you add the same amount each time to get the next term. That fixed amount is called the common difference.

$2, 5, 8, 11, \ldots$ is arithmetic with common difference $d = 3$.

Asymptote

A line that a graph gets closer and closer to but never actually touches. There are vertical, horizontal, and slant (oblique) asymptotes.

$f(x) = \dfrac{1}{x}$ has a vertical asymptote at $x = 0$ and a horizontal asymptote at $y = 0$.

Absolute value

The distance of a number from zero on the number line, always non-negative. Written $|x|$.

$|-7| = 7$ and $|3| = 3$.

Base (of an exponent)

The number being raised to a power. In $b^n$, the base is $b$.

In $2^5 = 32$, the base is $2$.

Base (of a logarithm)

The number at the bottom of a log. $\log_b(x)$ asks: "What power of $b$ gives me $x$?"

$\log_2(8) = 3$ because $2^3 = 8$.

Binomial theorem

A formula for expanding $(a + b)^n$ without multiplying it out step by step. It uses combinations ("choose" numbers) and powers of $a$ and $b$.

$(x + y)^3 = x^3 + 3x^2 y + 3xy^2 + y^3$.

Circle

The set of all points that are the same distance (the radius) from a center point. The standard equation is $(x - h)^2 + (y - k)^2 = r^2$.

$(x - 1)^2 + (y + 2)^2 = 9$ is a circle centered at $(1, -2)$ with radius $3$.

Combination

The number of ways to choose $r$ items from $n$ items when order doesn't matter. Written $\binom{n}{r}$ or $C(n, r)$.

$\binom{5}{2} = 10$ — there are $10$ ways to pick $2$ items from $5$.

Common difference

The constant amount you add to each term to get the next term in an arithmetic sequence. Usually called $d$.

In $4, 7, 10, 13, \ldots$ the common difference is $d = 3$.

Common log

A logarithm with base $10$. Written $\log(x)$ (no subscript). Your calculator's LOG button uses this.

$\log(1000) = 3$ because $10^3 = 1000$.

Common ratio

The constant factor you multiply by to get the next term in a geometric sequence. Usually called $r$.

In $3, 6, 12, 24, \ldots$ the common ratio is $r = 2$.

Completing the square

A technique for rewriting a quadratic expression $ax^2 + bx + c$ in the form $a(x - h)^2 + k$. This reveals the vertex of a parabola or helps solve equations.

$x^2 + 6x + 5 = (x + 3)^2 - 4$.

Composition (of functions)

Plugging one function into another. $(f \circ g)(x) = f(g(x))$ means: first apply $g$, then apply $f$ to the result.

If $f(x) = x^2$ and $g(x) = x + 1$, then $(f \circ g)(x) = (x+1)^2$.

Compound interest

Interest calculated on both the original amount and the accumulated interest. The formula is $A = P\left(1 + \dfrac{r}{n}\right)^{nt}$.

\1000$at$5\%$compounded monthly for$2$years:$A = 1000\left(1 + \frac{0.05}{12}\right)^{24}$.

Conic section

A curve you get by slicing a cone with a flat plane. The four types are circles, ellipses, parabolas, and hyperbolas.

Continuous compounding

Interest compounded infinitely often. The formula simplifies to $A = Pe^{rt}$, where $e \approx 2.718$.

\1000$at$5\%$continuously for$2$years:$A = 1000e^{0.05 \cdot 2} \approx \$1105.17$.

Cosecant

The reciprocal of sine: $\csc(\theta) = \dfrac{1}{\sin(\theta)}$. Undefined wherever $\sin(\theta) = 0$.

Cosine

In a right triangle, cosine is adjacent over hypotenuse. On the unit circle, $\cos(\theta)$ is the $x$-coordinate of the point at angle $\theta$.

$\cos(60°) = \cos\left(\frac{\pi}{3}\right) = \frac{1}{2}$.

Cotangent

The reciprocal of tangent: $\cot(\theta) = \dfrac{\cos(\theta)}{\sin(\theta)}$. Undefined wherever $\sin(\theta) = 0$.

Change of base formula

A way to compute any logarithm using a different base: $\log_b(x) = \dfrac{\ln(x)}{\ln(b)} = \dfrac{\log(x)}{\log(b)}$. Handy when your calculator only has LN and LOG.

$\log_5(20) = \frac{\ln 20}{\ln 5} \approx 1.861$.

Conjugate

The expression you get by flipping the sign between two terms. The conjugate of $a + b$ is $a - b$. Useful for rationalizing denominators.

The conjugate of $3 + \sqrt{2}$ is $3 - \sqrt{2}$.

Degree (of a polynomial)

The highest exponent on the variable in a polynomial. It tells you the most the graph can curve and the maximum number of roots.

$3x^4 - x^2 + 7$ has degree $4$.

Degree (angle)

A unit for measuring angles. A full rotation is $360°$. To convert degrees to radians, multiply by $\dfrac{\pi}{180}$.

$90° = \frac{\pi}{2}$ radians.

Discriminant

The expression $b^2 - 4ac$ under the square root in the quadratic formula. It tells you how many real solutions a quadratic equation has: positive → 2, zero → 1, negative → 0.

For $x^2 + 4x + 5 = 0$: discriminant $= 16 - 20 = -4 < 0$, so there are no real roots.

Domain

All the input values ($x$-values) you're allowed to plug into a function. Watch out for division by zero and square roots of negatives.

The domain of $f(x) = \sqrt{x - 3}$ is $[3, \infty)$.

Decay (exponential)

A quantity that shrinks by a fixed percentage over equal time intervals, modeled by $f(t) = a \cdot b^t$ where $0 < b < 1$.

A substance with half-life of $3$ hours: $f(t) = 100 \cdot \left(\frac{1}{2}\right)^{t/3}$.

Elimination (method)

A way to solve a system of equations by adding or subtracting the equations to cancel out one variable.

Adding $x + y = 5$ and $x - y = 1$ gives $2x = 6$, so $x = 3$.

Ellipse

An oval-shaped conic section. The standard form is $\dfrac{(x-h)^2}{a^2} + \dfrac{(y-k)^2}{b^2} = 1$. It has two foci inside it.

$\dfrac{x^2}{9} + \dfrac{y^2}{4} = 1$ is an ellipse centered at the origin.

Exponent

The small number written above and to the right of a base that tells you how many times to multiply the base by itself. Also called a power.

$2^5 = 2 \times 2 \times 2 \times 2 \times 2 = 32$.

End behavior

What happens to $f(x)$ as $x$ goes toward $+\infty$ or $-\infty$. Determined by the degree and leading coefficient of a polynomial.

For $f(x) = -2x^3 + \ldots$, as $x \to \infty$, $f(x) \to -\infty$.

Even function

A function where $f(-x) = f(x)$ for every $x$ in its domain. Its graph is symmetric about the $y$-axis.

$f(x) = x^2$ is even because $(-x)^2 = x^2$.

Factor

A number or expression that divides evenly into another. Factoring a polynomial means writing it as a product of simpler pieces.

$x^2 - 9 = (x - 3)(x + 3)$.

Factor theorem

If $f(c) = 0$, then $(x - c)$ is a factor of $f(x)$. And the reverse: if $(x - c)$ is a factor, then $f(c) = 0$.

Since $f(2) = 0$ for $f(x) = x^2 - 4$, we know $(x - 2)$ is a factor.

Factorial

The product of all positive integers up to $n$. Written $n!$. By convention, $0! = 1$.

$5! = 5 \times 4 \times 3 \times 2 \times 1 = 120$.

Focus/Foci

A special point (or pair of points) inside a conic section that helps define its shape. Ellipses and hyperbolas have two foci; parabolas have one.

Function

A rule that takes each input and gives exactly one output. If you plug in a value of $x$, you get back one and only one $y$.

$f(x) = 2x + 1$ is a function. For $x = 3$, the output is $f(3) = 7$.

Geometric sequence

A list of numbers where you multiply by the same amount each time to get the next term. That fixed multiplier is the common ratio.

$2, 6, 18, 54, \ldots$ is geometric with common ratio $r = 3$.

Growth (exponential)

A quantity that increases by a fixed percentage over equal time intervals, modeled by $f(t) = a \cdot b^t$ where $b > 1$.

A population doubling every $5$ years: $f(t) = 1000 \cdot 2^{t/5}$.

Hyperbola

A conic section with two separate branches that open away from each other. Standard form: $\dfrac{(x-h)^2}{a^2} - \dfrac{(y-k)^2}{b^2} = 1$.

$\dfrac{x^2}{4} - \dfrac{y^2}{9} = 1$ is a hyperbola opening left and right.

Horizontal line test

A visual check for whether a function has an inverse. If every horizontal line crosses the graph at most once, the function is one-to-one and has an inverse.

Identity (trigonometric)

An equation involving trig functions that is true for all valid inputs, not just specific angles. Used to simplify or prove expressions.

$\sin^2(\theta) + \cos^2(\theta) = 1$ is always true.

Inequality

A mathematical statement that compares two expressions using $<$, $>$, $\le$, or $\ge$ instead of $=$.

$2x + 1 > 5$ means $x > 2$.

Interval notation

A shorthand for describing a range of numbers. Use parentheses $(\,)$ when the endpoint is not included and brackets $[\,]$ when it is included.

$[2, 5)$ means all numbers from $2$ to $5$, including $2$ but not $5$.

Inverse function

A function that "undoes" another function. If $f(a) = b$, then $f^{-1}(b) = a$. You find it by swapping $x$ and $y$ and solving for $y$.

If $f(x) = 2x + 3$, then $f^{-1}(x) = \frac{x - 3}{2}$.

Law of cosines

A formula that relates the sides and angles of any triangle: $c^2 = a^2 + b^2 - 2ab\cos(C)$. It's a generalization of the Pythagorean theorem.

If $a = 5$, $b = 7$, $C = 60°$: $c^2 = 25 + 49 - 70\cos(60°) = 39$.

Law of sines

A formula for any triangle: $\dfrac{a}{\sin A} = \dfrac{b}{\sin B} = \dfrac{c}{\sin C}$. Useful when you know an angle and its opposite side.

If $A = 30°$, $a = 10$, $B = 45°$, then $\frac{10}{\sin 30°} = \frac{b}{\sin 45°}$.

Leading coefficient

The number in front of the highest-power term of a polynomial. It controls whether the ends of the graph go up or down.

In $-3x^4 + x^2 - 1$, the leading coefficient is $-3$.

Logarithm

The inverse of an exponent. $\log_b(x) = y$ means $b^y = x$. In plain terms: "What power of $b$ gives me $x$?"

$\log_3(81) = 4$ because $3^4 = 81$.

Midline

The horizontal line that a sinusoidal graph oscillates around. For $y = a\sin(bx) + d$, the midline is $y = d$.

$y = 3\sin(x) + 2$ has midline $y = 2$.

Natural log

A logarithm with base $e \approx 2.718$. Written $\ln(x)$. It's the most common log in calculus and shows up in continuous growth/decay.

$\ln(e^3) = 3$.

Odd function

A function where $f(-x) = -f(x)$ for every $x$ in its domain. Its graph has rotational symmetry about the origin.

$f(x) = x^3$ is odd because $(-x)^3 = -x^3$.

One-to-one function

A function that never gives the same output for two different inputs. Only one-to-one functions have inverses.

Parabola

The U-shaped (or upside-down U) graph of a quadratic function. It has a vertex (the highest or lowest point) and an axis of symmetry.

$y = x^2 - 4x + 3$ is a parabola opening upward with vertex at $(2, -1)$.

Period

The horizontal length it takes for a trig function to complete one full cycle. For $y = \sin(bx)$ or $y = \cos(bx)$, the period is $\dfrac{2\pi}{|b|}$.

$y = \sin(2x)$ has period $\frac{2\pi}{2} = \pi$.

Phase shift

A horizontal slide of a trig graph. For $y = \sin(b(x - c))$, the phase shift is $c$ units to the right.

$y = \cos\left(x - \frac{\pi}{4}\right)$ is shifted $\frac{\pi}{4}$ to the right.

Polynomial

An expression made of terms with whole-number exponents added together, like $a_n x^n + \cdots + a_1 x + a_0$. No fractions or negative exponents on the variable.

$4x^3 - 2x + 7$ is a polynomial of degree $3$.

Pythagorean identity

The most important trig identity: $\sin^2(\theta) + \cos^2(\theta) = 1$. Two variations follow from dividing by $\cos^2$ or $\sin^2$.

If $\sin(\theta) = \frac{3}{5}$, then $\cos^2(\theta) = 1 - \frac{9}{25} = \frac{16}{25}$.

Piecewise function

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

$f(x) = \begin{cases} x^2 & x < 0 \\ 2x + 1 & x \ge 0 \end{cases}$

Quadratic formula

The formula $x = \dfrac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ that solves any quadratic equation $ax^2 + bx + c = 0$.

For $x^2 - 5x + 6 = 0$: $x = \frac{5 \pm \sqrt{25 - 24}}{2} = \frac{5 \pm 1}{2}$, giving $x = 3$ or $x = 2$.

Radian

A unit for measuring angles based on the radius of a circle. One full rotation is $2\pi$ radians. To convert radians to degrees, multiply by $\dfrac{180}{\pi}$.

$\frac{\pi}{6}$ radians $= 30°$.

Range

All the output values ($y$-values) a function can produce. It's what comes out after plugging in every allowed $x$.

The range of $f(x) = x^2$ is $[0, \infty)$.

Rational expression

A fraction where the numerator and/or denominator are polynomials. Watch for values of $x$ that make the denominator zero — those are excluded from the domain.

$\dfrac{x + 1}{x^2 - 4}$ is undefined at $x = 2$ and $x = -2$.

Reference angle

The acute angle between the terminal side of an angle and the $x$-axis. It helps you find trig values in any quadrant.

The reference angle for $150°$ is $180° - 150° = 30°$.

Remainder theorem

When you divide a polynomial $f(x)$ by $(x - c)$, the remainder is $f(c)$. A quick way to evaluate polynomials.

For $f(x) = x^3 - 2x + 1$, the remainder when divided by $(x - 3)$ is $f(3) = 22$.

Secant (trig)

The reciprocal of cosine: $\sec(\theta) = \dfrac{1}{\cos(\theta)}$. Undefined wherever $\cos(\theta) = 0$.

Sequence

An ordered list of numbers following a pattern. Each number is called a term, and sequences can be finite or infinite.

$1, 4, 9, 16, 25, \ldots$ (the perfect squares).

Series

The sum of the terms in a sequence. A finite series adds up a fixed number of terms; an infinite series keeps going forever.

$1 + 2 + 3 + \cdots + 100 = 5050$.

Sine

In a right triangle, sine is opposite over hypotenuse. On the unit circle, $\sin(\theta)$ is the $y$-coordinate of the point at angle $\theta$.

$\sin(30°) = \sin\left(\frac{\pi}{6}\right) = \frac{1}{2}$.

Substitution (method)

A way to solve a system of equations by solving one equation for a variable and plugging that expression into the other equation.

From $y = 2x$ and $x + y = 9$: substitute to get $x + 2x = 9$, so $x = 3$.

Summation notation

A compact way to write a sum using the Greek letter sigma: $\displaystyle\sum_{i=1}^{n} a_i = a_1 + a_2 + \cdots + a_n$.

$\displaystyle\sum_{i=1}^{4} i^2 = 1 + 4 + 9 + 16 = 30$.

System of equations

Two or more equations that share the same variables. The solution is the set of values that makes all the equations true at the same time.

The system $x + y = 5$, $x - y = 1$ has solution $(3, 2)$.

Tangent (trig)

The ratio of sine to cosine: $\tan(\theta) = \dfrac{\sin(\theta)}{\cos(\theta)}$. Undefined wherever $\cos(\theta) = 0$.

$\tan(45°) = \frac{\sin 45°}{\cos 45°} = 1$.

Transformation

A change applied to a function's graph — shifting (translating), stretching, compressing, or reflecting it.

$f(x - 2) + 3$ shifts the graph of $f$ right $2$ and up $3$.

Unit circle

A circle with radius $1$ centered at the origin. It's the key tool for defining sine, cosine, and tangent for any angle.

At $\theta = \frac{\pi}{4}$, the point on the unit circle is $\left(\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2}\right)$.

Vertex

The highest or lowest point of a parabola, or the "corner" point of a conic section. For $y = a(x - h)^2 + k$, the vertex is $(h, k)$.

$y = 2(x - 3)^2 + 1$ has vertex $(3, 1)$.

Vertical line test

A visual check for whether a graph represents a function. If every vertical line crosses the graph at most once, it's a function.

Zero/Root

A value of $x$ that makes a function equal zero: $f(x) = 0$. Graphically, it's where the curve crosses or touches the $x$-axis.

The zeros of $f(x) = x^2 - 4$ are $x = 2$ and $x = -2$.