Precalculus Definitions

Precalculus vocabulary flashcards for exam review.

Function

A relation where each input has exactly one output.

Domain

The set of all possible input values for a function.

Range

The set of all possible output values of a function.

Piecewise Function

A function defined by different expressions over different intervals.

Even Function

A function symmetric about the y-axis.

Odd Function

A function symmetric about the origin.

One-to-One Function

A function where each output is paired with only one input.

Inverse Function

A function that reverses the output of the original function.

Transformation

An operation that moves or changes a graph.

Translation

A shift of a graph horizontally or vertically.

Reflection

A flip of a graph over a line.

Stretch

A graph pulled away from its axis.

Compression

A graph pushed toward its axis.

End Behavior

Describes the values of a function as x approaches ±∞.

Asymptote

A line a graph approaches but never touches.

Intercept

The point where a graph crosses an axis.

Relative Maximum

The highest point in a local region of a graph.

Relative Minimum

The lowest point in a local region of a graph.

Zero

An input value that results in an output of zero.

Increasing Interval

Where a function is rising as x increases.

Decreasing Interval

Where a function is falling as x increases.

Continuity

The property of a function without breaks, jumps, or holes.

Polynomial

An expression consisting of variables and coefficients.

Monomial

A polynomial with a single term.

Binomial

A polynomial with two terms.

Trinomial

A polynomial with three terms.

Degree

The highest power of the variable in a polynomial.

Coefficient

A number multiplied by a variable.

Factoring

Rewriting an expression as a product of simpler factors.

Discriminant

The expression under the square root in the quadratic formula.

Quadratic Equation

An equation of the form ax² + bx + c = 0.

Vertex Form

A quadratic function written as f(x) = a(x - h)² + k.

Standard Form

A polynomial written with terms in descending powers.

Synthetic Division

A shortcut for dividing a polynomial by a linear factor.

Remainder Theorem

If f(c) = r, then (x - c) divides f(x) with remainder r.

Rational Expression

A fraction where the numerator and/or denominator are polynomials.

Extraneous Solution

A solution that doesn't satisfy the original equation.

Radical Equation

An equation that contains a variable inside a radical.

Absolute Value Equation

An equation involving the absolute value of a variable.

Inequality

A relation showing one value is larger or smaller than another.

Compound Inequality

Two inequalities joined by 'and' or 'or'.

Interval Notation

A notation for describing intervals on a number line.

System of Equations

A set of equations with multiple variables.

Substitution Method

Solving one equation and substituting into another.

Elimination Method

Adding or subtracting equations to eliminate a variable.

Sine

Ratio of opposite side to hypotenuse in a right triangle.

Cosine

Ratio of adjacent side to hypotenuse in a right triangle.

Tangent

Ratio of opposite side to adjacent side in a right triangle.

Cosecant

Reciprocal of sine.

Secant

Reciprocal of cosine.

Cotangent

Reciprocal of tangent.

Unit Circle

A circle of radius 1 centered at the origin.

Radian

A unit of angle based on arc length.

Degree

A unit of angle equal to 1/360 of a full rotation.

Amplitude

Height from the midline to the peak of a wave.

Period

The length of one complete cycle of a wave.

Phase Shift

A horizontal shift in a trigonometric graph.

Vertical Shift

A vertical movement of a trigonometric graph.

Trigonometric Identity

An equation true for all values of the variable.

Pythagorean Identity

sin²x + cos²x = 1.

Double Angle Identity

Formulas for trig functions of 2x.

Half Angle Identity

Formulas for trig functions of x/2.

Inverse Trig Function

Gives the angle whose trig value is known.

Law of Sines

sinA/a = sinB/b = sinC/c in a triangle.

Law of Cosines

c² = a² + b² - 2ab cos(C).

Angle of Elevation

Angle above horizontal to an object.

Angle of Depression

Angle below horizontal to an object.

Exponent

A number indicating how many times a base is multiplied by itself.

Exponential Growth

A situation where the rate of change increases over time.

Exponential Decay

A situation where the rate of change decreases over time.

Base

The number that is raised to a power.

Power

The result of a base raised to an exponent.

Logarithm

The inverse of an exponential function.

Common Logarithm

A logarithm with base 10.

Natural Logarithm

A logarithm with base e.

Logarithmic Equation

An equation involving a logarithm.

Product Rule

logb(xy) = logb(x) + log_b(y).

Quotient Rule

logb(x/y) = logb(x) - log_b(y).

Power Rule

logb(x^n) = n logb(x).

Change of Base Formula

logb(x) = logc(x) / log_c(b).

e (Euler's Number)

An irrational number approximately equal to 2.718.

Inverse Properties

logb(b^x) = x and b^(logb(x)) = x.

Asymptote (Log)

The y-axis is a vertical asymptote for logarithmic functions.

Domain (Log)

x > 0 for log_b(x).

Range (Log)

All real numbers for log_b(x).

Exponential Function

A function with the form f(x) = a * b^x, where a ≠ 0 and b > 0, b ≠ 1.

Continuous Growth

Growth modeled using the natural exponential function.

Compound Interest Formula

A = P(1 + r/n)^(nt).

Continuous Compound Interest

A = Pe^(rt).

Sequence

An ordered list of numbers following a pattern.

Term

Each number in a sequence.

Arithmetic Sequence

A sequence with a constant difference between terms.

Geometric Sequence

A sequence with a constant ratio between terms.

Recursive Formula

Defines each term based on the previous one.

Explicit Formula

Defines any term without referencing previous ones.

Series

The sum of terms of a sequence.

Partial Sum

The sum of the first n terms of a sequence.

Arithmetic Series

A series whose terms form an arithmetic sequence.

Geometric Series

A series whose terms form a geometric sequence.

Infinite Series

A sum of infinitely many terms.