MB Pre-Calculus 40S key terms

A change made to the graph of a base function, such as shifting it, stretching or compressing it, or reflecting it, that produces a new graph with a related equation.

A rigid shift of a graph horizontally, vertically, or both, without changing its shape or orientation, produced by adding to or subtracting from the input or output.

A flip of a graph across a line such as the x-axis or y-axis, produced by negating either the output or the input of a function.

A stretch or compression of a graph away from or toward an axis, produced by multiplying the input or output by a factor other than one.

A compact way of writing how each point of a base graph moves to a new point under a transformation, for example (x, y) becoming (x plus h, ay plus k).

The set of ordered pairs obtained by swapping the coordinates of every point in a relation, which undoes the original correspondence between the variables.

A function that reverses another function, sending each output back to the input that produced it; it exists only when the original function is one-to-one.

A check in which a horizontal line is drawn across a graph; if it meets the graph more than once, the function is not one-to-one and has no inverse function.

A function whose rule contains a variable inside a root sign, such as a square root function, typically defined only where the radicand is non-negative.

The number or expression written beneath a radical sign, whose value determines whether the root is real and defined.

A value that appears when both sides of an equation are raised to a power but that does not actually satisfy the original equation, so it must be rejected.

A function written as one polynomial divided by another, whose graph may include vertical asymptotes, horizontal asymptotes, and holes.

A line that a graph approaches more and more closely as the input or output grows without bound, though the graph may never reach it.

A single input value at which a rational function is undefined because a common factor cancels, shown on the graph as a hole rather than an asymptote.

A limit placed on the input values a function may accept, often needed to keep a radicand non-negative, a denominator nonzero, or an inverse function valid.

A description of what happens to the outputs of a function as the inputs become very large positive or very large negative.

A unit for measuring angles based on arc length, where one radian is the angle subtended at the centre of a circle by an arc equal in length to the radius.

The circle of radius one centred at the origin, on which an angle in standard position has coordinates equal to the cosine and sine of that angle.

The acute angle formed between the terminal arm of an angle and the nearest horizontal axis, used to find exact trigonometric values in any quadrant.

The rotating ray of an angle in standard position, whose intersection with the unit circle gives the cosine and sine of the angle.

A function that repeats its outputs at regular intervals, so that adding a fixed constant to the input always produces the same output.

The smallest positive input interval over which a periodic function completes one full cycle and begins repeating.

A function whose graph has the shape of a sine wave, written with an amplitude, a period, a phase shift, and a vertical displacement.

Half the vertical distance between the maximum and minimum values of a sinusoidal function, measuring how far the graph rises above its middle line.

An equation involving trigonometric ratios that is true for every value of the variable for which both sides are defined, useful for simplifying expressions and solving equations.

A function in which the variable appears in the exponent, of the form base raised to a variable power, growing or decaying by a constant factor.

The exponent to which a fixed base must be raised to produce a given number, defined only for positive arguments and written as log base b of x.

A logarithm with base ten, often written without a base symbol, useful for calculations involving powers of ten.

A logarithm with base e, the irrational constant approximately equal to 2.718, written as ln and used to solve continuous growth problems.

A rule that rewrites a logarithm in one base as a quotient of logarithms in another base, allowing evaluation with a calculator.

An ordered arrangement of some or all of the items in a set, where changing the order produces a different arrangement.

A selection of items from a set in which order does not matter, so different orderings of the same items count as one selection.

The product of all positive integers from one up to a given whole number, written with an exclamation mark, with zero factorial defined as one.

A formula that expands a binomial raised to a whole-number power into a sum of terms whose coefficients are combinations, without repeated multiplication.

The number of ways to choose a subset of a given size from a larger set, appearing as a coefficient in the expansion of a binomial power.

A triangular array of numbers in which each entry is the sum of the two above it, whose rows give the coefficients of successive binomial expansions.

A curve obtained by slicing a double cone with a plane, giving a circle, ellipse, parabola, or hyperbola depending on how the plane cuts.

A fixed point used in the geometric definition of a conic section, such as one of two points whose distances to any point on an ellipse sum to a constant.

A fixed line used together with a focus to define a conic section, since points on the curve keep a constant distance ratio to the focus and the line.

A number describing how far a conic section departs from being circular, equal to the ratio of the distance to the focus over the distance to the directrix.

A sequence in which each term after the first is obtained by adding a constant difference to the previous term.

A sequence in which each term after the first is obtained by multiplying the previous term by a constant ratio.

The sum of the terms of an arithmetic sequence, which can be computed from the first term, the common difference, and the number of terms.

The sum of the terms of a geometric sequence, which has a finite formula and, when the ratio is small enough, a limiting infinite sum.

The constant amount added to each term of an arithmetic sequence to produce the next term, found by subtracting any term from the one after it.

The constant factor by which each term of a geometric sequence is multiplied to produce the next term, found by dividing any term by the one before it.