Precalculus
37 key words. Find a meaning, clear up a mix-up and return to the lesson.
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12 words · Choose a word to read its meaning.
A different-looking expression with the same value for every allowed input.
Exactly: An expression that agrees with another throughout their stated common domain.
Mix-up alert: Matching for just one input does not prove equivalence.
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Equivalent expression. 2 words link to it. It links to 1 word.
12 words shown.
A mathematical phrase made from numbers, variables and operations.
Exactly: A combination of numbers, variables and operations representing a value when defined.
Mix-up alert: An expression is not an equation; it has no equals sign.
A different-looking expression with the same value for every allowed input.
Exactly: An expression that agrees with another throughout their stated common domain.
Mix-up alert: Matching for just one input does not prove equivalence.
A number that tells you about a power.
Exactly: For a positive integer , is the product of factors of . Zero, negative and rational exponents extend this definition with restrictions.
Mix-up alert: , not .
A rule that gives one output for each allowed input.
Exactly: A function assigns exactly one value in its codomain to each element of its domain.
Mix-up alert: Different inputs can have the same output.
Use one function’s output as the next function’s input.
Exactly: The composite function satisfies . It is defined when is in the domain of and is in the domain of .
Mix-up alert: Composition is not multiplication, and order usually matters.
All the inputs you are allowed to use.
Exactly: The set of input values for which a function is defined.
Mix-up alert: A denominator cannot be zero; a real square root needs a nonnegative input.
How much the output changes per unit of input over an interval.
Exactly: For , the average rate of change of over is . Its units are output units per input unit.
Mix-up alert: Divide the output change by the input change, in the same order.
A rule that reverses a one-to-one function.
Exactly: The inverse of a one-to-one function reverses its input-output correspondence: on the domain of , and on its range.
Mix-up alert: The inverse function is not the reciprocal .
A graph can match itself across the vertical axis or after a half-turn about the origin.
Exactly: On a domain symmetric about zero, an even function satisfies ; an odd function satisfies .
Mix-up alert: A function can be neither. The zero function is both even and odd.
Vertical change divided by horizontal change.
Exactly: For a nonvertical line through distinct points and , its slope is .
Mix-up alert: A vertical line has undefined slope; a horizontal line has slope zero.
A function that changes by the same amount for each equal input step.
Exactly: A function of the form , with real constants and .
Mix-up alert: A straight vertical line is not a function of x.
A function that uses different rules for different inputs.
Exactly: A function defined by different expressions on specified parts of its domain, assigning one output to each allowed input.
Mix-up alert: Choose the rule by the input, not by the output.
A fractional power that combines a root with a power.
Exactly: For and integers with , . Negative bases require additional real-domain restrictions.
Mix-up alert: The denominator tells you the root; the numerator tells you the power.
A rule that multiplies by the same positive factor for equal input steps.
Exactly: A function , with , and .
Mix-up alert: A 20 percent increase uses a factor of 1.20, not 0.20.
Rewrite both sides as powers of the same base.
Exactly: If and , then implies .
Mix-up alert: The method needs equal bases and an injective exponential function.
One of the quantities multiplied to make a product.
Exactly: A number or expression that multiplies with another to give the specified number or expression.
Mix-up alert: You can cancel common factors, not terms joined by addition.
If a product is zero, at least one factor must be zero.
Exactly: For real numbers , implies or .
Mix-up alert: Make one side zero before using this law; it does not apply to a product equal to 6.
Rewrite a quadratic using a squared binomial.
Exactly: Rewrite , with a leading coefficient factored out or normalized first.
Mix-up alert: Adding a quantity to one side of an equation requires adding it to the other side.
Flip a graph across an axis.
Exactly: The graph of reflects across the horizontal axis; reflects across the vertical axis.
Mix-up alert: An outside minus changes outputs; an inside minus changes inputs.
The multiplier that stretches or compresses distances.
Exactly: For , with and , vertical distances scale by and horizontal distances by , with reflections for negative multipliers.
Mix-up alert: An inside factor 2 makes a graph half as wide, not twice as wide.
A change that moves, stretches or reflects a graph.
Exactly: A mapping of a function graph, such as , with and .
Mix-up alert: An inside shift x - h moves the graph right by h when h is positive.
A sum of terms with nonnegative whole-number powers of the variable.
Exactly: A finite sum of terms , with nonnegative integer exponents and real coefficients .
Mix-up alert: A variable in a denominator or a negative power of x does not make a polynomial.
How many times the same zero appears as a factor.
Exactly: A zero has multiplicity when divides the polynomial but does not.
Mix-up alert: Even multiplicity touches the axis; odd multiplicity crosses it.
The coefficient on the highest-power nonzero term.
Exactly: For a nonzero polynomial in descending degree order, the coefficient of its highest-degree term.
Mix-up alert: Use the highest degree, not the first term as originally written.
A polynomial divided by a polynomial.
Exactly: A quotient , where are polynomials and .
Mix-up alert: Simplifying does not restore inputs excluded by the original denominator.
What is left after polynomial division.
Exactly: In , the remainder is zero or has smaller degree than the nonzero divisor . For division by , the remainder is .
Mix-up alert: Use c, not -c, when dividing by x-c in synthetic division.
An input that the original expression does not allow.
Exactly: A value outside an expression’s domain, such as a zero of an original denominator.
Mix-up alert: A solution of a cleared-denominator equation can still be excluded.
A comparison of side lengths in a right triangle.
Exactly: For an acute angle in a right triangle, label opposite length , adjacent length , and hypotenuse . Then , , and .
Mix-up alert: Opposite and adjacent depend on which acute angle you chose.
An angle measured by comparing arc length with radius.
Exactly: One radian subtends an arc equal in length to the radius. In general, , where is arc length and is radius.
Mix-up alert: Use radians in . Convert a degree measure before using this formula.
A circle of radius one centered at the origin.
Exactly: The points satisfying . At angle , the point is .
Mix-up alert: Cosine is the horizontal coordinate; sine is the vertical coordinate.
An equation that works for every allowed input.
Exactly: An equality true for every value in the common domain of its two sides.
Mix-up alert: Checking one angle can disprove an identity but cannot prove it.
An equation with trigonometric expressions whose allowed solutions you must find.
Exactly: An equality involving trigonometric functions, solved on a stated interval or as a general solution.
Mix-up alert: Dividing by a factor can lose solutions where that factor is zero.
Half the distance between a wave’s highest and lowest outputs.
Exactly: For or , amplitude is , with .
Mix-up alert: Amplitude is always nonnegative and is not the vertical shift.
The horizontal line halfway between a wave’s high and low values.
Exactly: For or , the midline is .
Mix-up alert: The midline is an output value, while amplitude measures distance from it.
The shortest positive input shift that repeats the whole pattern.
Exactly: A fundamental period satisfies wherever defined, with no smaller positive period.
Mix-up alert: For a nonconstant transformed sine or cosine, the period is ; for tangent it is .
A horizontal shift of a repeating graph.
Exactly: In A sin(B(x-C))+D or A cos(B(x-C))+D, the phase shift is C, with B nonzero.
Mix-up alert: Factor before reading the shift in . The shift is , with .
The number of complete cycles per unit of input.
Exactly: For a sinusoid with period , ordinary frequency is cycles per input unit.
Mix-up alert: B is angular frequency in radians per input unit, not usually the number of cycles.
