A mathematical relation that maps a set of input values to a set of output values such that each input value is mapped to exactly one output value.
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Domain
The set of input values.
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Range
The set of output values.
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Increasing Function
As the input values increase, the output values always increase in the interval of its domain. That is, for all a and b in the interval, if a < b, then f(a) < f(b).
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Decreasing Function
As the input values increase, the output values always decrease in the interval of its domain. That is, for all a and b in the interval, if a < b, then f(a) > f(b).
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Concave up
When the average rate of change over equal-length input-value intervals is increasing for all small- length intervals.
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Concave down
When the average rate of change over equal-length input-value intervals is decreasing for all small-length intervals.
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Zeros
The graph intersects the x-axis when the output value is zero.
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Positive rate of change
This rate of change indicates that as one quantity increases, the other quantity does the same.
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Negative rate of change
This rate of change indicates that as one quantity increases, the other decreases.
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Polynomial Function
Any function representation that is equivalent to the analytical form (equation in picture), where n is a positive integer, a_i is a real number for each i from 1 to n, and a_n is nonzero.
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Local/Relative Maximum
The point where a polynomial function switches between increasing and decreasing. The included endpoint of a polynomial function with a restricted domain.
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Local/Relative Minimum
The point where a polynomial function switches between decreasing and increasing. The included endpoint of a polynomial function with a restricted domain.
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Global/Absolute Maximum
The greatest of all local maxima.
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Global/Absolute Minimum
The least of all local minima.
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Point of Inflection
Points of a polynomial function that occur at input values where the rate of change of the function changes from increasing to decreasing or vice versa.
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Multiplicity
When a linear factor (x - a) is repeated n times, the corresponding zero of the polynomial function is repeated n times.
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Degree of a Polynomial
The largest exponent of a polynomial function. Given a table of values, it can be found by examining the successive differences of the output values over equal-interval input values.
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Even function
Function symmetric over the line x = 0, or y-axis. It has the property f(−x) = f(x).
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Odd function
Function symmetric about the point (0,0), or the origin. It has the property f(−x) = −f(x).
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End Behavior of a polynomial function
As input values of a nonconstant polynomial function increase or decrease without bound, the output values will either increase or decrease without bound.
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Rational function
The quotient of two polynomial functions.
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Horizontal Asymptote
If the polynomial p(x) = ax^n+.. in the numerator has the same degree n as the denominator q(x) = bx^n+..., the equation of the asymptote is y = a/b . If a polynomial in the numerator has a degree smaller than that of the denominator, the equation of the asymptote is y = 0.
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Slant Asymptote
If a polynomial in the numerator has a degree one larger than the degree of denominator. You may use long or synthetic division, and as a result, you'll get a quotient of y = mx + b and a remainder.
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Vertical Asympote of a Rational Function
It occurs at x = a if the multiplicity of a as a real zero in the denominator is greater than its multiplicity as a real zero in the numerator.
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Zeros of a Rational Function
The real zeros of the numerator for such values in its domain.
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Holes
If the multiplicity of a real zero in the numerator is greater than or equal to its multiplicity in the denominator.
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Additive Transformation
Transformation of a function f that results in a vertical or horizontal translation of the graph of f.
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Multiplicative Transformation
Transformation of a function f that results in a vertical or horizontal dilation of the graph of f.
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