AP Precalculus Midterm Review

Function

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.

Domain

The set of input values.

Range

The set of output values.

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).

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).

Concave up

When the average rate of change over equal-length input-value intervals is increasing for all small- length intervals.

Concave down

When the average rate of change over equal-length input-value intervals is decreasing for all small-length intervals.

Zeros

The graph intersects the x-axis when the output value is zero.

Positive rate of change

This rate of change indicates that as one quantity increases, the other quantity does the same.

Negative rate of change

This rate of change indicates that as one quantity increases, the other decreases.

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.

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.

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.

Global/Absolute Maximum

The greatest of all local maxima.

Global/Absolute Minimum

The least of all local minima.

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.

Multiplicity

When a linear factor (x - a) is repeated n times, the corresponding zero of the polynomial function is repeated n times.

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.

Even function

Function symmetric over the line x = 0, or y-axis. It has the property f(−x) = f(x).

Odd function

Function symmetric about the point (0,0), or the origin. It has the property f(−x) = −f(x).

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.

Rational function

The quotient of two polynomial functions.

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.

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.

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.

Zeros of a Rational Function

The real zeros of the numerator for such values in its domain.

Holes

If the multiplicity of a real zero in the numerator is greater than or equal to its multiplicity in the denominator.

Additive Transformation

Transformation of a function f that results in a vertical or horizontal translation of the graph of f.

Multiplicative Transformation

Transformation of a function f that results in a vertical or horizontal dilation of the graph of f.