AP Precalculus Review: Polynomial, Exponential, and Logarithmic Functions

Vocabulary practice flashcards covering fundamental precalculus concepts including polynomial behavior, logarithm rules, function transformations, and exponential modeling.

Quotient Rule of Logarithms

The logarithmic property stating that the logarithm of a quotient equals the difference of the logarithms: $\log_b\left(\frac{u}{v}\right) = \log_b(u) - \log_b(v)$.

Power Rule of Logarithms

The property of logarithms stating that the exponent on an argument can be written as a multiplying coefficient: $\log_b(x^c) = c\log_b(x)$.

Multiplicity of a Zero

The number of times a factor is repeated in a polynomial; an even multiplicity causes the graph to bounce off the $x$-axis, while an odd multiplicity causes the graph to cross the $x$-axis.

Turning Point

A location where a polynomial graph shifts between increasing and decreasing; a polynomial function of degree $n$ has at most $n - 1$ turning points and at most $n$ real zeros.

Average Rate of Change

The ratio of the change in output values to the change in input values over an interval $[a, b]$, calculated using the formula $\frac{f(b) - f(a)}{b - a}$.

Point of Inflection

A point on a graph where the concavity changes; for a cubic function such as $f(x) = (x - 1)^3 + 2$, the inflection point is translated to $(1, 2)$.

Remainder Theorem

The principle stating that dividing a polynomial $p(x)$ by a linear binomial $(x - c)$ produces a remainder equal to evaluating the polynomial at that value, $p(c)$.

Rational Root Theorem

A theorem stating that any potential rational root of a polynomial with integer coefficients must take the form $\pm \frac{p}{q}$, where $p$ is a factor of the constant term and $q$ is a factor of the leading coefficient.

End Behavior

The trend of a graph as $x$ approaches positive or negative infinity; an odd-degree polynomial with a positive leading coefficient falls to the left and rises to the right.

Extraneous Solution

A solution derived algebraically from a rewritten logarithmic equation that must be rejected because it causes the argument of an original logarithm to be non-positive.

Odd Function

A function with origin symmetry where $(x, y)$ maps to $(-x, -y)$, causing a relative maximum at $(x, y)$ to correspond to a relative minimum at $(-x, -y)$.

Even Function

A function whose graph is symmetric about the $y$-axis ($x = 0$); a horizontal shift of $h$ units shifts this vertical line of symmetry to $x = h$.

Decay Factor

The base $b$ in an exponential decay function $V(t) = a \cdot b^t$, computed as $1 - r$, where $r$ is the percentage rate of decrease per time period.

Inverse Function of an Exponential

The logarithmic function found by interchanging the domain and range of an exponential function, turning $y = a \cdot b^x$ into $f^{-1}(x) = \log_b\left(\frac{x}{a}\right)$.