AP Precalculus 2.2: Change in Linear and Exponential Functions

Vocabulary flashcards comparing linear functions with arithmetic sequences and exponential functions with geometric sequences, covering standard forms, transformations using a known kth term, and domain differences.

Slope-intercept form of a linear function

The equation of a line written as $f(x) = mx + b$ (or $f(x) = b + mx$), where $b$ is the initial value and $m$ is the constant rate of change.

Arithmetic sequence (initial value form)

A sequence given by the formula $a_n = a_0 + dn$, where $a_0$ is the initial value and $d$ is the constant rate of change.

Point-slope form of a linear function

A transformed linear equation written as $f(x) = y_1 + m(x - x_1)$, where $(x_1, y_1)$ is a known point on the line and $m$ is the constant rate of change.

Arithmetic sequence with a known $k^{\text{th}}$ term

The formula $a_n = a_k + d(n - k)$, used to find any term when starting from a known term $(k, a_k)$ rather than the initial value.

Exponential function (standard form)

A function expressed as $f(x) = a \cdot b^x$, where $a$ represents the initial value and $b$ is the constant ratio or base.

Geometric sequence (initial value form)

A sequence defined by the formula $g_n = g_0 \cdot r^n$, where $g_0$ is the initial value and $r$ is the constant ratio.

Shifted exponential function

An exponential function written in the form $f(x) = y_1 \cdot r^{x - x_1}$, transformed using a known point $(x_1, y_1)$ rather than the initial value.

Geometric sequence with a known $k^{\text{th}}$ term

The formula $g_n = g_k \cdot r^{n - k}$, used to evaluate terms when given a specific known term $(k, g_k)$ and a constant ratio $r$.

Exponential Growth vs. Exponential Decay

Classification of exponential behavior where growth occurs when the base or ratio is greater than $1$ ($r > 1$), and decay occurs when the base or ratio is between $0$ and $1$ ($0 < r < 1$).

Sequence domain vs. Function domain

A sequence has a discrete domain (such as $\{0, 1, 2, 3, 4, 5, 6\}$), whereas its corresponding exponential or linear function has a continuous domain across all real numbers ($(-\infty, \infty)$).