AP Precalculus Mid Unit 2A Review

Comprehensive vocabulary flashcards reviewing arithmetic sequences, geometric sequences, exponential functions, regressions, rates of change, and asymptotes from the AP Precalculus Mid Unit 2A Review.

A sequence in which consecutive terms differ by a constant value called the common difference $d$, with explicit relation $a_n = a_k + (n - k)d$.

Arithmetic Sequence

Geometric Sequence

A sequence in which each term after the first is found by multiplying the previous term by a fixed, non-zero constant ratio $r$, with explicit relation $g_n = g_k \cdot r^{n - k}$.

$n^{\text{th}}$ Term of an Arithmetic Sequence from the $k^{\text{th}}$ Term

The formula $a_n = a_k + (n - k)d$, which gives the $n^{\text{th}}$ term of an arithmetic sequence using a known $k^{\text{th}}$ term and common difference $d$.

$n^{\text{th}}$ Term of a Geometric Sequence from the $k^{\text{th}}$ Term

The formula $g_n = g_k \cdot r^{n - k}$, which expresses the $n^{\text{th}}$ term of a geometric sequence using a known $k^{\text{th}}$ term and common ratio $r$.

Exponential Function

A function of the form $f(x) = a \cdot b^x$, where $a$ is a non-zero constant initial value and $b$ is the positive constant base with $b \neq 1$.

Constant Ratio Property of Exponential Tables

A property where equal increments in inputs (such as $\Delta x = 10$) result in constant multiplicative ratios between consecutive outputs, such as $\frac{20}{40} = \frac{10}{20} = 0.5$, confirming the function is exponential.

Increasing Exponential Function Condition

An exponential function $f(x) = a \cdot b^x$ is increasing when the initial value $a > 0$ and the base $b > 1$ (for example, $f(x) = 9(2)^x$).

Concavity of an Exponential Function

The curvature of an exponential function determined by the sign of the vertical coefficient $a$: it is concave down when $a < 0$ (such as $f(x) = -\frac{1}{5}(4)^x$) and concave up when $a > 0$.

End Behavior of $f(x) = -2(3)^x$

The asymptotic and unbounded values of the function as inputs grow infinitely large: $\lim_{x \to -\infty} f(x) = 0$ and $\lim_{x \to \infty} f(x) = -\infty$.

Exponential Regression

A statistical process of finding the best-fitting exponential curve of the form $y = a \cdot b^x$ that models a set of bivariate data points.

Time Unit Re-scaling in Exponential Models

The method of rewriting an exponential growth exponent to change units of time, such as converting a doubling period of $7$ days into hours $t$ via $d = \frac{t}{24}$, yielding $g(t) = A_0(2^{\frac{1}{168}})^t$.

Decreasing at an Increasing Rate

A rate of change behavior characteristic of a decreasing function that is concave up (such as $f(x) = 7\left(\frac{4}{5}\right)^x$), where consecutive differences are negative and increasing towards $0$.

Decreasing at a Decreasing Rate

A rate of change behavior characteristic of a decreasing function that is concave down (such as $g(x) = -6(5)^x$), where consecutive differences are negative and becoming more negative toward $-\infty$.

Geometric Sequence Expression for Graphed Sequence $g_n$

The closed-form sequence expression $g_n = 10\left(\frac{1}{2}\right)^{\frac{n}{2}}$, determined by initial term $g_0 = 10$ and terms halving every $2$ units of $n$ ($(0, 10)$, $(2, 5)$, $(4, 2.5)$, $(6, 1.25)$, $(8, 0.625)$).

Horizontal Asymptote of $H(x) = a \cdot b^x + c$

The horizontal line $y = c$ that the graph of $H(x)$ approaches as $x$ approaches either $\infty$ or $-\infty$, dictated by the vertical translation constant $c$.

Exponential Growth vs. Exponential Decay Factor

In the exponential term $b^x$, base $b$ represents growth if $b > 1$ and represents decay if $0 < b < 1$.