AP Precalculus Unit 2 Notes: Understanding Exponentials Through Patterns, Graphs, and Models

Sequence

An ordered list of numbers (terms) that follows a rule; a discrete way to describe patterns of change.

Term (of a sequence)

A number in a sequence; terms are indexed (e.g., a1, a2, …).

Arithmetic sequence

A sequence in which each term is found by adding/subtracting the same constant each step.

Common difference (d)

The constant amount added each step in an arithmetic sequence (found by subtracting consecutive terms).

Explicit formula for an arithmetic sequence

an = a1 + (n−1)d, where a1 is the first term and d is the common difference.

Consecutive differences test

A check for arithmetic behavior by computing $a_2-a_1$, $a_3-a_2$, …; if (approximately) constant, the sequence is arithmetic.

Linear behavior (discrete)

Change by a constant additive amount per step; arithmetic sequences are the discrete version of linear patterns.

Geometric sequence

A sequence in which each term is found by multiplying by the same constant factor each step.

Common ratio (r)

The constant multiplier between consecutive terms in a geometric sequence (found by dividing consecutive terms).

Explicit formula for a geometric sequence

$a_n = a_1 \times r^{(n-1)}$, where $a_1$ is the first term and $r$ is the common ratio.

Consecutive ratios test

A check for geometric behavior by computing a2/a1, a3/a2, …; if (approximately) constant, the sequence is geometric.

Exponential behavior (discrete)

Change by a constant multiplicative factor (often constant percent change); geometric sequences are the discrete version of exponential patterns.

Indexing (n−1) in sequences

In $a_n = a_1 \times r^{(n-1)}$ (or $a_n = a_1 + (n-1)d$), the exponent/term uses $n-1$ so that $n=1$ gives $a_1$.

Exponential function

A function where the variable appears in the exponent; commonly $f(x) = a \times b^x$ in precalculus.

Initial value (a) in f(x)=a·b^x

The value at $x = 0$; $f(0) = a$ (the y-intercept in the basic form).

Base (b) in f(x)=a·b^x

The constant multiplicative factor per 1 unit increase in x; determines growth vs decay.

Exponential growth

Exponential behavior with $b>1$ (or factor $1+r$ where $r>0$), increasing by a constant percent each step.

Exponential decay

Exponential behavior with $0<b<1$ (or factor $1-r$ where $0<r<1$), decreasing by a constant percent each step.

Y-intercept of an exponential (basic form)

For $f(x)=a \times b^x$, the y-intercept is $f(0)=a$.

Horizontal asymptote (transformed exponential)

For $f(x)=a \times b^{(x-h)}+k$, the horizontal asymptote is $y=k$ (the graph approaches but may not reach it).

Transformation parameters (h and k)

In f(x)=a·b^(x−h)+k: h shifts horizontally (right if h>0) and k shifts vertically, changing the asymptote to y=k.

Exponential model A(t)=A0·b^t

A modeling form where $A_0$ is the initial amount at $t = 0$ and $b$ is the growth/decay factor per unit time.

Growth/decay factor vs percent rate

If the percent rate per unit is $r$ (decimal), then $b=1+r$ for growth and $b=1-r$ for decay.

Residual

Observed − predicted; used to compare models (smaller residual magnitudes and no clear pattern indicates a better fit).

Competing model validation

A function where the variable appears in the exponent; commonly $f(x)=a \times b^x$ in precalculus.