AP Precalculus Unit 2: Exponential Functions Mastery Guide

Arithmetic Sequence

A sequence where each term is generated by adding a constant value to the previous term.

Common Difference (d)

The constant value added to each term in an arithmetic sequence.

Geometric Sequence

A sequence where each term is generated by multiplying the previous term by a constant non-zero value.

Common Ratio (r)

The constant value multiplied to each term in a geometric sequence.

Linear Functions

Functions that model arithmetic sequences, typically in the form y = mx + b.

Exponential Functions

Functions that model geometric sequences, typically in the form y = a * b^x.

Recursive Rule (Arithmetic)

un = u{n-1} + d, describes how to find the next term by adding the common difference.

Explicit Rule (Arithmetic)

un = u0 + n * d, gives a formula to find the nth term directly.

Horizontal Asymptote

A horizontal line that the graph of an exponential function approaches as x goes to infinity.

Y-Intercept

The point where the graph of a function crosses the y-axis, found by evaluating f(0).

Domain of Exponential Functions

The set of all possible input values, which is usually (-∞, ∞) for exponential functions.

Range of Exponential Functions

The set of possible output values; depends on the vertical shift (k).

Concavity of Exponential Functions

Exponential functions of the form f(x) = a * b^x (where a > 0) are always concave up.

Exponential Growth

Occurs when the base b of an exponential function is greater than 1.

Exponential Decay

Occurs when the base b of an exponential function is between 0 and 1.

Calculating Growth Rate

Using the formula f(t) = a(1 + r)^t, where r is the percentage rate of growth.

Calculating Decay Rate

Using the formula f(t) = a(1 - r)^t, where r is the percentage rate of decay.

First Differences

The differences between consecutive outputs, used to check for linear patterns.

Ratios of Outputs

The ratios of consecutive outputs, used to check for exponential patterns.

Residuals

The differences between observed and predicted values, used for visual model validation.

Positive Exponents

Always produce positive outcomes in exponential functions, regardless of the input.

Rate vs. Factor

Understanding that a growth rate of 15% indicates a factor of 1 + r = 1.15.

Concavity Confusion

Both exponential growth and decay graphs are concave up, even if one is decreasing.

Transformations and Asymptotes

Vertical shifts affect the horizontal asymptote and the range of the function.

Growth Factor (b)

In the exponential growth formula, a base greater than 1 that indicates increasing values.

Decay Factor (b)

In the exponential decay formula, a base less than 1 that indicates decreasing values.