Exponential Functions Vocabulary Flashcards

Vocabulary-style flashcards covering key concepts from the Exponential Functions lessons, including definitions of growth/decay, the standard form, shifts, intercepts, asymptotes, and representations.

Exponential Function

A function of the form f(x) = k a^x, with a > 0, a ≠ 1, k ≠ 0 and x real; can be represented by an equation, a table of values, and a graph.

Base (a) of an Exponential Function

The base a in f(x) = k a^x; a > 0 and a ≠ 1. If a > 1, the function shows growth; if 0 < a < 1, it shows decay.

Exponential Growth

An exponential function with a > 1 and k > 0; f(x) increases without bound as x increases.

Exponential Decay

An exponential function with 0 < a < 1 and k > 0; f(x) decreases as x increases.

General Form f(x) = k a^x

A standard exponential form where k shifts the graph vertically and a determines growth or decay.

Coefficient k

A scaling factor in f(x) = k a^x; determines the y-intercept of the base form (0, k).

y-intercept of base form

For f(x) = k a^x, the y-intercept is the point (0, k).

x-intercept of base form

For f(x) = k a^x with k ≠ 0, there is no x-intercept in the base form.

Horizontal Asymptote (base form)

For f(x) = k a^x, the horizontal asymptote is y = 0.

Vertical Asymptote

Exponential functions have no vertical asymptotes; their domain is all real numbers.

Vertical Shift

Shifting the graph by adding a constant: g(x) = k a^x + c moves the graph up by c (c > 0) or down by c (c < 0).

Horizontal Shift

Shifting the graph by changing the exponent: f(x) = k a^{x - b} + c moves left by |b| if b < 0 and right by |b| if b > 0.

y-intercept of shifted form

For f(x) = k a^x + c, the y-intercept is (0, k + c).

x-intercept condition for shifted form

If k and c have opposite signs (k c < 0), the function has an x-intercept.

Horizontal Asymptote of shifted form

For f(x) = k a^x + c, the horizontal asymptote is y = c.

Domain of Exponential Function

The domain is all real numbers (x ∈ (−∞, ∞)).

Representation of Exponential Functions

An exponential function can be represented by an equation, a table of values, and a graph.

Half-life

The time required for half of a radioactive substance to decay.

Compound Interest

Interest calculated on the original principal plus accumulated interest from previous periods.

0 < a < 1

Condition for exponential decay; the base is between 0 and 1.

a > 1

Condition for exponential growth; the base is greater than 1.

No Zeros / x-intercepts (base form)

Exponential base form f(x) = k a^x has no zeros (x-intercepts) when a > 0, a ≠ 1 and k ≠ 0.

Intercepts and Asymptotes in Graphing

Key features used to sketch exponential graphs: x-intercepts, y-intercepts, and horizontal asymptotes (y = 0 for base form; y = c for f(x) = k a^x + c).