Exponential Transformations Flashcard

Front

What is a horizontal translation in exponential functions?

Back

A horizontal translation shifts the graph of the function left or right along the x-axis. For example, a function of the form f(x) = a^(x - h) translates the graph h units to the right.

Front

What is a vertical translation in exponential functions?

Back

A vertical translation shifts the graph of the function up or down along the y-axis. For example, f(x) = a^x + k translates the graph k units up if k > 0 and down if k < 0.

Front

What is an asymptote in the context of exponential functions?

Back

An asymptote is a line that the graph of a function approaches but never touches. For exponential functions, the horizontal asymptote is typically y = 0.

Front

What is the difference between exponential growth and decay?

Back

Exponential growth occurs when the base of the exponential function is greater than 1 (e.g., f(x) = a^x where a > 1). Exponential decay occurs when the base is between 0 and 1 (e.g., f(x) = a^x where 0 < a < 1).

Front

What does it mean to reflect a function over the x-axis?

Back

Reflecting a function over the x-axis means that for every point (x, y) on the graph, there is a corresponding point (x, -y). This changes the sign of the output values.

Front

How do you identify a vertical translation in an exponential function?

Back

A vertical translation can be identified by a constant added or subtracted from the function, such as f(x) = a^x + k, where k indicates the vertical shift.

Front

What is the effect of a negative coefficient in front of an exponential function?

Back

A negative coefficient reflects the graph over the x-axis, changing the direction of growth or decay.

Front

What is the general form of an exponential function?

Back

The general form of an exponential function is f(x) = a * b^(x - h) + k, where (h, k) is the translation point, a is the vertical stretch or compression, and b is the base.

Front

What does the base of an exponential function determine?

Back

The base of an exponential function determines the rate of growth or decay. A base greater than 1 indicates growth, while a base between 0 and 1 indicates decay.

Front

What is the significance of the point (0, 1) in exponential functions?

Back

The point (0, 1) is significant because it represents the value of the function when x = 0 for any exponential function of the form f(x) = a^x, where a > 0.