Characteristics of Exponential Functions

Front

What is an exponential function?

Back

An exponential function is a mathematical function of the form f(x) = a * b^x, where 'a' is a constant, 'b' is a positive real number, and 'x' is the exponent.

Front

What is the general shape of the graph of an exponential function?

Back

The graph of an exponential function rises or falls rapidly, depending on whether the base 'b' is greater than 1 (increasing) or between 0 and 1 (decreasing).

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 domain of an exponential function?

Back

The domain of an exponential function is all real numbers, expressed as (-∞, ∞).

Front

What is the range of an exponential function with a positive base?

Back

The range of an exponential function with a positive base is all positive real numbers, expressed as (0, ∞).

Front

How does the value of 'a' affect the graph of an exponential function?

Back

The value of 'a' determines the vertical stretch or compression of the graph. If 'a' is positive, the graph is above the x-axis; if 'a' is negative, the graph is reflected across the x-axis.

Front

What happens to f(x) as x approaches negative infinity in an exponential function?

Back

As x approaches negative infinity, f(x) approaches 0 for exponential functions with a positive base.

Front

What is the y-intercept of an exponential function?

Back

The y-intercept of an exponential function is the point where the graph crosses the y-axis, which occurs at (0, a).

Front

What is the effect of changing the base 'b' in an exponential function?

Back

Changing the base 'b' affects the rate of growth or decay of the function. A larger base results in faster growth, while a base between 0 and 1 results in decay.

Front

What is the horizontal asymptote of the function f(x) = 2^x?

Back

The horizontal asymptote of the function f(x) = 2^x is y = 0.