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 the base (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 increases rapidly for positive values of x and approaches the horizontal asymptote for negative values of x.

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?

Back

The range of an exponential function is all real numbers greater than zero, expressed as (0, ∞) for functions of the form f(x) = a * b^x where a > 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 x-intercept of an exponential function?

Back

Exponential functions do not have an x-intercept because they never cross the x-axis; they only approach it.

Front

What happens to the value of an exponential function as x approaches negative infinity?

Back

As x approaches negative infinity, the value of an exponential function approaches zero.

Front

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

Back

Changing the base 'b' affects the growth rate of the function; a larger base results in a steeper increase.

Front

What is the vertical shift in an exponential function?

Back

A vertical shift occurs when a constant is added or subtracted from the function, affecting the horizontal asymptote.

Front

How do you determine the horizontal asymptote of an exponential function?

Back

The horizontal asymptote can be determined by analyzing the end behavior of the function as x approaches positive or negative infinity.