Fundamentals 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 difference between exponential growth and exponential decay?

Back

Exponential growth occurs when the base 'b' in f(x) = a * b^x is greater than 1, leading to an increase in value as 'x' increases. Exponential decay occurs when 'b' is between 0 and 1, resulting in a decrease in value as 'x' increases.

Front

What is the y-intercept of an exponential function?

Back

The y-intercept of an exponential function f(x) = a * b^x is the point where the graph intersects the y-axis, which occurs at (0, a).

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

How do you determine if a function is exponential?

Back

A function is exponential if it can be expressed in the form f(x) = a * b^x, where 'a' is a constant and 'b' is a positive real number.

Front

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

Back

The graph of an exponential growth function rises steeply to the right and approaches the horizontal asymptote as it moves to the left.

Front

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

Back

The graph of an exponential decay function falls steeply to the right and approaches the horizontal asymptote as it moves to the left.

Front

How do you find the y-intercept of the function y = 3^(x + 7) + 10?

Back

To find the y-intercept, substitute x = 0: y = 3^(0 + 7) + 10 = 3^7 + 10 = 2187 + 10 = 2197, so the y-intercept is (0, 2197).

Front

What does the base of an exponential function indicate?

Back

The base of an exponential function indicates 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 effect of changing the constant 'a' in the function f(x) = a * b^x?

Back

Changing the constant 'a' affects 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.