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. It represents growth or decay depending on the value of b.

Front

What does it mean for a function to exhibit exponential growth?

Back

Exponential growth occurs when a quantity increases by a consistent percentage over equal intervals of time, resulting in a rapid increase. For example, in the function f(t) = 12(1.015)^t, the population increases over time.

Front

What does it mean for a function to exhibit exponential decay?

Back

Exponential decay occurs when a quantity decreases by a consistent percentage over equal intervals of time, leading to a rapid decrease. For example, in the function f(t) = 12(0.95)^t, the population decreases over time.

Front

What is the y-intercept of an exponential function?

Back

The y-intercept of an exponential function is the value of the function when x = 0. It is represented by the constant 'a' in the function f(x) = a * b^x.

Front

How do you identify exponential growth from a graph?

Back

Exponential growth is identified on a graph by a curve that rises steeply to the right, indicating that the function's value increases rapidly as x increases.

Front

How do you identify exponential decay from a graph?

Back

Exponential decay is identified on a graph by a curve that falls steeply to the right, indicating that the function's value decreases rapidly as x increases.

Front

What is the common ratio in an exponential sequence?

Back

The common ratio in an exponential sequence is the factor by which each term is multiplied to get the next term. For example, in the sequence 28, 14, 7, 3.5, the common ratio is 1/2.

Front

What is the formula for calculating the population after t years in an exponential growth model?

Back

The formula is P(t) = P0 * (1 + r)^t, where P0 is the initial population, r is the growth rate, and t is the time in years.

Front

What is the difference between linear and exponential functions?

Back

Linear functions increase or decrease by a constant amount, while exponential functions increase or decrease by a constant percentage, leading to different growth patterns.

Front

What is the base of an exponential function?

Back

The base of an exponential function is the constant 'b' in the function f(x) = a * b^x, which determines the rate of growth or decay.