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 formula for compound interest?

Back

The formula for compound interest is A = P(1 + r/n)^(nt), where A is the amount of money accumulated after n years, P is the principal amount (the initial amount), r is the annual interest rate (decimal), n is the number of times that interest is compounded per year, and t is the number of years.

Front

What does the y-intercept represent in an exponential function?

Back

The y-intercept represents the initial value of the function when x = 0. In the function f(x) = a * b^x, the y-intercept is 'a'.

Front

How do you find the y-intercept of the function f(x) = 13(0.27)^x?

Back

To find the y-intercept, substitute x = 0 into the function: f(0) = 13(0.27)^0 = 13.

Front

What is the horizontal asymptote of an exponential function?

Back

The horizontal asymptote of an exponential function f(x) = a * b^x is typically y = 0, meaning the function approaches but never reaches this line as x approaches negative infinity.

Front

How do you rewrite a number in terms of a different base?

Back

To rewrite a number in terms of a different base, express the number as a power of the new base. For example, to rewrite 8 as a power of 2, you would write 8 = 2^3.

Front

What is the value of f(x) when x = 0 in the function f(x) = 2(3)^x?

Back

When x = 0, f(0) = 2(3)^0 = 2(1) = 2.

Front

What is the significance of the base in an exponential function?

Back

The base determines the rate of growth or decay of the function. If the base is greater than 1, the function grows; if the base is between 0 and 1, the function decays.

Front

How do you determine if a function is increasing or decreasing?

Back

A function is increasing if its base is greater than 1 (e.g., f(x) = 2^x). It is decreasing if its base is between 0 and 1 (e.g., f(x) = (1/2)^x).

Front

What is the relationship between exponential functions and logarithms?

Back

Exponential functions and logarithms are inverse functions. If y = b^x, then x = log_b(y). Logarithms can be used to solve for the exponent in exponential equations.