Writing 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 does the base of an exponential function represent?

Back

The base of an exponential function determines the rate of growth or decay. If the base is greater than 1, the function represents exponential growth; if the base is between 0 and 1, it represents exponential decay.

Front

Write the general form of an exponential growth function.

Back

f(x) = a * b^x, where a > 0 and b > 1.

Front

Write the general form of an exponential decay function.

Back

f(x) = a * b^x, where a > 0 and 0 < b < 1.

Front

How do you identify an exponential function from a table of values?

Back

An exponential function can be identified if the ratio of consecutive outputs (y-values) is constant when the inputs (x-values) are increased by the same amount.

Front

What is the formula to calculate the future value in exponential growth?

Back

Future Value = Present Value * (1 + r)^t, where r is the growth rate and t is the time.

Front

How do you write an exponential function from a word problem?

Back

Identify the initial value and the growth/decay rate, then use the formula f(t) = a * (1 + r)^t for growth or f(t) = a * (1 - r)^t for decay.

Front

What is the significance of the initial value in an exponential function?

Back

The initial value (a) represents the starting amount before any growth or decay occurs.

Front

How do you determine the growth rate from an exponential function?

Back

The growth rate can be determined from the base of the exponential function. If the function is in the form f(x) = a * b^x, then the growth rate is (b - 1) * 100%.

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).