Exponential Functions Context

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

How do you solve an exponential equation like 5^(2x-3) = 3125?

Back

To solve, express both sides with the same base. Here, 3125 can be written as 5^5. Then, set the exponents equal: 2x - 3 = 5.

Front

What is the formula for exponential growth?

Back

The formula for exponential growth is P(t) = P0 * (1 + r)^t, where P0 is the initial amount, r is the growth rate, and t is time.

Front

What is the annual growth rate in the function P(t) = 5000(1.02)^t?

Back

The annual growth rate is 2%, as indicated by the (1 + r) term in the function.

Front

How do you find the percent increase in an exponential function?

Back

To find the percent increase, subtract the initial value from the final value, divide by the initial value, and multiply by 100.

Front

What is the value of a population modeled by P(t) = 5000(1.02)^t after 10 years?

Back

P(10) = 5000(1.02)^10 ≈ 6104.1.

Front

If a quantity doubles every year, what is the growth factor?

Back

The growth factor is 2, as the quantity multiplies by 2 each year.

Front

What is the formula for exponential decay?

Back

The formula for exponential decay is P(t) = P0 * (1 - r)^t, where P0 is the initial amount, r is the decay rate, and t is time.

Front

How do you calculate the future value of an investment that doubles every year?

Back

Use the formula A = P * (2^t), where A is the amount after t years, P is the initial amount, and t is the number of years.