Exponential Functions

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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 Exponential Growth mean?

Back

Exponential growth occurs when the growth rate of a value is proportional to its current value, leading to a rapid increase over time. It can be represented by a function where the base 'b' is greater than 1.

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What does Exponential Decay mean?

Back

Exponential decay occurs when a quantity decreases at a rate proportional to its current value, leading to a rapid decrease over time. It can be represented by a function where the base 'b' is between 0 and 1.

Front

Identify the initial value in the function f(x) = 200(1.05)^x.

Back

The initial value (or y-intercept) is 200.

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What is the decay factor in the function f(x) = 500(0.9)^x?

Back

The decay factor is 0.9.

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How do you write an exponential decay equation for a quantity that decreases by 25% each year?

Back

The equation is y = a(0.75)^x, where 'a' is the initial amount.

Front

What is the formula for calculating the value of an investment after 't' years with an annual interest rate 'r'?

Back

The formula is A = P(1 + r)^t, where A is the amount after t years, P is the principal amount, and r is the interest rate.

Front

If a population of 1000 bacteria decreases by 20% each hour, what is the equation representing this situation?

Back

y = 1000(0.8)^x, where x is the number of hours.

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What is the y-intercept of the function f(x) = 50(2)^x?

Back

The y-intercept is 50.

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How do you determine if a model represents exponential growth or decay?

Back

If the base 'b' in the function f(x) = a(b)^x is greater than 1, it represents growth; if 'b' is between 0 and 1, it represents decay.