Skills Check 5.1a: Using Exponential Functions

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What is an exponential function?

Back

An exponential function is a mathematical function of the form y = a(b^x), where 'a' is a constant, 'b' is the base (a positive real number), and 'x' is the exponent. It represents growth or decay processes.

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What does the decay factor represent in exponential decay?

Back

The decay factor is the value by which the quantity decreases each time period. It is calculated as 1 minus the decay rate (in decimal form). For example, if the decay rate is 30%, the decay factor is 0.7.

Front

How do you write an equation for exponential decay?

Back

An equation for exponential decay can be written as y = a(1 - r)^x, where 'a' is the initial amount, 'r' is the decay rate, and 'x' is the time period.

Front

What is the difference between exponential growth and decay?

Back

Exponential growth occurs when a quantity increases by a constant percentage over time, while exponential decay occurs when a quantity decreases by a constant percentage over time.

Front

If a quantity decreases by 20% each hour, what is the decay factor?

Back

The decay factor is 0.8, calculated as 1 - 0.20.

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

Back

If the base of the exponential function is greater than 1, it represents growth. If the base is between 0 and 1, it represents decay.

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What is the initial amount in the equation y = 5000(0.7)^x?

Back

The initial amount is 5000, which represents the starting quantity before any decay occurs.

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How do you find the value of a car after several years of depreciation?

Back

You can use the formula y = a(1 - r)^t, where 'a' is the initial value, 'r' is the depreciation rate, and 't' is the number of years.

Front

What does the term 'depreciation' mean in the context of exponential functions?

Back

Depreciation refers to the reduction in value of an asset over time, often modeled using exponential decay.

Front

Write the equation for a quantity that starts at 1200 and decreases by 15% each year.

Back

The equation is y = 1200(0.85)^x.