Exponential Growth and Decay by a Factor

Exponential growth and decay describe quantities that change by being multiplied by the same number over and over again, instead of having the same amount added or subtracted each time. That repeated multiplier is called the growth factor or decay factor, and it is what separates exponential change from the straight-line, constant change you see in linear functions.

Both growth and decay share one formula:

\( y = a(b)^x \)

• \(a\) is the initial value (the amount when \(x = 0\))
• \(b\) is the growth or decay factor
• \(x\) is usually time, or however many times the multiplication has happened
• \(y\) is the amount after \(x\) steps

The only thing that changes between growth and decay is the size of \(b\).

When \(b > 1\), each time \(x\) increases by 1, \(y\) is multiplied by a number bigger than itself, so the value keeps getting larger. This is exponential growth.

For example, if a population starts at 2 and grows by a factor of 1.5 each year, the formula is \( y = 2(1.5)^x \). After one year there are 3, after two years there are 4.5, and the amount keeps climbing faster and faster as \(x\) increases.

Notice the curve rises slowly at first, then shoots upward. This increasingly steep climb is the signature shape of any growth factor greater than 1.

When \(0 < b < 1\), multiplying by \(b\) each time makes \(y\) smaller, since you're taking a fraction of the previous amount. This is exponential decay.

For example, suppose a sample starts at 100 grams and decays by a factor of 0.5 each hour: \( y = 100(0.5)^x \). After one hour there are 50 grams, after two hours 25 grams, and the amount keeps shrinking toward, but never quite reaching, zero.

A decay factor of exactly 0.5 is a special case worth knowing on its own, since it describes a quantity halving on a fixed schedule. You can see this idea developed further in half-life exponential decay.

Word problems rarely hand you \(b\) directly, so it helps to recognize the language that points to it:

• "Doubles every year" means \(b = 2\)
• "Triples every generation" means \(b = 3\)
• "Cuts in half every day" means \(b = \frac{1}{2}\)
• "Multiplied by 1.2 each cycle" means \(b = 1.2\)

If instead a problem describes a percent increase or decrease (like "grows by 8% per year"), you're working with the closely related idea of exponential growth and decay by percentage, where the factor is built from 1 plus or minus the percent as a decimal.

A bacteria colony starts with 40 cells and is multiplied by a factor of 3 every hour. Write the formula and find the population after 4 hours.

Step 1: Identify the values. \(a = 40\), \(b = 3\).

Step 2: Write the formula. \( y = 40(3)^x \).

Step 3: Substitute \(x = 4\). \( y = 40(3)^4 = 40(81) = 3240 \).

So after 4 hours the colony has 3240 cells. Because \(b = 3 > 1\), this is exponential growth, and the population grows faster with every hour that passes.

A car valued at \(\$18{,}000\) loses value by a factor of 0.85 each year. What is it worth after 3 years?

Step 1: Identify the values. \(a = 18000\), \(b = 0.85\).

Step 2: Write the formula. \( y = 18000(0.85)^x \).

Step 3: Substitute \(x = 3\). \( y = 18000(0.85)^3 \approx 18000(0.614125) \approx 11054.25 \).

The car is worth about \(\$11{,}054\) after 3 years. Since \(0 < 0.85 < 1\), the value shrinks a little less each year, which is exactly what a decay factor produces.

Because the factor \(b\) is applied again and again, the amount added or removed at each step is never the same, it depends on how large \(y\) already is. That's why exponential graphs curve instead of forming a straight line, and why they look so different from linear models. Once you're comfortable finding and interpreting the factor, you're ready to build and read full exponential graphs, including how graphing exponential functions and their transformations work, or how the same factor idea drives real formulas like compound interest.

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