Algebra 2 Exponential Growth and Decay Lesson (Free PDF)

Q: What is the growth factor of an exponential function?

The growth factor b is the number you multiply by for each one-unit step in x. In f(x) = a times b to the x, b is greater than 1 for growth and between 0 and 1 for decay.

Q: How do you find a percent growth factor?

Add the rate to 1 for growth, or subtract it from 1 for decay. A 6% increase has b = 1.06 and a 6% decrease has b = 0.94.

Q: What is the compound interest formula?

A = P(1 + r/n)^(nt), where P is the principal, r the annual rate, n the number of compoundings per year, and t the time in years.

Q: What is the number e?

The number e is about 2.71828. It is used for continuous growth, A = Pe^(rt), and it arises as the limit of compounding more and more often.

Q: How is an exponential function different from a linear function?

A linear function adds the same amount each step. An exponential function multiplies by the same factor each step, so it eventually grows faster than any linear function.

Algebra 2 math lesson: Exponential Growth and Decay – Maths PDF

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This algebra 2 exponential functions lesson for Algebra 2 covers exponential growth and decay (Common Core F-LE.A.2, F-LE.B.5, A-SSE.B.3c, F-IF.C.8b), with a free PDF to print.

An exponential function f(x) = a·b^x multiplies by the factor b for each step in x. It grows when b is greater than 1 and decays when b is between 0 and 1, and a percent rate r gives b = 1 + r or 1 – r. Compound interest and continuous growth are exponential models.

Algebra 2 · Common Core State Standards: F-LE.A.2, F-LE.B.5, A-SSE.B.3c, F-IF.C.8b

Key facts in this algebra 2 exponential functions lesson

Key facts

f(x) = a·b^x: a is the initial value, b the growth factor.
Growth factor b = 1 + r (growth) or 1 – r (decay).
Equal steps in x multiply the output by b.
Compound interest: A = P(1 + r/n)^(nt); continuous: A = Pe^(rt).
Rewrite b^x expressions to find equivalent rates.

What is an exponential function?

Exponential function

An exponential function has the form $f(x) = a \cdot b^{x}$ with $a \ne 0$, $b > 0$, and $b \ne 1$. The constant $a$ is the initial value (the y-intercept) and $b$ is the growth factor per unit of $x$. If $b > 1$ the function grows; if $0 < b < 1$ it decays.

Growth (left) and decay (right) both have a horizontal asymptote at y = 0.

In a linear function equal steps in $x$ add the same amount. In an exponential function equal steps in $x$ multiply by the same factor. A table with outputs $5, 10, 20, 40$ for $x = 0, 1, 2, 3$ shows ratio 2 each time, so $f(x) = 5 \cdot 2^{x}$.

How do percent rates turn into growth factors?

A quantity that changes by a percent $r$ each period has growth factor $b = 1 + r$ for growth and $b = 1 – r$ for decay. A town of 12,000 people growing 3.5% a year follows $P = 12{,}000(1.035)^{t}$. A car worth $24,000 that loses 15% each year follows $V = 24{,}000(0.85)^{t}$.

Growth factor = 1 + rate.

Exponential models are used whenever change is proportional to the current amount: bank balances, populations, medicine in the body, and the value of a car. To build a model, identify the starting value $a$ and the factor $b$ per time unit. To interpret a model, read $a$ as the amount at time 0 and convert $b$ back to a percent rate with $r = b – 1$. Always match the time units: if $t$ is in years, the factor is per year.

How does compound interest work?

When interest is added $n$ times a year at annual rate $r$, the balance after $t$ years is $A = P\left(1 + \frac{r}{n}\right)^{nt}$. With continuous compounding, $A = Pe^{rt}$, where $e \approx 2.71828$. For $2,000 at 4% compounded quarterly for 5 years, $A = 2000(1.01)^{20} \approx 2{,}440.38$.

Compound interest repeats a multiplication.

Annual, quarterly, and continuous compounding of $2,000 at 4%: more frequent compounding gives slightly more.

Exponential expressions can be rewritten to reveal other rates. For example $1.02^{12t} = (1.02^{12})^{t} \approx 1.268^{t}$: a 2% monthly rate is about 26.8% per year, not 24%. And $2^{3x} = (2^{3})^{x} = 8^{x}$.

Algebra 2 exponential functions lesson: questions and answers

What is the growth factor of an exponential function?

The growth factor b is the number you multiply by for each one-unit step in x. In f(x) = a times b to the x, b is greater than 1 for growth and between 0 and 1 for decay.

How do you find a percent growth factor?

Add the rate to 1 for growth, or subtract it from 1 for decay. A 6% increase has b = 1.06 and a 6% decrease has b = 0.94.

What is the compound interest formula?

A = P(1 + r/n)^(nt), where P is the principal, r the annual rate, n the number of compoundings per year, and t the time in years.

What is the number e?

The number e is about 2.71828. It is used for continuous growth, A = Pe^(rt), and it arises as the limit of compounding more and more often.

How is an exponential function different from a linear function?

A linear function adds the same amount each step. An exponential function multiplies by the same factor each step, so it eventually grows faster than any linear function.

How to use this algebra 2 exponential functions lesson

Read the definitions first, then follow the worked method line by line and redo the example on paper without looking. When every step makes sense, move on to the worksheet, then check yourself with the quiz.

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