This algebra 1 exponential functions lesson explains exponential growth and decay in one printable page, with definitions, a method and a worked example.
An exponential function has the form \(y=a\cdot b^{x}\): the starting value \(a\) is multiplied by the same factor \(b\) for every step in \(x\). When \(b>1\) the quantity grows, and when \(1>b>0\) it decays. Exponential growth eventually exceeds any linear growth, however large the line’s slope.
Algebra 1 Exponential Functions Lesson: key facts
- Linear functions add a constant difference; exponential functions multiply by a constant factor.
- In \(y=a\cdot b^{x}\), \(a\) is the value at \(x=0\) and \(b\) is the factor per step.
- Growth by \(r\) means \(b=1+r\); decay by \(r\) means \(b=1-r\).
- Compound interest: \(A=P(1+\frac{r}{n})^{nt}\).
- An exponential function eventually passes every linear function.
What is an exponential function?
An exponential function has the form \(y=a\cdot b^{x}\), where \(a\) is the starting value (the output at \(x=0\)) and \(b\) is the growth or decay factor. The factor \(b\) is positive and not equal to 1. The variable is in the exponent.
Linear: equal steps in the input add equal amounts to the output (a constant difference).
Exponential: equal steps in the input multiply the output by equal factors (a constant ratio).
How do percent rates become the factor b?
A quantity that grows by \(r\) each period has \(b=1+r\). A quantity that decays by \(r\) each period has \(b=1-r\). A 5% increase gives \(b=1.05\). A 12% decrease gives \(b=0.88\). If \(b>1\) the graph rises; if \(1>b>0\) it falls toward the \(x\)-axis.
How does compound interest work?
Interest that is added to the balance also earns interest, so the balance is exponential: \(A=P(1+\frac{r}{n})^{nt}\). Here \(P\) is the principal, \(r\) the yearly rate, \(n\) the compounds per year and \(t\) the years.
Example. $500 at 4% compounded yearly for 3 years: \(A=500(1.04)^{3}\approx\$562.43\).
How do you write an exponential function from a description?
- Find the starting value \(a\).
- Convert the percent rate to the factor \(b\).
- Write \(y=a\cdot b^{x}\) and check one known value.
Worked example. A town has 2,400 people and grows 3% per year. Then \(a=2400\), \(b=1.03\) and \(P(t)=2400(1.03)^{t}\). After 5 years, \(P(5)\approx 2,782\) people.
Does an exponential function always beat a linear one?
Yes, eventually. A line adds the same amount each step, but an exponential adds more each step. In the graph, \(y=2^{x}\) starts below \(y=10x+2\), yet it passes the line before \(x=6\) because \(2^{6}=64\) and \(10(6)+2=62\).
Exponential Growth and Decay: frequently asked questions
How can you tell whether a table is linear or exponential?
Check equal steps in \(x\). If the differences between outputs are equal, it is linear. If the ratios of consecutive outputs are equal, it is exponential. The outputs 3, 6, 12, 24 have a ratio of 2 each time.
What is the difference between a growth factor and a growth rate?
The rate is the percent change, such as 8%, written 0.08. The factor is what you multiply by, \(1+0.08=1.08\). Decay works the same way: a 20% loss has a factor of 0.80.
Why can b not be 1 or negative in an exponential function?
If \(b=1\), every output equals \(a\), so the function is constant. A negative \(b\) would make the outputs flip between positive and negative, which does not model steady growth or decay.
What does the number a mean in y = a · b^x?
It is the initial value, the output when \(x=0\), because \(b^{0}=1\). In \(y=250(0.9)^{x}\), the quantity starts at 250 and loses 10% each step.
Will an exponential function always exceed a linear function?
Eventually, yes. A line may stay ahead for a while, but each exponential step is larger than the last. For \(y=2^{x}\) and \(y=10x+2\), the exponential passes the line at \(x=6\).
How to use this algebra 1 exponential functions lesson
Use this algebra 1 exponential functions lesson before the worksheet, then check what you learned with the quiz.
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