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Introduction
Exponential Decay helps you connect a mathematical rule to a result you can explain and check. Write decay with a factor strictly between 0 and 1. The percent lost and the fraction remaining are different quantities. This lesson belongs to exponential functions and builds toward independent Algebra I problem solving.
Alignment uses 19 TAC §111.39(c). Review lessons sample their listed standards. Readiness and supporting labels describe STAAR emphasis, not whether a topic is required.
Learning objectives
- Identify the quantities and representations used in exponential decay.
- Apply the lesson's rule to solve a numerical or contextual problem.
- Explain each mathematical step using the correct vocabulary.
- Verify an answer and correct a likely error.
Prerequisite knowledge
Evaluate powers and convert percentages to decimals.
Previous lesson: Exponential Growth
Key mathematical vocabulary
- constant
- A value fixed within a problem.
- domain
- The set of permitted inputs.
- range
- The set of outputs produced.
- exponent
- A number specifying a power.
- factor
- An expression multiplied by another to form a product.
Formulas and rules
- f(x)=abx, b>0, b≠1
- Growth factor=1+r; decay factor=1−r
- A=P(1+r/n)nt
Exponential growth and decay
Write decay with a factor strictly between 0 and 1. The percent lost and the fraction remaining are different quantities.
An exponential model has the form f(t)=abt, where b>0 and b≠1 for a nonconstant exponential. The initial value is a=f(0). Each increase of 1 in t multiplies the output by b rather than adding a constant.
For a positive initial amount, b>1 models growth and 0<b<1 models decay. A rate r written as a decimal gives growth factor 1+r or decay factor 1−r. Compound interest is A=P(1+r/n)nt, with compatible units. For a≠0, the mathematical domain is all real numbers; the sign of a determines whether the range is y>0 or y<0. Context may restrict time to t≥0.
A method you can explain
- Read the question and identify the known information, unknown quantities, and any domain restrictions. Select a suitable representation and tool.
- Apply the relationship explained above. The initial amount is 420; the multiplicative factor is 0.5.
- Write each calculation or logical inference on a separate line. Keep units and signs visible. When two methods are possible, compare which one uses the given information most directly.
- Check the result in the ORIGINAL conditions, not merely the last simplified line. Divide by the accumulated factor to recover the initial amount.
- Interpret what the answer means. Decide whether its size, sign, and precision make sense; identify a restriction or assumption that would change the conclusion.
Five worked examples
Example 1 · Beginner
For f(t)=420(0.5)t, find f(3) and identify growth or decay.
- The initial amount is 420; the multiplicative factor is 0.5.
- After 3 periods, multiply by 0.5 a total of 3 times: 420·0.53.
- 0.53=0.125; the output is 52.5. A factor between 0 and 1 means decay.
Answer: 52.5; decay
Verify: Divide by the accumulated factor to recover the initial amount.
Avoid this mistake: Using the percent rate itself as the growth factor.
Example 2 · Basic
For f(t)=430(0.5)t, find f(3) and identify growth or decay.
- The initial amount is 430; the multiplicative factor is 0.5.
- After 3 periods, multiply by 0.5 a total of 3 times: 430·0.53.
- 0.53=0.125; the output is 53.75. A factor between 0 and 1 means decay.
Answer: 53.75; decay
Verify: Divide by the accumulated factor to recover the initial amount.
Avoid this mistake: Adding the same amount each period in an exponential model.
Example 3 · Intermediate
For f(t)=440(0.5)t, find f(3) and identify growth or decay.
- The initial amount is 440; the multiplicative factor is 0.5.
- After 3 periods, multiply by 0.5 a total of 3 times: 440·0.53.
- 0.53=0.125; the output is 55. A factor between 0 and 1 means decay.
Answer: 55; decay
Verify: Divide by the accumulated factor to recover the initial amount.
Avoid this mistake: Including zero in the range of an unshifted nonzero exponential.
Example 4 · Advanced
A lab starts with 330 cells. The population halves each hour. How many are predicted after 4 hours?
- The initial amount is 330; the multiplicative factor is 0.5.
- After 4 periods, multiply by 0.5 a total of 4 times: 330·0.54.
- 0.54=0.0625; the output is 20.625. A factor between 0 and 1 means decay.
Answer: 20.625; decay
Verify: Divide by the accumulated factor to recover the initial amount.
Avoid this mistake: Using the percent rate itself as the growth factor.
Example 5 · Challenge
For f(t)=340(0.5)t, find f(4) and identify growth or decay.
- The initial amount is 340; the multiplicative factor is 0.5.
- After 4 periods, multiply by 0.5 a total of 4 times: 340·0.54.
- 0.54=0.0625; the output is 21.25. A factor between 0 and 1 means decay.
Answer: 21.25; decay
Verify: Divide by the accumulated factor to recover the initial amount.
Avoid this mistake: Adding the same amount each period in an exponential model.
Common mistakes and corrections
- Using the percent rate itself as the growth factor.
- Adding the same amount each period in an exponential model.
- Including zero in the range of an unshifted nonzero exponential.
Real-world applications
- Model a population with a constant percent change. Define the input and its units, apply the mathematical relationship, and check that the result fits the setting.
- Predict a repeated depreciation or compound-interest balance. Define the input and its units, apply the mathematical relationship, and check that the result fits the setting.
Topic summary
Write decay with a factor strictly between 0 and 1. The percent lost and the fraction remaining are different quantities.
An exponential model has the form f(t)=abt, where b>0 and b≠1 for a nonconstant exponential. The initial value is a=f(0). Each increase of 1 in t multiplies the output by b rather than adding a constant.
Frequently asked questions
What should I know before exponential decay?
Evaluate powers and convert percentages to decimals.
How can I check my work?
Divide by the accumulated factor to recover the initial amount. A numerical check can detect an error; when claiming an identity for all inputs, also explain the algebraic property that makes it true.
Which mistakes should I watch for?
Using the percent rate itself as the growth factor. Adding the same amount each period in an exponential model. Including zero in the range of an unshifted nonzero exponential.
Can I use another valid method?
Yes. Explain your assumptions, show enough intermediate work to justify the result, and verify it in the original problem. Equivalent expressions and valid alternative methods receive credit.
Mathematical process practice · A.1A–A.1G
- A.1A: Model a population with a constant percent change.
- A.1B: Plan, solve, check, and interpret the written response.
- A.1C: Select paper, a calculator, a table, or a graph; explain the choice.
- A.1D: Connect symbols with a verbal explanation and another representation.
- A.1E: Organize the solution with a table, diagram, or labeled equations.
- A.1F: Explain how the representation and the algebraic rule express the same relationship.
- A.1G: Justify each operation and the reasonableness of the conclusion.
Quick revision notes
- Write decay with a factor strictly between 0 and 1. The percent lost and the fraction remaining are different quantities.
- Using the percent rate itself as the growth factor.
- Adding the same amount each period in an exponential model.
- Including zero in the range of an unshifted nonzero exponential.
Practice and explain
Try all four multiple-choice questions, then show your reasoning in the written response. Objective scores and AI feedback assess different skills.
Continue learning
Original Cambridge Veritas practice. These are not official STAAR items. Cambridge Veritas is not affiliated with or endorsed by TEA.
Official Algebra I TEKS (§111.39) · TEA assessed curriculum · STAAR blueprint · TEA instructional materials