BetterGrades Precalculus · Unit 7 · Lesson

Growth and decay models

Construct exponential models from an initial value, rate, multiplier, two points, or a known doubling or half-life.

Opening

Start with the situation

Exponential models can be written from an initial value and rate, multiplier, doubling time, or half-life.

Multiplicative models describe repeated percentage change, while logarithms recover the time or exponent hidden inside that process. Together they support growth, decay, finance, regression, and bounded models.

Before you begin

Prerequisite check

Use exponent laws.
Interpret function parameters.
Distinguish exact and approximate values.

Core explanation

Explanation

Choose the form matching the information, state the time unit and domain, and interpret the repeated multiplier.

Growth ignores resource limits and decay approaches zero without becoming negative.

A symbolic answer is not complete by itself. In this lesson, the same claim must also be readable through model-form selector, doubling-half-life timelines, or another equivalent representation.

Conceptual reading

What the idea is really doing

Exponential change multiplies over equal input steps, while logarithms answer the inverse question: what exponent produces a given output? Parameters must be interpreted as an initial value, a multiplier, a rate, or a long-run bound—not as decoration.

This lesson narrows that lens to one goal: construct exponential models from an initial value, rate, multiplier, two points, or a known doubling or half-life. The point is not to memorize an isolated trick; it is to know what evidence makes the conclusion valid and how a second representation can check it.

Reusable method

A reliable route through the problem

Choose the form matching the information.
State the time unit.
Domain.
Interpret the repeated multiplier.

Verification: Test the model at input zero and one step later, confirm the multiplier or inverse relationship, and state whether the domain and long-run behavior make sense in context.

Foundation walkthrough

Plan before calculating

Problem

$240$ mg with half-life $6$ hours.

Plan: Start by identifying the mathematical structure in the prompt. Then use the lesson method rather than guessing from appearance: Choose the form matching the information, state the time unit and domain, and interpret the repeated multiplier.
Conclusion: $A(t)=240(\frac{1}{2})^(\frac{t}{6})$
Why the check works: Each six-hour interval halves the amount.

Worked examples

See the idea in three forms

foundation example

$240$ mg with half-life $6$ hours.

Solution $A(t)=240(\frac{1}{2})^(\frac{t}{6})$

Each six-hour interval halves the amount.

representation example

$900$ decays $4\%$ .

Solution $900(0.96)^t$

This example expresses growth and decay models in a second form.

transfer example

$50$ doubles every $8$ days.

Solution $50\cdot 2^(\frac{t}{8})$

Growth ignores resource limits and decay approaches zero without becoming negative.

Model-form selector. Follow the foundation example from its given information to the conclusion. The labels identify the mathematical feature that makes the result valid: Each six-hour interval halves the amount. — **Anchor figure · Model-form selector**
Follow the foundation example from its given information to the conclusion. The labels identify the mathematical feature that makes the result valid: Each six-hour interval halves the amount.

Doubling-half-life timelines. Read the numbered reasoning path in order. Each stage preserves the quantities, restrictions, or structural conditions needed for growth and decay models. — **Mechanism figure · Doubling-half-life timelines**
Read the numbered reasoning path in order. Each stage preserves the quantities, restrictions, or structural conditions needed for growth and decay models.

Model-domain warning. Compare the valid path with the tempting shortcut. The figure shows why mixing time units or using 6 instead of 1.06 for 6 percent growth leads to a false conclusion. — **Comparison and error figure · Model-domain warning**
Compare the valid path with the tempting shortcut. The figure shows why mixing time units or using $6$ instead of $1.06$ for $6$ percent growth leads to a false conclusion.

Common mistake

Find the first invalid move

A frequent error is mixing time units or using $6$ instead of $1.06$ for $6$ percent growth.

Check yourself

$1200$ grows $6\%$ .

Write a complete attempt before opening the exact answer.

Attempt once to unlock the answer

Complete a substantive attempt before revealing the server-held answer.

Practice

Ten concrete questions

Practice 101

$1200$ grows $6\%$ .

Write a complete attempt before opening the exact answer.

Attempt once to unlock the answer

Complete a substantive attempt before revealing the server-held answer.

Practice 202

$900$ decays $4\%$ .

Write a complete attempt before opening the exact answer.

Attempt once to unlock the answer

Complete a substantive attempt before revealing the server-held answer.

Practice 303

$50$ doubles every $8$ days.

Write a complete attempt before opening the exact answer.

Attempt once to unlock the answer

Complete a substantive attempt before revealing the server-held answer.

Practice 404

After one half-life.

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Practice 505

Explain why this conclusion is valid: $A(t)=240(\frac{1}{2})^(\frac{t}{6})$ . Use the foundation problem as evidence: $240$ mg with half-life $6$ hours.

Write a complete attempt before opening the exact answer.

Attempt once to unlock the answer

Complete a substantive attempt before revealing the server-held answer.

Practice 606

Solve the representation example, then name the feature of growth and decay models that it illustrates: $900$ decays $4\%$ .

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Attempt once to unlock the answer

Complete a substantive attempt before revealing the server-held answer.

Practice 707

Correct this reasoning and identify the first unsafe assumption: A frequent error is mixing time units or using $6$ instead of $1.06$ for $6$ percent growth.

Write a complete attempt before opening the exact answer.

Attempt once to unlock the answer

Complete a substantive attempt before revealing the server-held answer.

Practice 808

Connect two representations for this example: $240$ mg with half-life $6$ hours. Describe what a graph, table, mapping, or algebraic form would have to show.

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Practice 909

Create a nearby example by changing one number or condition in this prompt: $50$ doubles every $8$ days. Predict the effect, solve your new example, and compare it with the original.

Write a complete attempt before opening the exact answer.

Attempt once to unlock the answer

Complete a substantive attempt before revealing the server-held answer.

Practice 1010

Write a short verification checklist for growth and decay models, then apply it to one worked example from this lesson.

Write a complete attempt before opening the exact answer.

Attempt once to unlock the answer

Complete a substantive attempt before revealing the server-held answer.

Lesson close

Connect forward

The next lesson, Compound interest and effective rate, uses this result as part of a larger structure.

Source record

Original BetterGrades manuscript, rights-separated references.

Yoshiwara, Modeling, Functions, and Graphs
Lippman and Rasmussen, Precalculus Volume 1
Stitz and Zeager, Precalculus

No long source passage is reproduced.

240240240 mg with half-life 666 hours.

240240240 mg with half-life 666 hours.

900900900 decays 4%4\%4%.

505050 doubles every 888 days.

Original BetterGrades manuscript, rights-separated references.

$240$ mg with half-life $6$ hours.

$240$ mg with half-life $6$ hours.

$900$ decays $4\%$ .

$50$ doubles every $8$ days.