Unit 5 · FunctionsF5 · Exponentials & Logs · Guidance

Exponential Growth & Decay

Build and interpret exponential models from percent changes and repeated factors.

30 min
Unlocks: Burn Scheduler
Desmos way

Ready when you are

7 steps · about 30 minutes

Start with the warm-up.

Start

Step 1 of 7Warm-upDone

Warm-up

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Key terms

growth factor: $1+r$ for growth rate $r$ written as a decimal

Step 2 of 7Try it firstDone

Try it first

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Try this before the explanation. Commit to a setup, even if you are unsure.

Step 3 of 7LearnDone

The big idea

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The decision that matters

The initial value is at exponent 0; the factor repeats once per period. Write that decision before calculating. It turns a crowded ACT stem into a familiar structure and makes the likely distractors easier to spot.

Key idea

Structure before arithmetic

The initial value is at exponent 0; the factor repeats once per period. Keep exact values through the setup; round only when the stem asks.

Core relationship

A(t)=A_0(1+r)^t

Use the formula only after matching every symbol to the diagram or equation.

Algebra or Desmos?

Build the repeated multiplier before using a calculator: $1+r$ for growth and $1-r$ for decay. Desmos then evaluates the model or shows its value at a specified time.

Explore in Desmos

Evaluate three periods of 15% growth

Type the model and x=3, then click the intersection to read the amount.

The calculator is live: change anything and see what happens.

Evaluate three periods of 15% growth

Degree mode

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Trap

The main trap

A 12% decrease uses

0.88

, not

0.12

Step 4 of 7Worked examplesDone

Worked examples

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Skip check · 2 questions

Already know this? Get both right on the first try and you can skip the worked examples.

Worked example

Example 1

Problem

A culture starts at 800 organisms and grows 15% per hour. About how many organisms are present after 3 hours?

Use factor

1.15

Which multiplier represents retaining 100% and adding 15% each hour?

A\approx1,217

organisms.

Why is the factor $1.15$ raised to the third power?

Worked example

Example 2: ACT twist

Problem

A car worth

24{,}000

loses 18% of its value each year. About what is it worth after 4 years?

Use decay factor

0.82

Which yearly multiplier represents an 18% loss?

V\approx10,851

Why does the model use $0.82$ rather than $0.18$ ?

Faded example · you finish it

Example 3: You finish it

Problem

A town of 12,000 grows 4% annually. About what is its population after 5 years?

12000(1.04)^5

Complete the calculation.

Desmos way

Read the culture size after 3 hours

Type the model and x=3, then click the intersection to read the amount.

Read the culture size after 3 hours

Degree mode

Loading Desmos…

Step 5 of 7PracticeDone

Practice

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Step 6 of 7Spot the mistakeDone

Spot the mistake

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Spot the mistake

Where did it go wrong?

One line has an error. Click the line where it first goes wrong.

Step 7 of 7Exit ticketDone

Exit ticket

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Optional · 4 questions · Hardest levelChallenge set: the kind of question that decides a 36

These are last-ten-questions hard. Take your time, use Desmos when it helps, and expect to miss some: that's where the learning is.