Exponential Functions: Growth, Decay, Rates

Lesson 1 of 2

In this lesson:

• Reading growth and decay directly from the base
• Rewriting exponential expressions to reveal different time-scale rates

What You Will Learn Today

By the end of this lesson:

1. Identify initial value and factor in
2. Find percent rate: (growth) or (decay)
3. Classify as growth or decay from base
4. Convert time scales using

The Base Tells the Story

In :

• = initial value (value at )
• = growth or decay factor per unit of

If : exponential growth

If : exponential decay

If : no change (constant function)

Extracting the Percent Rate from b

• Growth: → rate
• Decay: → rate

Standard Examples from the CCSS

Both assume initial value (or 100% of starting amount).

Reading an Investment Growth Function

• Initial: → $5,000 starting investment
• Factor: → multiplies by each year
• Rate: per year

"Grows by 8% annually from a $5,000 start."

Reading a Car Depreciation Function

• Initial: $28,000 purchase price
• Factor: → remains each year
• Decay rate: per year

"Loses 15% of its value each year."

Quick Check: Classify and Find the Rate

Where is population and is years.

Identify initial value, classify as growth or decay, and state the percent rate.

Quick Check Verified: 3.5% Annual Growth

• Initial value: → population is 3,000
• Base: → exponential growth
• Rate: per year

"The population grows by 3.5% per year, starting at 3,000."

What If t Is Not in Years?

Some exponential functions use units other than years:

• : in years, but rate is per month (12 months/year)
• : in years, but rate is per decade (10 years)

Challenge: what is the per-year rate?

Tool: Use the exponent rule

Converting Monthly Rate to Annual Rate

Use in reverse:

Annual rate: — not 12%!

Why 1% Monthly Does Not Equal 12% Annually

If 1% monthly were simply additive:

Naive: interest → balance

With actual compounding:

The extra 0.68 is interest earned on interest — compounding.

Converting Decade Rate to Annual Rate

Rewrite:

Annual rate: per year

"20% per decade averages to only about 1.84% per year."

Daily Rate to Annual Rate

Rewrite:

"0.01% daily → not 3.65% annually, but 3.77% annually."

Quick Check: Convert This Rate

Rewrite in the form and state the annual growth rate.

Quick Check Verified: 6.17% Annual Rate

Annual rate:

"0.5% per month is approximately 6.17% per year, not 6%."

Compounding vs. Naive: The Gap Always Grows

Higher rates → bigger gap between naive and actual.

Practice: Classify and Convert These Functions

1. — classify and state rate
2. — rewrite as annual form and state rate
3. — rewrite as annual form and state rate

Apply the classification rule and the exponent conversion for each.

Practice Answers: Rates Found and Converted

1. : decay, per year; initial 2000

2. : ; annually

3. : ; decay/year

Lesson 1 Summary: Base and Time-Scale Rules

Classification: → growth; → decay

Rate extraction: growth rate ; decay rate

Time-scale conversion: ; isolate first

Factor ≠ rate. Compounding makes rates non-additive.

Coming Up: Complex Forms and Full Interpretation

In Lesson 2, you will:

• Rewrite exponential expressions with negative and fractional exponents
• Interpret half-life and doubling-time expressions
• Write complete contextual interpretations of all parameters

Next: what does the base of a complex expression reveal?