Recognizing Language and Writing Functions

Lesson 2 of 2: Classification and Function Writing

In this lesson:

• Recognize constant percent rate language in verbal descriptions
• Distinguish from constant rate (linear) descriptions
• Write and interpret from descriptions

What You Will Learn Today

1. Identify verbal signals for constant percent rate: percent, doubles, half-life
2. Distinguish constant rate from constant percent rate in paired descriptions
3. Write an exponential function from initial value and base
4. Interpret the initial value and base in context
5. Adjust the exponent for non-unit period lengths

The Key Signal: Percent or Multiplier Word

Constant percent rate signals: percent sign · "doubles" · "halves" · "half-life" · "factor of"

Pair One: Linear vs. Exponential City Growth

• "The city adds 500 residents per year." → fixed amount → linear

• "The city grows by 2% per year." → percent → exponential

Pair Two: Linear vs. Exponential Car Depreciation

• "Car loses $3,000 per year." → fixed amount → linear

• "Car loses 18% of its value per year." → percent → exponential

Both describe loss. The linear model eventually reaches $0; the exponential never does.

Pair Three: Viral Social Media Post

• "$1,000 in donations each month." → fixed amount → linear
• "Views double every day." → doubles → exponential

"Doubles" = 100% growth = factor 2.

, where = days.

Disguised Rates: Doubles, Triples, Halves

"Doubles every period" = 100% growth per period = factor 2

"Triples every period" = 200% growth per period = factor 3

"Halves every period" = 50% decay = factor 0.5 = "half-life of 1 period"

These phrases describe exponential behavior with NO percent sign visible.

Quick Check: Classify Three Descriptions

1. "A salary increases by $2,000 per year."
2. "A radioactive sample loses 8% per hour."
3. "A social media account gains 500 followers each month."

Classify each: linear or exponential? Identify the signal.

From Classification to Function Writing

Once you've classified, you need two pieces:

• = initial value (at time 0)
• = growth or decay factor (from percent rate)

Then write:

Verify: . If not, check your initial value.

Example One: Town Population Growth

"A town of 8,000 grows by 3% per year."

• (initial population)
• (3% growth → factor 1.03)
• , = years

Prediction: people.

Interpret Parameters a and b in Context

• : starting population (at )
• : each year, population is 1.03× the previous — 3% growth

Rate = .

Medication Half-Life: Period Adjustment in Practice

"Initial dose 200 mg, half-life of 4 hours."

• "Half-life of 4 hours" → halves every 4 hours → factor 0.5 per 4-hour period
• , = hours

Prediction: mg remaining.

Period Adjustment: Divide by Period Length

Factor applies every units → divide exponent by :

• "Doubles every 3 years" →
• "Half-life 6 hours" →

Check: at , value = (one full period).

Compound Interest: Growth Over Twenty Years

"A $2,000 account grows at 5% compounded annually."

• ;

— initial investment grows 2.65× in 20 years.

Quick Check: Write the Exponential Function

"A 500-gram sample decays at 20% per hour."

1. Identify and the decay factor .
2. Write with variable names.
3. Verify:
4. Predict:

Try before the next slide.

Practice: Write Five Exponential Functions

1. Town: 12,000 people, grows 4%/yr.
2. Car: $25,000 value, loses 15%/yr.
3. Bacteria: 1,000 colony, doubles/2 hr.
4. Sample: 800 mg, half-life 3 hr.
5. Account: $3,500 at 6%/yr.

Write (or adjusted). State and .

Answers to Five Exponential Function Problems

1. ; ,
2. ; ,
3. ; , , period 2 hr
4. ; , , period 3 hr
5. ; ,

Key Takeaways from Lesson Two

✓ Percent, "doubles," "half-life" → exponential; fixed amount → linear
✓ Initial value = ; growth/decay factor = in the function
✓ Non-unit period: divide exponent by period length

Factor ≠ rate: 1.06 means 6% growth, not 6%
"Doubles every 3 years" needs , not
Exponential decay never reaches zero

What Comes Next in Functions

You've completed HSF.LE.A.1.c.

Coming up:

• HSF.LE.A.2: Construct linear and exponential models from data
• HSF.LE.A.3: Observe exponential eventually exceeds linear
• HSF.LE.B.5: Interpret parameters in linear and exponential models