From Factor to Rate and Mixed Interpretation

Lesson 2 of 2: Percent Rate and Full Sentences

In this lesson:

• Convert between growth/decay factor and percent rate
• Interpret both and in real-world contexts
• Write complete interpretive sentences for mixed problems

What You Will Learn Today

By the end of this lesson, you can:

5. Translate between growth/decay factor and percent rate
6. Write complete interpretive sentences: value, meaning, and units

Why Do We Need the Percent Rate?

— "the quantity is multiplied by 1.08 each year"

That's accurate. But which is clearer to most people?

• "The balance is multiplied by 1.08 each year."
• "The balance increases by 8% each year."

Both are correct. The percent rate is often more intuitive.

Growth and Decay Factor-to-Rate Formulas

Growth (): , so

Decay (): , so

Growth Example: Converting to

Interpretation: "The quantity increases by 8% per period."

Check: — that's a gain of , which is of . ✓

Decay Example: Converting to

Interpretation: "The quantity decreases by 15% per period."

Check: — that's a loss of , which is of . ✓

Quick Check: Convert These Factors

A: — what is the percent rate and type?

B: — what is the percent rate and type?

Use the formulas: for growth, for decay.

Check Answer: 4.5% Growth and 8% Decay

A: : increase

B: : decrease

These are the rates you state in your interpretation.

Watch Out: Means 8% Increase

Wrong: "The population grew by 108% each year."

Right: "The population grew by 8% each year."

If you grew by 108%, you'd have more than double in one year!

Check: — grew by , not .

Putting It All Together: Bank Account Model

— balance (dollars), years

Interpret : "The initial deposit was $1,200."

Interpret : "The balance is multiplied by 1.045 each year."

Interpret : "The account earns 4.5% interest per year."

Mixed Practice: Start Identifying Function Type

— monthly phone bill (dollars), minutes

This is a linear function ( form).

Interpret : "Each additional minute costs $0.15."

Interpret : "The base monthly charge is $45."

Mixed Practice: Exponential in Context

— magazine subscribers, years after 2015

This is an exponential function ( form).

Interpret : "There were 800 subscribers in 2015."

Interpret , : "The subscriber count decreases by 8% each year."

Comparison: Linear vs. Exponential Interpretations

Both functions have two parameters. The difference is how they change.

Guided Practice: Interpret

= years after 2020; = town population

Interpret each parameter:

1. What does mean in context?
2. What does mean?
3. What is the percent rate ? What does it mean?

Write complete sentences before advancing.

Answers: Interpreting All Three Parameters

— population, years after 2020

• : population in 2020 was 1,500
• : multiplied by 1.04 each year
• : grows 4% per year

Misconception: Linear Slope Is Not a Percent Rate

— does mean per period?

• at period 1
• at period 2

Fixed amount each step, not a fixed percent.

Check: Is 50 a Percent Rate Here?

Does 50 represent a 50% increase per period?

Think about what , , and tell you.

Answer: No — Fixed Amount, Not Fixed Percent

, ,

• Gain each period: always (fixed amount) ✓
• Percent each period: , then , then (not fixed) ✗

means "50 units per period" — an additive rate, not a percent rate.

Five Misconceptions to Watch Out For

1. No context — interpret direction, amount, units
2. means — not
3. y-intercept ≠ first coefficient — evaluate
4. Decay factor is positive (), not negative
5. Linear slope = fixed amount — not a fixed percent

Preview: Constructing and Modeling Functions

In the next lessons (HSF.BF.A.1):

• Write a linear or exponential function from a verbal description
• Choose which function type fits a given context
• Interpret the parameters you chose — applying today's skills

Interpretation is the final step of modeling — you've now built that skill.