From Description to Function: Linear Modeling

Lesson 2 of 2: Writing Functions and Evaluating Models

In this lesson:

• Write linear functions from constant rate descriptions
• Interpret rate and initial value with correct units
• Evaluate whether a constant rate model is reasonable

What You Will Learn Today

1. Extract slope and y-intercept from a verbal description
2. Write a linear function with correct variable names and units
3. Verify the function using the initial condition
4. State the contextual domain
5. Evaluate whether the constant rate assumption is reasonable

Two-Step Strategy: Rate Then Initial Value

Step 1 — Find the rate (slope ): Look for "per/each/every + amount." This is .

Step 2 — Find the initial value (): Look for the starting amount, fee, deposit, or value at .

Then write: and verify .

Example One: The Plumber Problem

"A plumber charges a $75 service fee plus $50 per hour."

• Rate: $50 per hour →
• Initial value: $75 fee →
• Function: , = hours, = cost (dollars)

Verify and Check the Plumber Function

Verify with initial condition: ✓

This is the service fee — the cost before any hours of work.

Check a value: — fee plus 2 hours of labor. ✓

Example Two: The Draining Tank

"A 500-gallon tank drains at 8 gallons per minute."

• Rate: 8 gal/min draining →
• Initial value: 500 gal →
• ; domain

Negative Rates: Drops, Loses, Decreases

"Drops," "loses," "decreases," "falls," "drains" → negative slope

• "Water drops 3 in/hr" →
• "Value falls $200/month" →
• "Temperature decreases 4°/hr" →

Check: after 1 unit, . If not, fix the sign.

Example Three: No Initial Value

"A phone plan costs $40 per month with no activation fee."

• Rate: $40 per month →
• Initial value: no fee →
• Function:

— passes through the origin. ✓

Quick Check: Write the Function

"A parking garage charges $3 per hour with a $5 flat fee."

1. Rate: $m = $ ?
2. Initial value: $b = $ ?
3. Write with units.
4. Verify:

Try before the next slide.

Practice: Write Three Linear Functions

1. Taxi: $2.50/mile + $3 boarding fee.
2. Barrel: 120 liters, drains 4 L/hr.
3. Gym: $25/month, no enrollment fee.

Write each function, state the domain, verify .

Pause before the next slide.

Answers: Three Linear Function Problems

1. ; = miles; ✓
2. ; domain ; ✓
3. ; = months; ✓

From Writing Models to Evaluating Them

You can now build a linear function from any constant rate description.

The next question: is the model actually reasonable?

A linear model assumes the rate is truly constant. In reality, rates often change.

Every Linear Model Is a Simplification

A constant rate model is valid when:

• The rate is physically controlled (factory output, utility billing)
• The time period is short relative to feedback effects
• The domain is limited to where the assumption holds

Ask: over what range of inputs is this model reasonable?

Candle Example: Domain Limits the Model

"A candle burns at 2 cm per hour."

, where = initial candle length

• Valid for the first few hours? Likely yes.
• Valid past the candle's full length? No — the candle has a finite length.
• Domain:

Population Growth: When Linear Falls Short

"A city grows by 1,000 people per year."

• Reasonable for 5 years? Perhaps.
• Reasonable for 100 years? Likely not — growth may accelerate or stall.

A constant percent growth model may be more realistic long-term.

When Linear Models Break Down

Linear models fail when the rate is NOT truly constant:

• Feedback: rate increases as quantity grows
• Physical limits: tank can't overflow; temp can't fall below zero
• Changing conditions: speed varies with traffic or grade

Ask: is constant rate reasonable for this context?

Quick Check: Write and Evaluate a Model

"A car depreciates by $2,000 per year from an initial value of $24,000."

1. Write — include slope, intercept, variable names.
2. State the domain.
3. Is the constant rate assumption reasonable? For how long?

Think before the next slide.

Key Takeaways from Lesson Two

✓ Rate → slope; starting value → y-intercept; write the linear function
✓ Decreasing context → negative slope
✓ Verify initial value; state the domain

Omitting the y-intercept ignores the initial value
Decreasing context → slope is NEGATIVE
Linear models assume constant rate — evaluate the domain

What Comes Next in Functions

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

Coming up:

• HSF.LE.A.1.c: Recognize constant percent rate situations
• HSF.LE.A.2: Construct linear and exponential models from data