Question:Let \(\mathrm{C(d) = 0.25d + 4.25}\). The function C gives the total cost, in dollars, of renting an electric scooter...

1. TRANSLATE the function components

• Given function: \(\mathrm{C(d) = 0.25d + 4.25}\)
• This is in the linear form \(\mathrm{y = mx + b}\) where:
  • 0.25 is the coefficient of d (the slope)
  • 4.25 is the constant term (the y-intercept)

2. INFER what the constant term means

• In any linear function \(\mathrm{y = mx + b}\), the constant b represents the y-value when x = 0
• For our function: when \(\mathrm{d = 0}\), \(\mathrm{C(0) = 0.25(0) + 4.25 = 4.25}\)
• This means $4.25 is the cost when zero minutes are used

3. TRANSLATE this mathematical meaning into context

• $4.25 when zero minutes are used = the upfront cost before any riding time
• This is like a base fee or startup charge for using the scooter
• The 0.25 represents the additional cost per minute of riding

4. APPLY CONSTRAINTS to eliminate wrong interpretations

• Choice (A): Claims $4.25 is the per-minute increase, but that's what 0.25 represents
• Choice (B): Correctly describes 0.25, not 4.25
• Choice (D): Claims $4.25 is the cost after 1 minute, but \(\mathrm{C(1) = 4.50}\), not 4.25

Answer: C

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students mix up which number represents the rate versus the initial amount.

They see both 0.25 and 4.25 and don't carefully track which coefficient goes with which part of the context. Since 4.25 is the larger number, they might assume it must be the "per minute" charge, leading them to select Choice (A) ($4.25 per minute).

Second Most Common Error:

Inadequate INFER reasoning: Students don't connect the constant term to "when d = 0."

They recognize that 0.25 is the per-minute rate but can't figure out what 4.25 represents in context. Without understanding that the constant term equals the cost at \(\mathrm{d = 0}\), they might guess or select Choice (D) thinking 4.25 must happen "after some time" rather than "before any time."

The Bottom Line:

Linear function interpretation requires carefully matching each mathematical component (slope vs y-intercept) to its contextual meaning (rate vs initial value). The key insight is that the constant term always represents what happens when the input variable equals zero.