The function f defined by \(\mathrm{f(t) = 14t + 9}\) gives the estimated length, in inches, of a vine plant...

1. TRANSLATE the function components

• Given: \(\mathrm{f(t) = 14t + 9}\) represents vine length after t months
• This is a linear function in the form \(\mathrm{f(t) = mt + b}\) where:
  • 14 is the coefficient of t
  • 9 is the constant term

2. INFER what the constant represents

• In any linear function \(\mathrm{f(t) = mt + b}\), the constant b tells us the value when \(\mathrm{t = 0}\)
• Since t represents months after purchase, \(\mathrm{t = 0}\) means "at the time of purchase"
• Therefore: \(\mathrm{f(0) = 14(0) + 9 = 9}\) inches

3. Connect to the context

• The 9 represents the vine's length when Tavon first bought it
• The 14 represents how much it grows each month

Answer: D. The estimated length of the vine plant was 9 inches when Tavon purchased it.

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students confuse the growth rate with the initial value

They see both numbers (14 and 9) but don't systematically think about which represents what. They might think "the plant grows 9 inches per month" because 9 is prominently mentioned in the question.

This may lead them to select Choice B (The vine plant is expected to grow 9 inches each month)

Second Most Common Error:

Poor TRANSLATE reasoning: Students don't connect \(\mathrm{t = 0}\) to "when purchased"

They understand that 9 is the constant but don't realize that \(\mathrm{t = 0}\) corresponds to the purchase time. Without this connection, they can't properly interpret what the 9 means in context.

This leads to confusion and guessing among the remaining choices.

The Bottom Line:

This problem tests whether students understand that the constant term in a linear function represents the starting value, not just abstract mathematical knowledge but its real-world meaning when \(\mathrm{t = 0}\).