\(\mathrm{f(x) = 8x + 4}\). The function f gives the estimated height, in feet, of a willow tree x years...

1. TRANSLATE the problem information

• Given information:
  • \(\mathrm{f(x) = 8x + 4}\)
  • \(\mathrm{f(x)}\) = estimated height in feet
  • \(\mathrm{x}\) = years after height was first measured

• What this tells us: We have a linear function where we need to interpret what the constant 4 means in context.

2. INFER the mathematical structure

• This is a linear function in the form \(\mathrm{f(x) = mx + b}\)
• Here: \(\mathrm{m = 8}\) (slope) and \(\mathrm{b = 4}\) (y-intercept)
• The y-intercept occurs when \(\mathrm{x = 0}\)

3. INFER what x = 0 means in context

• Since x represents "years after height was first measured"
• \(\mathrm{x = 0}\) means "0 years after first measurement" = when it was first measured
• So \(\mathrm{f(0)}\) gives us the height when the tree was first measured

4. Calculate f(0)

• \(\mathrm{f(0) = 8(0) + 4 = 4}\)
• This means the estimated height was 4 feet when first measured

Answer: D. The estimated height of the tree was 4 feet when it was first measured.

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Confusing slope with y-intercept meanings in context

Students often think "4 feet" relates to growth rather than initial height. They see the number 4 and incorrectly associate it with the rate of change, leading them to think the tree grows 4 feet each year.

This may lead them to select Choice C (The estimated height of the tree increased by 4 feet each year)

Second Most Common Error:

Poor TRANSLATE reasoning: Misunderstanding what \(\mathrm{x = 0}\) represents

Students may not clearly connect that "x years after first measurement" means \(\mathrm{x = 0}\) is the time of first measurement. Without this connection, they cannot properly interpret what the y-intercept represents in context.

This leads to confusion and guessing among the remaining choices.

The Bottom Line:

Linear function interpretation requires understanding both the mathematical structure (slope vs y-intercept) and what the input variable represents in context. The key insight is recognizing that the y-intercept always tells us the output value when the input is zero.