The function \(\mathrm{B(m) = 118m + 195}\) gives the estimated balance, in dollars, of Sarah's savings account m months after...

1. TRANSLATE the function components

• Given: \(\mathrm{B(m) = 118m + 195}\)
• This represents the balance in dollars after m months
• We need to interpret what 195 means in this context

2. INFER the mathematical structure

• This is a linear function in the form \(\mathrm{y = mx + b}\)
• In \(\mathrm{B(m) = 118m + 195}\):
  • 118 is the coefficient (rate of change per month)
  • 195 is the constant term (y-intercept)

3. INFER what the y-intercept means

• The y-intercept occurs when the input variable equals 0
• When \(\mathrm{m = 0}\) (meaning 0 months after she started, or when she started):
  \(\mathrm{B(0) = 118(0) + 195}\)
  \(\mathrm{B(0) = 195}\)
• This means her account had $195 when she started her job

4. INFER which answer choice matches

• Looking at the choices, (B) states "The estimated balance was 195 dollars when she started her part-time job"
• This exactly matches our mathematical interpretation

Answer: B

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students confuse which number represents what in the linear function.

Many students see both 118 and 195 as dollar amounts and think "195 dollars each month" sounds reasonable for monthly savings. They don't connect the mathematical structure (coefficient vs constant term) to the real-world meaning, leading them to select Choice D (195 dollars each month).

Second Most Common Error:

Poor TRANSLATE reasoning: Students don't properly interpret what \(\mathrm{m = 0}\) means in the context.

Without clearly understanding that "m months after she started" means \(\mathrm{m = 0}\) is the starting point, students can't connect the constant term to the initial balance. This leads to confusion and random guessing between the remaining choices.

The Bottom Line:

Success requires connecting abstract function components to real-world timeline meaning - the constant term represents the "starting point" value when time = 0.