\(\mathrm{y = 2(-x^2 + 8x + 25)}\) The given equation models a small business's monthly profit over 10 months, where...

1. TRANSLATE the question requirements

• We need to find the y-intercept of \(\mathrm{y = 2(-x^2 + 8x + 25)}\)
• Y-intercept means the point where the graph crosses the y-axis
• This happens when \(\mathrm{x = 0}\)

2. SIMPLIFY by substituting x = 0

• Substitute \(\mathrm{x = 0}\) into the equation:
  \(\mathrm{y = 2(-0^2 + 8(0) + 25)}\)
• Simplify inside parentheses first:
  \(\mathrm{y = 2(0 + 0 + 25)}\)
  \(\mathrm{y = 2(25)}\)
  \(\mathrm{y = 50}\)

3. TRANSLATE the mathematical result back to context

• \(\mathrm{y = 50}\), but y represents profit in hundreds of dollars
• Convert to actual dollars: \(\mathrm{50 \times \$100 = \$5000}\)
• The context states x is "months after the end of March 2023"
• When \(\mathrm{x = 0}\), we're at 0 months after March's end = end of March 2023

Answer: (B) At the end of March 2023, the profit was $5000.

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students misinterpret the time reference point, thinking \(\mathrm{x = 0}\) represents April instead of March.

The problem states "x months after the end of March 2023." Students might think this means \(\mathrm{x = 0}\) corresponds to April, since it's the first month after March ends. However, "0 months after" means we're still at the reference point (end of March), not one month later.

This may lead them to select Choice (D) (At the end of April 2023, the profit was $5000)

Second Most Common Error:

Poor TRANSLATE execution: Students forget that y is in hundreds of dollars and report \(\mathrm{y = 50}\) as $50 instead of $5000.

They correctly find \(\mathrm{y = 50}\) and correctly identify the time as end of March, but miss the unit conversion requirement stated in the problem.

This may lead them to select Choice (A) (At the end of March 2023, the profit was $0) because none of the choices show $50, causing confusion and potentially guessing toward the $0 option.

The Bottom Line:

This problem tests careful reading of context more than mathematical calculation. The arithmetic is straightforward, but students must accurately interpret both the time reference and the units to avoid the trap answers.