Question:\(\mathrm{P(x) = -2x^2 + 150x + 45,000}\)The given function P models the monthly profit, in dollars, of a manufacturing company,...

1. TRANSLATE the question requirements

• We need to find the y-intercept of \(\mathrm{P(x) = -2x^2 + 150x + 45,000}\)
• The y-intercept occurs when \(\mathrm{x = 0}\)
• We need to interpret this value in the given context

2. INFER the mathematical approach

• To find the y-intercept, substitute \(\mathrm{x = 0}\) into the function
• The result will be the y-coordinate of the y-intercept

3. Calculate P(0)

• \(\mathrm{P(0) = -2(0)^2 + 150(0) + 45,000}\)
• \(\mathrm{P(0) = 0 + 0 + 45,000 = 45,000}\)

4. TRANSLATE the mathematical result to contextual meaning

• We found that when \(\mathrm{x = 0}\), \(\mathrm{P(x) = 45,000}\)
• Since x represents "units produced above baseline production level"
• When \(\mathrm{x = 0}\), the company is producing exactly at their baseline level
• Therefore, $45,000 is the monthly profit when producing at baseline level

Answer: B

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students misinterpret what \(\mathrm{x = 0}\) represents in context

Many students read "x represents the number of units produced above their baseline production level" and incorrectly think \(\mathrm{x = 0}\) means the company produces zero units total (no production at all). They miss that \(\mathrm{x = 0}\) actually means zero units above baseline, which means they're producing at the baseline level, not producing nothing.

This confusion leads them to think the y-intercept represents something like fixed costs when no production occurs, potentially selecting Choice D ($45,000 in fixed costs).

Second Most Common Error:

Poor INFER reasoning: Students confuse y-intercept with maximum value

Some students see the large coefficient (45,000) and assume this must be the maximum profit without actually analyzing what the y-intercept means. Since this is a downward-opening parabola, the maximum occurs at the vertex, not the y-intercept.

This may lead them to select Choice A (maximum monthly profit is $45,000).

The Bottom Line:

Success on this problem requires carefully translating the variable definition (what \(\mathrm{x = 0}\) actually means) rather than just finding the mathematical y-intercept. The key insight is understanding that "above baseline" means \(\mathrm{x = 0}\) represents baseline production, not zero production.