\(\mathrm{f(x) = (x - 44)(x - 46)}\) The function f is defined by the given equation. For what value of...

1. TRANSLATE the problem information

• Given information:
  • \(\mathrm{f(x) = (x - 44)(x - 46)}\)
  • Need to find the x-value where f(x) reaches its minimum

2. INFER the mathematical approach

• This is a quadratic function in factored form
• Since it's the product of two linear terms, when expanded it will be \(\mathrm{ax^2 + bx + c}\) with \(\mathrm{a \gt 0}\)
• This means the parabola opens upward and has a minimum value at its vertex
• Strategy: Find the x-coordinate of the vertex

3. INFER the most efficient method

• From the factored form (x - 44)(x - 46), we can see the zeros are at \(\mathrm{x = 44}\) and \(\mathrm{x = 46}\)
• For any parabola, the vertex lies exactly halfway between the zeros
• Midpoint = \(\mathrm{\frac{44 + 46}{2} = 45}\)

4. Verify using the vertex formula (optional)

• SIMPLIFY by expanding: \(\mathrm{f(x) = x^2 - 90x + 2024}\)
• APPLY vertex formula: \(\mathrm{x = \frac{-b}{2a} = \frac{-(-90)}{2 \times 1} = 45}\)

Answer: B. 45

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students recognize the zeros at \(\mathrm{x = 44}\) and \(\mathrm{x = 46}\) but think one of these must be the minimum point, not realizing the minimum occurs between the zeros.

This leads them to select Choice A (46) or Choice C (44) because these are the more obvious x-values from the factored form.

Second Most Common Error:

Inadequate SIMPLIFY execution: Students expand the function incorrectly or make arithmetic errors when using the vertex formula, leading to calculation mistakes.

This causes confusion and may lead to guessing among the available choices.

The Bottom Line:

The key insight is recognizing that for a quadratic function, the minimum (or maximum) never occurs at the zeros, but rather at the vertex, which is the midpoint between the zeros for a parabola.