Question:\(\mathrm{g(x) = -x^2 + 36x + 76}\)What is the maximum value of the given function?183244001048

1. INFER the problem strategy

• Given: \(\mathrm{g(x) = -x^2 + 36x + 76}\)
• Since the coefficient of \(\mathrm{x^2}\) is negative (-1), this parabola opens downward
• For downward-opening parabolas, the maximum value occurs at the vertex
• Strategy: Find vertex coordinates, then evaluate the function at that point

2. SIMPLIFY to find the vertex x-coordinate

• Use vertex formula: \(\mathrm{x = -\frac{b}{2a}}\) where \(\mathrm{a = -1, b = 36}\)
• \(\mathrm{x = -\frac{36}{2(-1)}}\)
• \(\mathrm{x = -\frac{36}{-2}}\)
• \(\mathrm{x = 18}\)
• The vertex occurs at \(\mathrm{x = 18}\)

3. SIMPLIFY to find the maximum value

• Evaluate \(\mathrm{g(18)}\):
• \(\mathrm{g(18) = -(18)^2 + 36(18) + 76}\)
• \(\mathrm{g(18) = -324 + 648 + 76}\)
• \(\mathrm{g(18) = 324 + 76 = 400}\)

Answer: C) 400

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students find the correct x-coordinate of the vertex (\(\mathrm{x = 18}\)) but think this IS the maximum value, rather than understanding they need to evaluate the function at this point.

This leads them to select Choice A (18) instead of evaluating \(\mathrm{g(18)}\) to get the actual maximum.

Second Most Common Error:

Poor SIMPLIFY execution: Students make sign errors when computing the vertex formula or arithmetic mistakes when evaluating \(\mathrm{g(18)}\), particularly with the negative terms.

Common mistake: Computing \(\mathrm{-(18)^2 + 36(18) + 76}\) as \(\mathrm{+324 + 648 + 76 = 1048}\) (forgetting the negative sign), leading them to select Choice D (1048).

The Bottom Line:

This problem tests whether students understand the difference between the x-coordinate where the maximum occurs versus the actual maximum value of the function. Many students stop halfway through the solution process.