An object is thrown from a platform. The equation \(\mathrm{h = -16(t - 2)^2 + 128}\) represents the height h...

1. INFER what the problem is asking

• The question asks for 'the height from which the object was thrown'
• This means we need the height at the very beginning - when \(\mathrm{t = 0}\) seconds
• The equation \(\mathrm{h = -16(t - 2)^2 + 128}\) gives height at any time t

2. SIMPLIFY by substituting \(\mathrm{t = 0}\)

• Substitute \(\mathrm{t = 0}\) into the equation:
  \(\mathrm{h = -16(0 - 2)^2 + 128}\)
• Work inside the parentheses first:
  \(\mathrm{h = -16(-2)^2 + 128}\)
• Calculate the exponent: \(\mathrm{(-2)^2 = 4}\)
  \(\mathrm{h = -16(4) + 128}\)
• Multiply: \(\mathrm{-16(4) = -64}\)
  \(\mathrm{h = -64 + 128}\)
• Add: \(\mathrm{h = 64}\)

Answer: B) 64

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students don't recognize that 'height from which object was thrown' means the initial height at \(\mathrm{t = 0}\). Instead, they might think the answer is the maximum height of 128 feet, which occurs at the vertex when \(\mathrm{t = 2}\).

This reasoning makes sense to them because 128 is prominently displayed in the equation and represents the highest point the object reaches. However, the platform height (where it started) is different from the maximum height it achieves.

This may lead them to select Choice D (128).

Second Most Common Error:

Poor SIMPLIFY execution: Students make calculation errors, particularly with \(\mathrm{(-2)^2 = 4}\). They might incorrectly think \(\mathrm{(-2)^2 = -4}\), or make arithmetic mistakes in the final steps.

These calculation errors can lead to various incorrect values and cause confusion about which answer choice to select.

The Bottom Line:

This problem tests whether students can distinguish between different meaningful points in a motion problem - the starting height versus the maximum height. The key insight is that 'thrown from' always refers to the initial conditions at \(\mathrm{t = 0}\).