An object is thrown upward from a platform. The equation \(\mathrm{h = -4(t - 3)^2 + 68}\) represents this situation,...

1. TRANSLATE the problem information

• Given information:
  • Equation: \(\mathrm{h = -4(t - 3)^2 + 68}\)
  • h = height above ground (feet)
  • t = seconds after thrown
  • Need: height from which object was thrown

• What this tells us: We need the height at the moment of throwing, which occurs at \(\mathrm{t = 0}\).

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

• Substitute \(\mathrm{t = 0}\) into the equation:

\(\mathrm{h = -4(0 - 3)^2 + 68}\)

• Calculate step by step:

• First: \(\mathrm{(0 - 3) = -3}\)
• Next: \(\mathrm{(-3)^2 = 9}\)
• Then: \(\mathrm{-4(9) = -36}\)
• Finally: \(\mathrm{-36 + 68 = 32}\)

Answer: C (32 feet)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE reasoning: Students misinterpret what "height from which the object was thrown" means and think it refers to the maximum height the object reaches.

Looking at the equation \(\mathrm{h = -4(t - 3)^2 + 68}\), they see 68 as the largest number and assume this must be the starting height, not recognizing that 68 represents the maximum height (vertex) of the parabolic path.

This leads them to select Choice D (68).

Second Most Common Error:

Poor SIMPLIFY execution: Students correctly identify that they need \(\mathrm{t = 0}\), but make calculation errors, particularly with \(\mathrm{(-3)^2}\) or confusing the sign when calculating \(\mathrm{-4(9)}\).

Common mistakes include:

• Thinking \(\mathrm{(-3)^2 = -9}\) instead of +9
• Getting confused with the negative signs in \(\mathrm{-4(9)}\)

This causes calculation errors that may lead to incorrect answer choices or confusion and guessing.

The Bottom Line:

Success requires recognizing that "initial height" means \(\mathrm{t = 0}\), then carefully executing the arithmetic with negative numbers and order of operations.