A quadratic function models the height, in feet, of an object above the ground in terms of the time, in...

1. TRANSLATE the problem information

• Given information:
  • Object launched from elevated surface
  • Initial height: 10 feet (when \(\mathrm{t = 0}\))
  • Maximum height: 1,034 feet (when \(\mathrm{t = 8}\) seconds)
  • Need to find: height when \(\mathrm{t = 10}\) seconds

• What this tells us: We have a quadratic function with vertex at (8, 1034)

2. INFER the best approach

• Since we know the vertex (maximum point), vertex form is most efficient
• Vertex form: \(\mathrm{h(t) = a(t - 8)² + 1,034}\)
• We need to find the value of 'a' using the initial condition

3. TRANSLATE the initial condition into an equation

• At \(\mathrm{t = 0}\), height = 10 feet
• Substitute into vertex form: \(\mathrm{10 = a(0 - 8)² + 1,034}\)

4. SIMPLIFY to find the coefficient 'a'

• \(\mathrm{10 = a(-8)² + 1,034}\)
• \(\mathrm{10 = 64a + 1,034}\)
• Subtract 1,034: \(\mathrm{-1,024 = 64a}\)
• Divide by 64: \(\mathrm{a = -16}\)

5. SIMPLIFY to find height at \(\mathrm{t = 10}\)

• Complete equation: \(\mathrm{h(t) = -16(t - 8)² + 1,034}\)
• Substitute \(\mathrm{t = 10}\): \(\mathrm{h(10) = -16(10 - 8)² + 1,034}\)
• \(\mathrm{h(10) = -16(2)² + 1,034}\)
• \(\mathrm{h(10) = -16(4) + 1,034 = -64 + 1,034 = 970}\)

Answer: C. 970

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE reasoning: Students may not recognize that "maximum height at 8 seconds" gives them the vertex of the parabola, instead trying to use standard form unnecessarily.

They might attempt to set up \(\mathrm{y = ax² + bx + c}\) and use the three conditions, leading to a much more complex system of equations. This makes the problem unnecessarily difficult and increases chances of arithmetic errors, potentially causing them to select an incorrect choice or abandon the systematic approach entirely.

Second Most Common Error:

Poor SIMPLIFY execution: Students correctly identify the vertex form approach but make sign errors, particularly when dealing with negative coefficients or when squaring negative values.

For example, they might calculate \(\mathrm{a = +16}\) instead of \(\mathrm{a = -16}\), or make errors in the final substitution like \(\mathrm{h(10) = -16(2)² + 1,034 = -16(-4) + 1,034 = 64 + 1,034 = 1,098}\). This leads to confusion since 1,098 isn't among the choices, causing them to guess randomly.

The Bottom Line:

This problem tests whether students can efficiently use vertex form when given vertex information, rather than forcing themselves through unnecessarily complex algebraic manipulations with standard form.