A quadratic function models a projectile's height, in meters, above the ground in terms of the time, in seconds, after...

1. TRANSLATE the problem information

• Given information:
  • Projectile modeled by quadratic function
  • Initial height: 7 meters at t = 0
  • Maximum height: 51.1 meters at t = 3 seconds
  • Need to find: When height returns to 7 meters

2. INFER the mathematical structure

• Since we know the maximum height occurs at t = 3, this tells us the vertex is at (3, 51.1)
• For quadratic functions, vertex form is most useful: \(\mathrm{f(x) = a(x - h)^2 + k}\)
• We have h = 3 and k = 51.1, so \(\mathrm{f(x) = a(x - 3)^2 + 51.1}\)
• We need to find coefficient 'a' using the initial condition

3. SIMPLIFY to find the coefficient

• Use the initial condition \(\mathrm{f(0) = 7}\):
  \(\mathrm{7 = a(0 - 3)^2 + 51.1}\)
  \(\mathrm{7 = 9a + 51.1}\)
  \(\mathrm{-44.1 = 9a}\)
  \(\mathrm{a = -4.9}\) (use calculator)
• Complete function: \(\mathrm{f(x) = -4.9(x - 3)^2 + 51.1}\)

4. SIMPLIFY to solve for return time

• Set height equal to 7:
  \(\mathrm{7 = -4.9(x - 3)^2 + 51.1}\)
  \(\mathrm{-44.1 = -4.9(x - 3)^2}\)
  \(\mathrm{9 = (x - 3)^2}\) (use calculator for -44.1 ÷ (-4.9))

5. CONSIDER ALL CASES from the square root

• Taking square root: \(\mathrm{±3 = x - 3}\)
• This gives us: \(\mathrm{x - 3 = 3}\) OR \(\mathrm{x - 3 = -3}\)
• So: \(\mathrm{x = 6}\) OR \(\mathrm{x = 0}\)
• Since \(\mathrm{x = 0}\) represents the launch time, \(\mathrm{x = 6}\) is when it returns to 7 meters

Answer: B. 6

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students don't recognize the connection between "maximum height at 3 seconds" and the vertex form parameters. They might try to guess a general quadratic form like \(\mathrm{f(x) = ax^2 + bx + c}\) instead of using the more efficient vertex form. This leads to a much more complex system of equations that's difficult to solve, causing confusion and potentially leading them to select Choice A (3) thinking that's when the maximum occurs.

Second Most Common Error:

Poor CONSIDER ALL CASES execution: Students correctly set up and solve the equation up to \(\mathrm{(x - 3)^2 = 9}\), but when they take the square root, they only consider the positive solution \(\mathrm{x - 3 = 3}\), giving them \(\mathrm{x = 6}\). However, they fail to check this makes sense or don't realize there should be two times when the projectile is at 7 meters. This incomplete analysis might make them doubt their answer and guess, or they might select Choice B (6) for the right reasons but with incomplete understanding.

The Bottom Line:

This problem tests whether students can efficiently use the vertex form of quadratic functions and understand the physical meaning of projectile motion symmetry. The key insight is recognizing that the given information directly provides the vertex coordinates, making vertex form the natural choice rather than standard form.