The function \(\mathrm{f(x) = \frac{1}{2}(x - 7)^2 + 3}\) gives a metal ball's height above the ground \(\mathrm{f(x)}\), in inches,...

1. TRANSLATE the mathematical form

• Given: \(\mathrm{f(x) = \frac{1}{2}(x - 7)^2 + 3}\)
• This is in vertex form: \(\mathrm{y = a(x - h)^2 + k}\)
• Comparing our function: \(\mathrm{a = \frac{1}{2}, h = 7, k = 3}\)
• The vertex is at point \(\mathrm{(h, k) = (7, 3)}\)

2. INFER what the vertex represents

• Since \(\mathrm{a = \frac{1}{2} \gt 0}\), the parabola opens upward
• For upward-opening parabolas, the vertex is the lowest point
• This means the vertex represents the minimum value of the function

3. TRANSLATE back to the physical context

• The vertex (7, 3) tells us:
  • At x = 7 seconds, the function reaches its minimum value
  • The minimum value is \(\mathrm{f(7) = 3}\) inches
• Since f(x) represents height above ground, the minimum height is 3 inches

Answer: A. The metal ball's minimum height was 3 inches above the ground.

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students confuse which coordinate represents what information in the vertex.

They see the vertex (7, 3) and incorrectly think "the minimum height was 7 inches" because 7 is the larger number or they mix up x and y coordinates. They don't properly connect that in the vertex (x, y), the y-coordinate represents the function's output value (height), while the x-coordinate represents the input value (time).

This may lead them to select Choice B (7 inches).

Second Most Common Error:

Poor TRANSLATE reasoning: Students misunderstand what "when it started moving" means in the context.

They think the vertex represents the starting conditions rather than recognizing it represents the minimum point during the motion. They might calculate f(0) or just assume the vertex describes the initial state.

This may lead them to select Choice C or D or causes confusion leading to guessing.

The Bottom Line:

This problem requires students to fluently move between the mathematical representation of a vertex and its real-world interpretation, while clearly distinguishing between coordinates and their physical meanings.