Question:\(\mathrm{h(t) = -4t^2 + 36t + 5}\)A small drone is launched from a rooftop. The equation above models the drone's...

1. TRANSLATE the problem setup

• Given: \(\mathrm{h(t) = -4t^2 + 36t + 5}\) models drone height above ground
• Need: Interpret what the vertex of this graph means in context
• This is a quadratic function in standard form \(\mathrm{ax^2 + bx + c}\)

2. INFER the significance of the quadratic structure

• Since \(\mathrm{a = -4 \lt 0}\), the parabola opens downward
• A downward-opening parabola has its highest point at the vertex
• Therefore, the vertex represents the maximum height of the drone

3. SIMPLIFY to find the vertex coordinates

• Use vertex formula: \(\mathrm{t = -b/(2a)}\)
• Here: \(\mathrm{a = -4, b = 36}\), so \(\mathrm{t = -36/(2 \times (-4)) = 4.5}\) seconds
• Substitute back: \(\mathrm{h(4.5) = -4(4.5)^2 + 36(4.5) + 5}\)
• \(\mathrm{h(4.5) = -4(20.25) + 162 + 5 = 86}\) meters (use calculator for \(\mathrm{4.5^2}\))

4. TRANSLATE the vertex meaning to real-world context

• Vertex \(\mathrm{(4.5, 86)}\) means: at \(\mathrm{t = 4.5}\) seconds, \(\mathrm{h = 86}\) meters
• In context: The drone reaches its maximum height of 86 meters at 4.5 seconds after launch

Answer: B

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE reasoning: Students confuse different features of the quadratic function and their real-world meanings.

Many students see "5" in the equation and think the vertex relates to the initial height, since \(\mathrm{h(0) = 5}\) gives the launch height. They don't recognize that the vertex (the turning point) represents something different from the y-intercept (the starting point). This may lead them to select Choice A (initial height of 5 meters).

Second Most Common Error:

Inadequate INFER skill: Students don't connect the mathematical concept of "vertex of a downward parabola" to "maximum height in the real world."

They might correctly calculate that something happens at \(\mathrm{t = 4.5}\), but think this represents when the drone hits the ground rather than when it reaches maximum height. This may lead them to select Choice C (hits ground at 4.5 seconds).

The Bottom Line:

This problem requires students to bridge abstract mathematical concepts (vertex, parabola orientation) with real-world interpretations (maximum height, timing). Success depends on recognizing that the vertex of a downward parabola represents the peak of the drone's flight path.