\(\mathrm{h(t) = -16t^2 + 110t + 72}\). The function above models the height h, in feet, of an object above...

1. TRANSLATE the question into mathematical terms

• The question asks what the number 72 represents in \(\mathrm{h(t) = -16t^2 + 110t + 72}\)
• Need to determine which physical quantity this constant term represents

2. INFER the mathematical approach

• To understand what a constant term means in a function, evaluate the function at key points
• Since t represents time after launch, \(\mathrm{t = 0}\) represents the initial moment
• The initial height would be \(\mathrm{h(0)}\)

3. SIMPLIFY by evaluating h(0)

• \(\mathrm{h(0) = -16(0)^2 + 110(0) + 72}\)
• \(\mathrm{h(0) = 0 + 0 + 72 = 72}\) feet

4. TRANSLATE the result back to context

• At \(\mathrm{t = 0}\) seconds (launch moment), height = 72 feet
• Therefore, 72 represents the initial height of the object

Answer: A. The initial height, in feet, of the object

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students may confuse which coefficient represents which physical quantity. They might think the coefficient of the linear term (110) represents initial height, or confuse speed with height units.

This confusion about the structure of quadratic functions may lead them to select Choice C (The initial speed, in feet per second, of the object) because 110 is larger and seems more "initial-like."

Second Most Common Error:

Poor INFER reasoning: Students may not recognize that to find what the constant term represents, they should evaluate the function at \(\mathrm{t = 0}\). Instead, they might try to find the maximum of the function or make assumptions about what each term "should" represent.

This leads to confusion and guessing among the remaining choices.

The Bottom Line:

This problem tests whether students understand that in a quadratic function modeling a real-world situation, the constant term represents the function's value when the variable equals zero—in this case, the initial height when time equals zero.