\(\mathrm{g(t) + t^2 = 49}\) The given equation relates the variable t and the value of \(\mathrm{g(t)}\) for the function...

1. TRANSLATE the constraint into function form

• Given equation: \(\mathrm{g(t) + t^2 = 49}\)
• TRANSLATE this to isolate the function: \(\mathrm{g(t) = 49 - t^2}\)
• What this tells us: We now have g as an explicit function of t

2. INFER the type of function and its behavior

• INFER that \(\mathrm{g(t) = -t^2 + 49}\) is a quadratic function
• Since the coefficient of \(\mathrm{t^2}\) is -1 (negative), this parabola opens downward
• INFER that a downward-opening parabola has a maximum value at its vertex

3. SIMPLIFY to find the vertex location

• For quadratic \(\mathrm{g(t) = at^2 + bt + c}\), vertex occurs at \(\mathrm{t = -b/(2a)}\)
• Here: \(\mathrm{a = -1, b = 0, c = 49}\)
• SIMPLIFY: \(\mathrm{t = -0/(2(-1)) = 0}\)

4. SIMPLIFY to calculate the maximum value

• The maximum occurs at \(\mathrm{t = 0}\)
• SIMPLIFY: \(\mathrm{g(0) = 49 - (0)^2 = 49}\)

Answer: D) 49

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students may struggle to isolate \(\mathrm{g(t)}\) from the constraint equation, either leaving it as \(\mathrm{g(t) + t^2 = 49}\) or making algebraic errors when rearranging.

Without properly isolating the function, they cannot proceed with finding the maximum. This leads to confusion and guessing among the answer choices.

Second Most Common Error:

Conceptual confusion about parabola orientation: Students might remember that parabolas have maximum or minimum values but confuse which occurs when the coefficient is negative vs. positive.

If they incorrectly think \(\mathrm{g(t) = -t^2 + 49}\) has a minimum (instead of maximum), they might look for the smallest possible value. Since \(\mathrm{t^2 \geq 0}\), the smallest value would be \(\mathrm{g(t) = 49 - \infty = -\infty}\), leading them to select Choice A (-49) as the most negative option available.

The Bottom Line:

This problem tests whether students can convert a constraint equation into an explicit function form and then apply their knowledge of quadratic function properties. The algebraic manipulation is straightforward, but students need to recognize what they're being asked to find and how parabola orientation determines maximum vs. minimum values.