The function \(\mathrm{g(t)} = -\frac{2}{5}(\mathrm{t} - 6)^2 + 15\) models the brightness of a stage light, in lumens, t seconds...

1. TRANSLATE the function into vertex form components

• Given function: \(\mathrm{g(t) = -(2/5)(t - 6)^2 + 15}\)
• This matches vertex form: \(\mathrm{f(x) = a(x - h)^2 + k}\)
• Identifying components:
  • \(\mathrm{a = -2/5}\)
  • \(\mathrm{h = 6}\)
  • \(\mathrm{k = 15}\)
• Therefore, the vertex is \(\mathrm{(6, 15)}\)

2. INFER what the vertex represents

• Since \(\mathrm{a = -2/5 \lt 0}\), the parabola opens downward
• When a parabola opens downward, the vertex is the highest point (maximum)
• So the vertex \(\mathrm{(6, 15)}\) represents a maximum brightness of 15 lumens occurring at \(\mathrm{t = 6}\) seconds

3. Match interpretation to answer choices

• Looking for the choice that correctly states: maximum brightness of 15 lumens at \(\mathrm{t = 6}\) seconds
• Choice A matches exactly: "The stage light reaches a maximum brightness of 15 lumens at \(\mathrm{t = 6}\) seconds"

Answer: A

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Not understanding the relationship between coefficient sign and parabola direction

Students know vertex form but don't connect that \(\mathrm{a \lt 0}\) means the parabola opens downward, making the vertex a maximum rather than minimum. They might think "vertex = extreme value" but guess whether it's maximum or minimum.

This may lead them to select Choice B (minimum brightness of 15 lumens at \(\mathrm{t = 6}\) seconds)

Second Most Common Error:

Poor TRANSLATE reasoning: Misinterpreting the vertex coordinates

Students confuse which coordinate represents what value. They see the vertex \(\mathrm{(6, 15)}\) and think the 6 represents a brightness value rather than the time coordinate. They incorrectly conclude the brightness was 6 lumens initially.

This may lead them to select Choice C (brightness was 6 lumens when turned on)

The Bottom Line:

Success requires both accurately extracting vertex coordinates from the function AND correctly interpreting what a negative leading coefficient means for the parabola's orientation. The vertex tells you the extreme value and when it occurs, but you must know parabola properties to determine if it's a maximum or minimum.