Question:A quadratic function g has a maximum value of m. Which of the following equivalent forms of g shows the...

1. TRANSLATE the problem requirements

• Given: A quadratic function g with maximum value m
• Need to find: Which form shows m "as a coefficient or constant term"
• This means: The maximum value appears explicitly as a number in the equation

2. INFER which form reveals maximum values

• Different quadratic forms reveal different features:
  • Standard form \(\mathrm{(ax^2 + bx + c)}\): Shows y-intercept
  • Factored form \(\mathrm{(a(x - r_1)(x - r_2))}\): Shows x-intercepts (roots)
  • Vertex form \(\mathrm{(a(x - h)^2 + k)}\): Shows vertex coordinates \(\mathrm{(h,k)}\)
• Since maximum value occurs at the vertex, vertex form is the key

3. INFER the parabola direction

• All forms have coefficient -2 for the \(\mathrm{x^2}\) term
• Since \(\mathrm{a = -2 \lt 0}\), the parabola opens downward
• This means the vertex represents a maximum point

4. TRANSLATE the vertex form

• Choice (C): \(\mathrm{g(x) = -2(x - 3)^2 + 8}\)
• This is vertex form with \(\mathrm{a = -2, h = 3, k = 8}\)
• Vertex coordinates: \(\mathrm{(3, 8)}\)
• Maximum value \(\mathrm{m = 8}\) (the k-value)

5. Verify the maximum value appears explicitly

• The number 8 appears as the constant term in choice (C)
• This directly shows the maximum value \(\mathrm{m = 8}\)

Answer: C

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students don't recognize that different quadratic forms reveal different features of the function.

They might think standard form shows the maximum because it's the most familiar, or assume they need to calculate the maximum from any form rather than recognizing which form displays it directly. This leads them to select Choice (A) \(\mathrm{(-2x^2 + 12x - 10)}\) thinking the constant term -10 represents something significant about the maximum.

Second Most Common Error:

Poor TRANSLATE reasoning: Students misinterpret "shows the value of m as a coefficient or constant term" and look for the number that appears to be largest or most prominent.

They might select Choice (B) \(\mathrm{(-2(x - 1)(x - 5))}\) thinking the factored form must show important values, or get confused about what "coefficient or constant term" means in context. This causes them to get stuck and guess among the wrong choices.

The Bottom Line:

Success requires connecting the concept of vertex form with maximum value location. Students who don't understand that vertex form \(\mathrm{g(x) = a(x - h)^2 + k}\) directly displays the maximum/minimum as the k-value will struggle to identify which form explicitly shows this information.