Question:The graph of the equation y = -2x^2 + 20x - 42 in the xy-plane is a parabola. The vertex...

1. TRANSLATE the problem information

• Given: \(\mathrm{y = -2x^2 + 20x - 42}\) is a parabola
• Need to find: The value of k where the vertex is at point (h, k)

2. INFER the approach needed

• To find vertex coordinates, we need the equation in vertex form: \(\mathrm{y = a(x - h)^2 + k}\)
• Since we have standard form \(\mathrm{y = ax^2 + bx + c}\), we must complete the square
• The k value will be the constant term in vertex form

3. SIMPLIFY by factoring out the leading coefficient

• Start with: \(\mathrm{y = -2x^2 + 20x - 42}\)
• Factor -2 from the x-terms: \(\mathrm{y = -2(x^2 - 10x) - 42}\)

4. SIMPLIFY by completing the square

• Inside the parentheses we have \(\mathrm{x^2 - 10x}\)
• Take half the x-coefficient: \(\mathrm{-10 ÷ 2 = -5}\)
• Square this value: \(\mathrm{(-5)^2 = 25}\)
• Add and subtract 25: \(\mathrm{y = -2(x^2 - 10x + 25 - 25) - 42}\)
• Rewrite: \(\mathrm{y = -2(x^2 - 10x + 25) - 2(-25) - 42}\)
• Simplify: \(\mathrm{y = -2(x - 5)^2 + 50 - 42}\)
• Final form: \(\mathrm{y = -2(x - 5)^2 + 8}\)

5. INFER the vertex coordinates

• Comparing \(\mathrm{y = -2(x - 5)^2 + 8}\) to \(\mathrm{y = a(x - h)^2 + k}\)
• We see that \(\mathrm{h = 5}\) and \(\mathrm{k = 8}\)

Answer: C) 8

Why Students Usually Falter on This Problem

Most Common Error Path:

Poor SIMPLIFY execution: Sign errors when distributing the leading coefficient through the completed square.

When students write \(\mathrm{y = -2(x^2 - 10x + 25) - 2(-25) - 42}\), they might incorrectly calculate \(\mathrm{-2(-25)}\) as -50 instead of +50, leading to \(\mathrm{y = -2(x - 5)^2 + (-50 - 42) = -2(x - 5)^2 - 92}\). This gives \(\mathrm{k = -92}\), which isn't among the choices, causing confusion and guessing.

Second Most Common Error:

Conceptual confusion about vertex form: Mixing up which value represents k in the final equation.

Some students correctly complete the square to get \(\mathrm{y = -2(x - 5)^2 + 8}\), but then think the vertex is at \(\mathrm{(-5, 8)}\) instead of \(\mathrm{(5, 8)}\), or confuse whether k is the value inside or outside the squared term. This may lead them to select Choice B (5) thinking that \(\mathrm{h = 5}\) means \(\mathrm{k = 5}\).

The Bottom Line:

This problem requires both strategic thinking (recognizing the need for vertex form) and careful algebraic execution (completing the square without sign errors). The multi-step nature means small mistakes compound into wrong final answers.