The equation y = x^2 - 14x + 39 represents a parabola in the xy-plane. What are the coordinates of...

1. INFER the approach needed

• Given: \(\mathrm{y = x^2 - 14x + 39}\) in standard form
• Need: Vertex coordinates (h, k)
• Strategy: Convert to vertex form \(\mathrm{y = a(x - h)^2 + k}\) using completing the square

2. SIMPLIFY by completing the square

• Start with the x-terms: \(\mathrm{x^2 - 14x}\)
• Take half the coefficient of x: \(\mathrm{-14/2 = -7}\)
• Square this value: \(\mathrm{(-7)^2 = 49}\)
• Add and subtract 49: \(\mathrm{y = (x^2 - 14x + 49) - 49 + 39}\)

3. SIMPLIFY further to get vertex form

• Factor the perfect square: \(\mathrm{y = (x - 7)^2 - 49 + 39}\)
• Combine constants: \(\mathrm{y = (x - 7)^2 - 10}\)
• This is vertex form with \(\mathrm{h = 7, k = -10}\)

Answer: B. (7, -10)

Alternative Quick Method: Use the vertex formula \(\mathrm{x = -b/(2a)}\) to get \(\mathrm{x = 7}\), then substitute back to find \(\mathrm{y = -10}\).

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Students make sign errors when completing the square, especially with the constant terms. They might write \(\mathrm{y = (x - 7)^2 + 49 + 39}\) instead of \(\mathrm{y = (x - 7)^2 - 49 + 39}\), forgetting that they added 49 so must subtract it back out.

This leads to \(\mathrm{y = (x - 7)^2 + 88}\), giving vertex (7, 88) instead of (7, -10). While this exact value isn't among the choices, the confusion often leads to selecting Choice C (7, 39) because they remember the x-coordinate is 7.

Second Most Common Error:

Poor INFER reasoning: Students try to read the vertex directly from standard form without converting it, or they confuse which coordinate comes from which part of the vertex form. They might think the constant term 39 is automatically the y-coordinate of the vertex.

This may lead them to select Choice C (7, 39) by correctly finding \(\mathrm{x = 7}\) but incorrectly using the original constant.

The Bottom Line:

This problem tests precision in algebraic manipulation. The vertex isn't obvious from standard form - students must systematically convert to vertex form or use the vertex formula, then carefully track positive and negative signs throughout their calculations.