Question:In the xy-plane, the graph of the equation y = x^2 - 6x + 50 intersects the line y =...

1. INFER the geometric relationship

• Given: Parabola \(\mathrm{y = x^2 - 6x + 50}\) intersects line \(\mathrm{y = k}\) at exactly one point
• Key insight: "Exactly one point" means the line is tangent to the parabola
• Since the parabola opens upward (positive \(\mathrm{x^2}\) coefficient), the line must touch at the vertex (minimum point)

2. INFER which vertex formula to use

• For any quadratic \(\mathrm{y = ax^2 + bx + c}\), the vertex occurs at \(\mathrm{x = \frac{-b}{2a}}\)
• This gives us the x-coordinate where the parabola reaches its minimum/maximum

3. SIMPLIFY to find the vertex x-coordinate

• From \(\mathrm{y = x^2 - 6x + 50}\): \(\mathrm{a = 1, b = -6, c = 50}\)
• \(\mathrm{x = \frac{-(-6)}{2 \times 1} = \frac{6}{2} = 3}\)

4. SIMPLIFY to find the vertex y-coordinate

• Substitute \(\mathrm{x = 3}\) into the original equation:
• \(\mathrm{y = (3)^2 - 6(3) + 50}\)
• \(\mathrm{y = 9 - 18 + 50 = 41}\)

5. INFER the final answer

• The vertex is at \(\mathrm{(3, 41)}\)
• For the line \(\mathrm{y = k}\) to be tangent at this point: \(\mathrm{k = 41}\)

Answer: B) 41

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students don't recognize that "exactly one intersection point" means tangency at the vertex.

Instead, they might try to solve \(\mathrm{x^2 - 6x + 50 = k}\) algebraically, setting the discriminant equal to zero, but without knowing what k should be. This approach becomes circular and leads to confusion. This causes them to abandon systematic solution and guess.

Second Most Common Error:

Inadequate SIMPLIFY execution: Students correctly identify the need to find the vertex but make arithmetic errors in the calculation.

For example, when computing \(\mathrm{y = 9 - 18 + 50}\), they might get 32 instead of 41 (perhaps calculating \(\mathrm{9 + 50 - 18 = 59 - 18 = 41}\) incorrectly as \(\mathrm{9 + 32 = 41}\), arriving at 32). This may lead them to select Choice C (32).

The Bottom Line:

This problem tests whether students understand the geometric meaning of "exactly one intersection point" between a parabola and horizontal line. The key breakthrough is recognizing this describes tangency at the vertex, not just setting up algebraic equations.