In the xy-plane, the graph of the equation y = -x^2 + 9x - 100 intersects the line y =...

1. INFER the geometric relationship

• Given: Parabola \(\mathrm{y = -x^2 + 9x - 100}\) intersects horizontal line \(\mathrm{y = c}\) at exactly one point
• Key insight: A parabola and horizontal line intersect at exactly one point only when the line passes through the vertex of the parabola
• Therefore: c must equal the y-coordinate of the parabola's vertex

2. SIMPLIFY to find the vertex x-coordinate

• Use the vertex formula: For \(\mathrm{y = ax^2 + bx + c}\), the x-coordinate is \(\mathrm{-b/(2a)}\)
• Here: \(\mathrm{a = -1, b = 9, c = -100}\)
• x-coordinate of vertex = \(\mathrm{-9/(2(-1))}\) = \(\mathrm{-9/(-2)}\) = \(\mathrm{9/2}\)

3. SIMPLIFY to find the vertex y-coordinate

• Substitute \(\mathrm{x = 9/2}\) into the original equation:
• \(\mathrm{y = -(9/2)^2 + 9(9/2) - 100}\)
• \(\mathrm{y = -81/4 + 81/2 - 100}\)
• Convert to common denominator: \(\mathrm{y = -81/4 + 162/4 - 400/4}\)
• \(\mathrm{y = (-81 + 162 - 400)/4 = -319/4}\)

4. APPLY CONSTRAINTS to select the answer

• The vertex is at \(\mathrm{(9/2, -319/4)}\)
• Therefore \(\mathrm{c = -319/4}\)

Answer: C. \(\mathrm{-319/4}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students don't recognize what "intersects at exactly one point" means geometrically. They might try to set the equations equal (\(\mathrm{-x^2 + 9x - 100 = c}\)) and solve for when the discriminant equals zero, but get confused about how to proceed without knowing c.

This leads to confusion and guessing among the answer choices.

Second Most Common Error:

Poor SIMPLIFY execution: Students find the correct x-coordinate (\(\mathrm{9/2}\)) but make arithmetic errors when calculating the y-coordinate, especially when working with fractions like \(\mathrm{-81/4 + 162/4 - 400/4}\).

Common calculation errors include:

• Forgetting to convert 100 to 400/4
• Sign errors when combining fractions
• Incorrect evaluation of \(\mathrm{(9/2)^2}\)

This may lead them to select Choice A (\(\mathrm{-481/4}\)) or Choice B (\(\mathrm{-100}\)) depending on the specific arithmetic mistake.

The Bottom Line:

This problem tests whether students can connect the geometric concept of tangency (one intersection point) to the algebraic concept of vertex location, then execute multi-step fraction arithmetic accurately.