In the xy-plane, the graph of the function \(\mathrm{f(x) = -x^2 + 8x - 7}\) is a parabola. A horizontal...

1. TRANSLATE the problem information

• Given information:
  • Function: \(\mathrm{f(x) = -x^2 + 8x - 7}\) (a parabola)
  • A horizontal line \(\mathrm{y = k}\) intersects this parabola at its vertex
  • Need to find the value of k
• What this tells us: Since \(\mathrm{y = k}\) is horizontal and intersects at the vertex, k must be the y-coordinate of the vertex (the highest point since the parabola opens downward).

2. INFER the solution approach

• To find the y-coordinate of the vertex, we first need the x-coordinate of the vertex
• Then we'll substitute that x-value into the original function to get the y-value (which is k)
• For any parabola \(\mathrm{y = ax^2 + bx + c}\), the vertex x-coordinate is \(\mathrm{x = -\frac{b}{2a}}\)

3. TRANSLATE the coefficients and apply the vertex formula

• From \(\mathrm{f(x) = -x^2 + 8x - 7}\): \(\mathrm{a = -1, b = 8, c = -7}\)
• x-coordinate of vertex = \(\mathrm{-\frac{b}{2a}}\)
  \(\mathrm{= -\frac{8}{2(-1)}}\)
  \(\mathrm{= -\frac{8}{-2}}\)
  \(\mathrm{= 4}\)

4. SIMPLIFY to find the y-coordinate

• Substitute \(\mathrm{x = 4}\) into the original function:
• \(\mathrm{k = f(4) = -(4)^2 + 8(4) - 7}\)
• \(\mathrm{k = -16 + 32 - 7}\)
• \(\mathrm{k = 16 - 7 = 9}\)

Answer: C) 9

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students may not understand that "intersects at the vertex" means finding the vertex's coordinates. They might try to solve \(\mathrm{-x^2 + 8x - 7 = k}\) without realizing they need to find what k actually represents first.

This leads to confusion about what they're supposed to calculate, causing them to get stuck and guess randomly among the answer choices.

Second Most Common Error:

Poor SIMPLIFY execution: Students correctly identify they need \(\mathrm{f(4)}\) but make arithmetic errors with the negative signs. They might calculate \(\mathrm{f(4) = -16 + 32 - 7}\) as \(\mathrm{-16 + 32 + 7 = 23}\) (wrong sign) or as \(\mathrm{16 + 32 - 7 = 41}\) (wrong sign on first term).

Since neither of these incorrect values appears in the answer choices, this leads to confusion and guessing.

The Bottom Line:

This problem tests whether students understand the geometric relationship between a function and horizontal lines, not just mechanical formula application. The key insight is recognizing that "intersects at the vertex" translates to "find the vertex coordinates."