A quadratic function is defined by \(\mathrm{h(x) = (x + 2)(x - 10)}\). The graph of \(\mathrm{y = h(x)}\) in...

1. INFER the most efficient approach

• Given: \(\mathrm{h(x) = (x + 2)(x - 10)}\) in factored form
• Key insight: The factored form makes it easy to find the roots, and the vertex lies exactly halfway between the roots of any parabola

2. TRANSLATE the factored form to find roots

• Set \(\mathrm{h(x) = 0}\): \(\mathrm{(x + 2)(x - 10) = 0}\)
• Using Zero Product Property:
  • \(\mathrm{x + 2 = 0}\) → \(\mathrm{x = -2}\)
  • \(\mathrm{x - 10 = 0}\) → \(\mathrm{x = 10}\)

3. SIMPLIFY to find the vertex x-coordinate

• The vertex x-coordinate is the average of the two roots:
• \(\mathrm{x\text{-coordinate} = \frac{\text{first root} + \text{second root}}{2}}\)
• \(\mathrm{x\text{-coordinate} = \frac{-2 + 10}{2}}\)
  \(\mathrm{= \frac{8}{2}}\)
  \(\mathrm{= 4}\)

Answer: B) 4

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students recognize they need to find roots but confuse the roots themselves with the vertex coordinate.

They correctly find that the roots are \(\mathrm{x = -2}\) and \(\mathrm{x = 10}\), but then think one of these IS the vertex x-coordinate. This leads them to select Choice A (-2) or Choice D (10) instead of recognizing that the vertex is halfway between these points.

Second Most Common Error:

Poor SIMPLIFY execution: Students understand they need the midpoint but make arithmetic errors.

They might compute \(\mathrm{\frac{-2 + 10}{2}}\) as \(\mathrm{\frac{12}{2} = 6}\) (adding instead of finding the difference) or make sign errors. While 6 isn't an answer choice, this confusion might lead them to select Choice C (8) as the closest value.

The Bottom Line:

This problem tests whether students truly understand the geometric relationship between roots and vertex, not just mechanical formula application. The key insight is that parabolas are symmetric, so the vertex must be exactly centered between the two roots.