The function f is defined by \(\mathrm{f(x) = (x + 3)(x + 1)}\). The graph of f in the xy-plane...

1. SIMPLIFY to find the x-intercepts

To find where the parabola crosses the x-axis, set \(\mathrm{f(x) = 0}\):

• \(\mathrm{(x + 3)(x + 1) = 0}\)
• Using the zero product property: \(\mathrm{x + 3 = 0}\) OR \(\mathrm{x + 1 = 0}\)
• Therefore: \(\mathrm{x = -3}\) OR \(\mathrm{x = -1}\)

2. INFER the vertex location strategy

• Key insight: For any parabola, the vertex lies exactly halfway between the x-intercepts
• This means the x-coordinate of the vertex = (sum of x-intercepts)/2

3. SIMPLIFY the midpoint calculation

• x-coordinate of vertex = \(\mathrm{\frac{-3 + (-1)}{2}}\)
• \(\mathrm{= \frac{-4}{2}}\)
• \(\mathrm{= -2}\)

4. APPLY CONSTRAINTS to select the correct interval

• We need the interval that contains \(\mathrm{x = -2}\)
• Checking each option:
  • A: \(\mathrm{-4 \lt x \lt -3}\) → No, since \(\mathrm{-2 \gt -3}\)
  • B: \(\mathrm{-3 \lt x \lt 1}\) → Yes, since \(\mathrm{-3 \lt -2 \lt 1}\) ✓
  • C: \(\mathrm{1 \lt x \lt 3}\) → No, since \(\mathrm{-2 \lt 1}\)
  • D: \(\mathrm{3 \lt x \lt 4}\) → No, since \(\mathrm{-2 \lt 3}\)

Answer: B

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students don't connect the vertex location to the x-intercepts. Instead, they might try to expand \(\mathrm{f(x) = (x + 3)(x + 1)}\) to get \(\mathrm{f(x) = x^2 + 4x + 3}\), then attempt to complete the square or use the vertex formula \(\mathrm{x = \frac{-b}{2a}}\). While this approach works, it's more complex and creates opportunities for algebraic errors that lead them to calculate the wrong x-coordinate.

This computational confusion may cause them to select Choice A (\(\mathrm{-4 \lt x \lt -3}\)) or get stuck and guess.

Second Most Common Error:

Poor APPLY CONSTRAINTS reasoning: Students correctly find that the vertex x-coordinate is \(\mathrm{-2}\), but then incorrectly determine which interval contains this value. They might misread the inequality symbols or make sign errors when checking \(\mathrm{-3 \lt -2 \lt 1}\).

This may lead them to select Choice A (\(\mathrm{-4 \lt x \lt -3}\)) by incorrectly thinking \(\mathrm{-2}\) falls in this range.

The Bottom Line:

This problem tests whether students recognize the elegant connection between x-intercepts and vertex location, rather than forcing them through lengthy algebraic manipulations. The key insight is that parabola symmetry makes the vertex lie exactly between the x-intercepts.