When the quadratic function f is graphed in the xy-plane, where \(\mathrm{y = f(x)}\), its vertex is \((-3, 6)\). One...

1. TRANSLATE the problem information

• Given information:
  • Vertex: \((-3, 6)\)
  • One x-intercept: \((-17/4, 0)\)
  • Need: the other x-intercept

2. INFER the key relationship

• Since parabolas are symmetric about their line of symmetry (which passes through the vertex), the x-coordinate of the vertex is exactly halfway between the two x-intercepts
• The vertex x-coordinate (-3) is the midpoint of the x-intercepts
• This means: \(-3 = (\mathrm{x}_1 + \mathrm{x}_2)/2\), where \(\mathrm{x}_1\) and \(\mathrm{x}_2\) are the x-coordinates of the intercepts

3. SIMPLIFY to find the unknown intercept

• Set up the equation: \(-3 = (-17/4 + \mathrm{x}_2)/2\)
• Multiply both sides by 2: \(-6 = -17/4 + \mathrm{x}_2\)
• Add 17/4 to both sides: \(\mathrm{x}_2 = -6 + 17/4\)
• Convert -6 to fourths: \(-6 = -24/4\)
• Combine: \(\mathrm{x}_2 = -24/4 + 17/4 = -7/4\)

Answer: B. \((-7/4, 0)\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students don't recognize the symmetry property of parabolas and instead try to reconstruct the entire quadratic equation using the vertex form or other complex methods.

Without understanding that the vertex sits exactly between the x-intercepts, they might attempt to use \(\mathrm{y} = \mathrm{a(x - h)}^2 + \mathrm{k}\) with the vertex \((-3, 6)\) and the point \((-17/4, 0)\) to find the equation first, then solve for the other intercept. This creates unnecessary complexity and increases chances for errors.

This leads to confusion and abandoning systematic solution for guessing.

Second Most Common Error:

Poor SIMPLIFY execution: Students understand the symmetry concept but make arithmetic errors with fractions, particularly when converting -6 to -24/4 or when adding fractions with different denominators.

For example, they might incorrectly calculate \(-6 + 17/4\) as \(-6 + 4.25 = -1.75\), then convert this incorrectly back to a fraction, leading them to select an incorrect answer choice.

The Bottom Line:

This problem rewards understanding parabola symmetry over complex algebraic manipulation. The key insight that makes everything simple is recognizing that the vertex's x-coordinate is the average of the x-intercepts.