The function \(\mathrm{g(x) = (x - 3)(x + 7)}\) represents a parabola graphed in the xy-plane. What is the y-coordinate...

1. TRANSLATE the problem information

• Given: \(\mathrm{g(x) = (x - 3)(x + 7)}\) in factored form
• Find: y-coordinate of the vertex

2. INFER the solution approach

• To find the vertex y-coordinate, we first need the vertex x-coordinate
• The vertex lies on the axis of symmetry, which is halfway between the x-intercepts
• Strategy: Find x-intercepts → Find axis of symmetry → Evaluate function there

3. TRANSLATE to find the x-intercepts

• Set \(\mathrm{g(x) = 0}\): \(\mathrm{(x - 3)(x + 7) = 0}\)
• This gives us \(\mathrm{x = 3}\) or \(\mathrm{x = -7}\)

4. INFER the axis of symmetry

• The axis of symmetry is the midpoint of the x-intercepts
• x-coordinate of vertex = \(\mathrm{\frac{3 + (-7)}{2}}\)
  \(\mathrm{= \frac{-4}{2}}\)
  \(\mathrm{= -2}\)

5. SIMPLIFY to find the y-coordinate

• Evaluate \(\mathrm{g(-2)}\): \(\mathrm{g(-2) = (-2 - 3)(-2 + 7)}\)
  \(\mathrm{= (-5)(5)}\)
  \(\mathrm{= -25}\)

Answer: -25

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students don't recognize the systematic approach needed and try to expand the factored form first, then complete the square or use the vertex formula.

This leads to unnecessary algebraic work: \(\mathrm{g(x) = (x-3)(x+7) = x^2 + 4x - 21}\), then trying to find the vertex as \(\mathrm{(-\frac{4}{2}, g(-2)) = (-2, g(-2))}\). While this can work, it's much more prone to algebraic errors and takes longer. Students often make mistakes in the expansion or in applying the vertex formula, leading to incorrect answers.

Second Most Common Error:

Poor SIMPLIFY execution: Students find the correct x-coordinate of the vertex (\(\mathrm{x = -2}\)) but make arithmetic errors when evaluating \(\mathrm{g(-2)}\).

Common mistake: \(\mathrm{g(-2) = (-2-3)(-2+7)}\)
\(\mathrm{= (-5)(5)}\)
\(\mathrm{= 25}\) (forgetting the negative sign)
This leads them to answer 25 instead of -25.

The Bottom Line:

This problem rewards recognizing that factored form gives you direct access to x-intercepts, and that the vertex x-coordinate is simply their average. Students who try to expand first or who rush through the final arithmetic evaluation typically struggle the most.