\(\mathrm{f(x) = (x - 10)(x + 13)}\)The function f is defined by the given equation. For what value of x...

1. TRANSLATE the problem information

• Given: \(\mathrm{f(x) = (x - 10)(x + 13)}\)
• Find: The x-value where \(\mathrm{f(x)}\) reaches its minimum
• This asks for the x-coordinate, not the minimum value itself

2. INFER the mathematical approach

• This is a quadratic function in factored form
• When expanded, the coefficient of \(\mathrm{x^2}\) will be positive (1), so the parabola opens upward
• Upward-opening parabolas have their minimum at the vertex
• We need the x-coordinate of the vertex

3. SIMPLIFY using the vertex formula approach

• First expand to standard form:
  \(\mathrm{f(x) = (x - 10)(x + 13) = x^2 + 3x - 130}\)
• For \(\mathrm{ax^2 + bx + c}\), vertex x-coordinate = \(\mathrm{-b/(2a)}\)
• Here: \(\mathrm{a = 1, b = 3}\), so \(\mathrm{x = -3/(2·1) = -3/2}\)

4. Alternative: SIMPLIFY using midpoint of roots

• Find where \(\mathrm{f(x) = 0}\): \(\mathrm{(x - 10)(x + 13) = 0}\)
• Roots are \(\mathrm{x = 10}\) and \(\mathrm{x = -13}\)
• Vertex x-coordinate = midpoint = \(\mathrm{(10 + (-13))/2 = -3/2}\)

Answer: D. \(\mathrm{-3/2}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE reasoning: Students confuse what the question is asking for and calculate the minimum value instead of the x-coordinate where it occurs.

They correctly find the vertex at \(\mathrm{(-3/2, y)}\), then substitute to find

\(\mathrm{f(-3/2) = (-3/2 - 10)(-3/2 + 13)}\)

\(\mathrm{= (-23/2)(23/2)}\)

\(\mathrm{= -529/4 ≈ -130}\)

and incorrectly select Choice A (\(\mathrm{-130}\)).

Second Most Common Error:

Poor INFER execution: Students recognize they need to find roots but mistake one of the roots for the answer.

They solve \(\mathrm{(x - 10)(x + 13) = 0}\) and get \(\mathrm{x = 10}\) or \(\mathrm{x = -13}\), then pick the negative root without realizing they need the midpoint. This leads them to select Choice B (\(\mathrm{-13}\)).

The Bottom Line:

This problem tests whether students understand the difference between finding a coordinate versus finding a function value, and whether they know that the vertex of a parabola lies at the midpoint between its roots.