Question:The quadratic function f is defined by \(\mathrm{f(x) = 3x(x - 10)}\). In the xy-plane, the graph of \(\mathrm{y =...

1. TRANSLATE the problem information

• Given: \(\mathrm{f(x) = 3x(x - 10)}\) in factored form
• Need to find: k, which is the y-coordinate of the vertex \(\mathrm{(h,k)}\)

2. INFER the most efficient approach

• Since we have factored form, we can easily find the roots
• The vertex lies on the axis of symmetry, which is halfway between the roots
• This gives us a direct path: roots → vertex x-coordinate → vertex y-coordinate

3. SIMPLIFY to find the roots

• Set \(\mathrm{f(x) = 0}\): \(\mathrm{3x(x - 10) = 0}\)
• This gives us two factors: \(\mathrm{3x = 0}\) or \(\mathrm{(x - 10) = 0}\)
• So \(\mathrm{x = 0}\) or \(\mathrm{x = 10}\)
• The roots are at \(\mathrm{(0, 0)}\) and \(\mathrm{(10, 0)}\)

4. INFER the vertex location

• The axis of symmetry is the midpoint between \(\mathrm{x = 0}\) and \(\mathrm{x = 10}\)
• \(\mathrm{h = (0 + 10)/2 = 5}\)

5. SIMPLIFY to find k

• Substitute \(\mathrm{x = 5}\) into the original function:
• \(\mathrm{k = f(5) = 3(5)(5 - 10)}\)
• \(\mathrm{k = 3(5)(-5)}\)
• \(\mathrm{k = 15(-5)}\)
• \(\mathrm{k = -75}\)

Answer: B (-75)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Students correctly find \(\mathrm{h = 5}\) but make arithmetic errors when calculating \(\mathrm{f(5) = 3(5)(5-10)}\).

The most frequent mistake is with the negative sign: calculating \(\mathrm{3(5)(5-10) = 15(-5)}\) incorrectly as +75 instead of -75, or making errors in the step-by-step arithmetic. Some students might write \(\mathrm{3(5)(-5) = -15(-5) = 75}\), incorrectly distributing the negative.

This may lead them to select Choice C (5) if they confuse h and k, or get a positive result instead of negative.

Second Most Common Error:

Poor INFER reasoning: Students attempt to expand the factored form to standard form \(\mathrm{f(x) = 3x^2 - 30x}\), then use the vertex formula \(\mathrm{h = -b/(2a)}\), creating unnecessary complexity and opportunities for algebraic errors.

While this approach can work, the expansion and subsequent calculations (like finding \(\mathrm{3(25) - 30(5)}\)) introduce more steps where arithmetic errors can occur, especially when they rush through the algebra.

This leads to confusion and potential computational mistakes that could result in any of the wrong answer choices.

The Bottom Line:

This problem rewards recognizing that factored form gives you a direct route to the vertex through the roots and axis of symmetry, but demands careful arithmetic with negative numbers in the final calculation step.