\(\mathrm{f(x) = x^2 - 18x - 360}\). If the given function f is graphed in the xy-plane, where \(\mathrm{y =...

1. TRANSLATE the problem information

• Given information:
  • \(\mathrm{f(x) = x^2 - 18x - 360}\)
  • We need an x-intercept of the graph where \(\mathrm{y = f(x)}\)

• What this tells us: X-intercepts are points where the graph crosses the x-axis, meaning \(\mathrm{y = 0}\), so we need \(\mathrm{f(x) = 0}\)

2. TRANSLATE to set up the equation

• Since x-intercepts occur where \(\mathrm{f(x) = 0}\):
  • \(\mathrm{0 = x^2 - 18x - 360}\)

3. SIMPLIFY by factoring the quadratic

• We need two numbers that multiply to -360 and add to -18
• Testing factor pairs of 360: The pair 12 and -30 works:
  • \(\mathrm{(12)(-30) = -360}\) ✓
  • \(\mathrm{12 + (-30) = -18}\) ✓
• Factor: \(\mathrm{0 = (x + 12)(x - 30)}\)

4. SIMPLIFY using zero product property

• If \(\mathrm{(x + 12)(x - 30) = 0}\), then either factor equals zero:
  • \(\mathrm{x + 12 = 0}\) → \(\mathrm{x = -12}\)
  • \(\mathrm{x - 30 = 0}\) → \(\mathrm{x = 30}\)

5. CONSIDER ALL CASES to identify the answer

• The x-intercepts are \(\mathrm{(-12, 0)}\) and \(\mathrm{(30, 0)}\)
• Looking at the answer choices, only \(\mathrm{(-12, 0)}\) is listed as option A

Answer: A. (-12, 0)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE reasoning: Students may not connect "x-intercept" with the condition that \(\mathrm{y = 0}\) or \(\mathrm{f(x) = 0}\). Instead, they might try to substitute answer choices back into the original function, or they might think they need to find the vertex or some other feature of the parabola.

This leads to confusion and guessing among the answer choices.

Second Most Common Error:

Poor SIMPLIFY execution: Students set up \(\mathrm{f(x) = 0}\) correctly but make errors in factoring the quadratic \(\mathrm{x^2 - 18x - 360}\). They might attempt to use the quadratic formula incorrectly, or factor incorrectly, leading to wrong values for x.

This may lead them to select Choice D (12, 0) if they confuse signs, or other incorrect choices based on their factoring mistakes.

The Bottom Line:

This problem requires students to bridge the conceptual understanding of x-intercepts (where graphs cross the x-axis) with the algebraic skill of solving quadratic equations. The key insight is recognizing that "finding x-intercepts" means "solving \(\mathrm{f(x) = 0}\)."