The function f is defined by \(\mathrm{f(x) = 7x - 84}\). What is the x-intercept of the graph of \(\mathrm{y...

1. TRANSLATE the problem information

• Given: \(\mathrm{f(x) = 7x - 84}\)
• Find: x-intercept of \(\mathrm{y = f(x)}\)
• Key insight: The x-intercept is where the graph crosses the x-axis, which means \(\mathrm{y = 0}\) (or \(\mathrm{f(x) = 0}\))

2. TRANSLATE to set up the equation

• Since we need \(\mathrm{f(x) = 0}\):
  \(\mathrm{7x - 84 = 0}\)

3. SIMPLIFY by solving the linear equation

• Add 84 to both sides:
  \(\mathrm{7x - 84 + 84 = 0 + 84}\)
  \(\mathrm{7x = 84}\)
• Divide both sides by 7:
  \(\mathrm{x = 84 ÷ 7 = 12}\)

4. TRANSLATE back to coordinate form

• The x-intercept is the point \(\mathrm{(12, 0)}\)

Answer: D. \(\mathrm{(12, 0)}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Confusing x-intercept with y-intercept

Students might try to find where \(\mathrm{x = 0}\) instead of where \(\mathrm{y = 0}\). They would substitute \(\mathrm{x = 0}\) into \(\mathrm{f(x) = 7x - 84}\), getting \(\mathrm{f(0) = 7(0) - 84 = -84}\), and incorrectly think the answer involves the point \(\mathrm{(0, -84)}\) or just the value \(\mathrm{-84}\).

This leads to confusion and guessing among the given choices.

Second Most Common Error:

Poor SIMPLIFY execution: Arithmetic errors in division

Students correctly set up \(\mathrm{7x - 84 = 0}\) and get to \(\mathrm{7x = 84}\), but then make an error dividing 84 by 7. They might get \(\mathrm{x = -12}\) (from incorrect sign handling) or \(\mathrm{x = 7}\) (from switching numerator and denominator).

This may lead them to select Choice A \(\mathrm{(-12, 0)}\) or Choice C \(\mathrm{(7, 0)}\).

The Bottom Line:

This problem tests whether students understand what an x-intercept means geometrically and can translate that understanding into the algebraic condition \(\mathrm{f(x) = 0}\). The arithmetic is straightforward once the setup is correct.