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 information:
  • Function: \(\mathrm{f(x) = 7x - 84}\)
  • Need to find: x-intercept of \(\mathrm{y = f(x)}\)

• What this tells us: The x-intercept occurs where the graph crosses the x-axis, meaning \(\mathrm{y = 0}\) or \(\mathrm{f(x) = 0}\)

2. TRANSLATE to mathematical equation

• Since x-intercept means \(\mathrm{f(x) = 0}\), we need to solve:
  \(\mathrm{7x - 84 = 0}\)

3. SIMPLIFY to solve for x

• Add 84 to both sides:
  \(\mathrm{7x = 84}\)

• Divide both sides by 7:
  \(\mathrm{x = 12}\)

4. Express as coordinate point

• 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: Students confuse x-intercept with y-intercept concept

Instead of setting \(\mathrm{f(x) = 0}\), they might set \(\mathrm{x = 0}\) to find \(\mathrm{f(0) = 7(0) - 84 = -84}\), thinking the intercept is at \(\mathrm{(-84, 0)}\) or getting confused about which coordinate should be zero. This leads to confusion and guessing among the given choices.

Second Most Common Error:

Poor SIMPLIFY execution: Arithmetic errors when dividing 84 by 7

Students correctly set up \(\mathrm{7x - 84 = 0}\) and get to \(\mathrm{7x = 84}\), but then make calculation errors. They might compute \(\mathrm{84 \div 7}\) incorrectly as 7 or -12, leading them to select Choice C \(\mathrm{(7, 0)}\) or Choice A \(\mathrm{(-12, 0)}\).

The Bottom Line:

This problem tests whether students understand what an x-intercept represents and can execute basic linear equation solving. The key insight is recognizing that "x-intercept" means "where y equals zero."