Question:The function g is defined by \(\mathrm{g(x) = -4x + 28}\). What is the x-coordinate of the x-intercept of the...

1. TRANSLATE the problem information

• Given: \(\mathrm{g(x) = -4x + 28}\)
• Need to find: x-coordinate of the x-intercept
• What this means: Find where the graph crosses the x-axis (where \(\mathrm{y = 0}\))

2. TRANSLATE the x-intercept condition into mathematics

• At the x-intercept, \(\mathrm{y = g(x) = 0}\)
• So we need to solve: \(\mathrm{-4x + 28 = 0}\)

3. SIMPLIFY by solving the linear equation

• Start with: \(\mathrm{-4x + 28 = 0}\)
• Subtract 28 from both sides: \(\mathrm{-4x = -28}\)
• Divide both sides by -4: \(\mathrm{x = 7}\)

Answer: 7

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students confuse x-intercept with y-intercept

They mistakenly set \(\mathrm{x = 0}\) instead of \(\mathrm{y = 0}\), leading them to calculate \(\mathrm{g(0) = -4(0) + 28 = 28}\). This leads to confusion since the question asks for an x-coordinate but they found a y-value, causing them to get stuck and guess.

Second Most Common Error:

Poor SIMPLIFY execution: Students make sign errors during algebraic manipulation

When solving \(\mathrm{-4x + 28 = 0}\), they might incorrectly get \(\mathrm{-4x = 28}\) (forgetting to change the sign when moving 28), leading to \(\mathrm{x = -7}\) instead of \(\mathrm{x = 7}\). Since this isn't an integer that appears reasonable as an x-intercept for the given function, this leads to confusion and guessing.

The Bottom Line:

This problem tests whether students truly understand what an intercept means mathematically and can execute basic algebraic steps without sign errors. The conceptual leap from "x-intercept" to "set \(\mathrm{y = 0}\)" is the key insight that unlocks the entire solution.