A linear function g is defined by \(\mathrm{g(x) = mx + c}\), where m and c are constants. The graph...

1. TRANSLATE the problem information

• Given information:
  • \(\mathrm{g(x) = mx + c}\) (linear function)
  • The graph of \(\mathrm{y = g(x-2)}\) has y-intercept at \(\mathrm{(0, 11)}\)
  • \(\mathrm{m + c = 29}\)
  • Need to find: value of m

2. INFER the approach

• To find m, I need to use both pieces of information about the transformed function
• The y-intercept condition will give me one equation involving m and c
• Combined with \(\mathrm{m + c = 29}\), I'll have a system to solve

3. TRANSLATE the function transformation

• Start with \(\mathrm{g(x) = mx + c}\)
• For \(\mathrm{y = g(x-2)}\), substitute (x-2) wherever I see x:
  \(\mathrm{y = g(x-2) = m(x-2) + c = mx - 2m + c}\)

4. INFER how to use the y-intercept

• The y-intercept occurs when \(\mathrm{x = 0}\)
• Substitute \(\mathrm{x = 0}\) into the transformed function:
  \(\mathrm{y = m(0) - 2m + c = -2m + c}\)
• Since the y-intercept is at \(\mathrm{(0, 11)}\): \(\mathrm{-2m + c = 11}\)

5. SIMPLIFY the system of equations

• Now I have two equations:
  • Equation 1: \(\mathrm{-2m + c = 11}\)
  • Equation 2: \(\mathrm{m + c = 29}\)
• Subtract Equation 1 from Equation 2:
  \(\mathrm{(m + c) - (-2m + c) = 29 - 11}\)
  \(\mathrm{m + c + 2m - c = 18}\)
  \(\mathrm{3m = 18}\)
  \(\mathrm{m = 6}\)

Answer: A) 6

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students often confuse \(\mathrm{g(x-2)}\) with \(\mathrm{g(x) - 2}\), thinking the transformation affects the output rather than the input. They might write \(\mathrm{y = mx + c - 2}\) instead of \(\mathrm{y = m(x-2) + c}\).

This fundamental misunderstanding of function transformation notation leads them to set up incorrect equations, making systematic solution impossible. This leads to confusion and guessing.

Second Most Common Error:

Poor INFER reasoning: Students correctly transform the function but fail to recognize that finding the y-intercept requires setting \(\mathrm{x = 0}\) in the transformed equation. They might try to use the original function \(\mathrm{g(x)}\) or get confused about which equation represents the y-intercept condition.

Without the correct y-intercept equation \(\mathrm{(-2m + c = 11)}\), they can't form the proper system of equations, causing them to get stuck and randomly select an answer.

The Bottom Line:

This problem tests whether students truly understand function transformations (input vs. output changes) and can systematically work with the relationships they create. The algebra is straightforward once the correct equations are established.