For the linear function f, the table shows three values of x and their corresponding values of \(\mathrm{f(x)}\). If \(\mathrm{h(x)...

1. TRANSLATE the problem information

• Given information:
  • \(\mathrm{f(x)}\) is linear with three coordinate points
  • \(\mathrm{h(x) = f(x) - 13}\)
  • Need to find equation for \(\mathrm{h(x)}\)

2. INFER the solution approach

• Since we need \(\mathrm{h(x)}\) but only have points for \(\mathrm{f(x)}\), we must find \(\mathrm{f(x)}\) first
• Linear functions have form \(\mathrm{f(x) = mx + b}\), so we need to find \(\mathrm{m}\) and \(\mathrm{b}\)
• Use any two points to create a system of equations

3. SIMPLIFY to find the slope and y-intercept

• Using point (-4, 0): \(\mathrm{0 = m(-4) + b}\)
  \(\mathrm{b = 4m}\)
• Substitute into \(\mathrm{f(x)}\): \(\mathrm{f(x) = mx + 4m}\)
• Using point (-19/5, 1): \(\mathrm{1 = m(-19/5) + 4m}\)
• Combine terms: \(\mathrm{1 = -19m/5 + 20m/5}\)
  \(\mathrm{1 = m/5}\)
• Therefore: \(\mathrm{m = 5}\) and \(\mathrm{b = 4(5) = 20}\)
• So \(\mathrm{f(x) = 5x + 20}\)

4. TRANSLATE the transformation to find h(x)

• \(\mathrm{h(x) = f(x) - 13}\)
  \(\mathrm{h(x) = (5x + 20) - 13}\)
  \(\mathrm{h(x) = 5x + 7}\)

Answer: B. \(\mathrm{h(x) = 5x + 7}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER reasoning: Students try to work directly with \(\mathrm{h(x)}\) without realizing they need to find \(\mathrm{f(x)}\) first. They might attempt to use the given points as if they were points on \(\mathrm{h(x)}\), leading to completely incorrect slope and y-intercept values. This leads to confusion and random guessing among the answer choices.

Second Most Common Error:

Poor TRANSLATE execution: Students correctly find \(\mathrm{f(x) = 5x + 20}\) but misinterpret the transformation \(\mathrm{h(x) = f(x) - 13}\). They might subtract 13 from only the constant term, getting \(\mathrm{h(x) = 5x + 7}\), or add instead of subtract, getting \(\mathrm{h(x) = 5x + 33}\). Neither matches the answer choices exactly, but they might select Choice D (\(\mathrm{h(x) = 5x + 20}\)) thinking they've found the right answer.

The Bottom Line:

This problem tests whether students understand the two-step process: first find the original function using the given data, then apply the given transformation. The key insight is recognizing that you can't shortcut directly to \(\mathrm{h(x)}\).