In the linear function h, \(\mathrm{h(0) = 41}\) and \(\mathrm{h(1) = 40}\). Which equation defines h?

1. TRANSLATE the problem information

• Given information:
  • h is a linear function
  • \(\mathrm{h(0) = 41}\) (when input is 0, output is 41)
  • \(\mathrm{h(1) = 40}\) (when input is 1, output is 40)

2. INFER the approach

• Since h is linear, it has the form \(\mathrm{h(x) = mx + b}\)
• We need to find the values of \(\mathrm{m}\) (slope) and \(\mathrm{b}\) (y-intercept)
• We can substitute our two known points to create a system of equations

3. SIMPLIFY to find the y-intercept

• Substitute \(\mathrm{x = 0}\) and \(\mathrm{h(0) = 41}\) into \(\mathrm{h(x) = mx + b}\):
  \(\mathrm{41 = m(0) + b}\)
  \(\mathrm{41 = 0 + b}\)
  \(\mathrm{b = 41}\)

4. SIMPLIFY to find the slope

• Now we know \(\mathrm{h(x) = mx + 41}\)
• Substitute \(\mathrm{x = 1}\) and \(\mathrm{h(1) = 40}\):
  \(\mathrm{40 = m(1) + 41}\)
  \(\mathrm{40 = m + 41}\)
  \(\mathrm{m = 40 - 41 = -1}\)

5. Write the final equation

• \(\mathrm{h(x) = -1x + 41 = -x + 41}\)

Answer: A. \(\mathrm{h(x) = -x + 41}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE reasoning: Students may confuse which values correspond to inputs versus outputs, or mix up the conditions \(\mathrm{h(0) = 41}\) and \(\mathrm{h(1) = 40}\).

For example, they might think \(\mathrm{h(0) = 41}\) means "41 times 0" or misinterpret the function notation entirely. This leads to setting up incorrect equations and potentially guessing among the answer choices.

Second Most Common Error:

Poor SIMPLIFY execution: Students correctly set up the equations but make sign errors when solving \(\mathrm{m + 41 = 40}\).

They might incorrectly calculate \(\mathrm{m = 41 - 40 = 1}\) instead of \(\mathrm{m = 40 - 41 = -1}\). This leads them to think \(\mathrm{h(x) = x + 41}\), which isn't among the choices, causing confusion and potentially selecting Choice B (\(\mathrm{h(x) = -x}\)) as the "closest" option.

The Bottom Line:

This problem tests whether students can systematically use the linear function structure to work backwards from outputs to the defining equation. The key insight is that function evaluation gives you concrete equations to solve for the unknown parameters.