A linear function h has slope 5, and \(\mathrm{h(4) = 31}\). What is the value of \(\mathrm{h(0)}\)?

1. TRANSLATE the problem information

• Given information:
  • h is a linear function with slope = 5
  • \(\mathrm{h(4) = 31}\)
  • Need to find \(\mathrm{h(0)}\)

• This tells us we have a linear function with known slope and one point

2. INFER the approach

• Since we have a linear function with known slope, we should use slope-intercept form: \(\mathrm{h(x) = mx + b}\)
• We can substitute our known point to find the y-intercept b
• Once we have b, we know that \(\mathrm{h(0) = b}\)

3. TRANSLATE into slope-intercept form

• With slope 5: \(\mathrm{h(x) = 5x + b}\)
• We need to find the value of b (the y-intercept)

4. SIMPLIFY using the known point

• Substitute the point (4, 31):
  \(\mathrm{31 = 5(4) + b}\)
  \(\mathrm{31 = 20 + b}\)
  \(\mathrm{b = 31 - 20 = 11}\)

5. INFER the final answer

• Now we know: \(\mathrm{h(x) = 5x + 11}\)
• Therefore: \(\mathrm{h(0) = 5(0) + 11 = 11}\)

Answer: C. 11

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE reasoning: Students may confuse what \(\mathrm{h(0)}\) represents and try to substitute 0 for the slope or mix up the given point coordinates.

They might think \(\mathrm{h(0)}\) means "when the slope is 0" rather than "when x equals 0." This conceptual confusion about function notation can lead them to misinterpret the entire problem setup and select Choice A (-11) through incorrect calculations.

Second Most Common Error:

Poor SIMPLIFY execution: Students correctly set up \(\mathrm{31 = 5(4) + b}\) but make arithmetic errors when solving for b.

Common calculation mistakes include: \(\mathrm{31 = 20 + b}\) → \(\mathrm{b = 31 + 20 = 51}\) instead of \(\mathrm{b = 31 - 20 = 11}\). Since none of the choices match 51, this confusion may lead them to select Choice D (20) thinking that's what they calculated along the way.

The Bottom Line:

This problem tests whether students truly understand that the y-intercept of a linear function is the output value when the input is zero, and whether they can accurately manipulate the slope-intercept form to find that value.