Question:A linear function h is such that for every increase of 1 in x, the value of \(\mathrm{h(x)}\) increases by...

1. TRANSLATE the problem information

• Given information:
  • h is a linear function
  • For every increase of 1 in x, \(\mathrm{h(x)}\) increases by \(0.25\)
  • \(\mathrm{h(8)} = -1\)
  • Need to find: y-intercept of \(\mathrm{y} = \mathrm{h(x)}\)

2. TRANSLATE the rate of change into slope

• "For every increase of 1 in x, \(\mathrm{h(x)}\) increases by \(0.25\)" means:
  • When x changes by +1, \(\mathrm{h(x)}\) changes by +\(0.25\)
  • This gives us \(\mathrm{slope} = 0.25 = \frac{1}{4}\)

3. INFER the approach needed

• To find the y-intercept, we need the complete equation of the line
• We have: \(\mathrm{slope} = \frac{1}{4}\) and one point \(\mathrm{(8, -1)}\)
• Strategy: Use point-slope form, then evaluate at \(\mathrm{x} = 0\)

4. SIMPLIFY using point-slope form

• Start with: \(\mathrm{h(x)} - \mathrm{y_1} = \mathrm{m}(\mathrm{x} - \mathrm{x_1})\)
• Substitute our values: \(\mathrm{h(x)} - (-1) = \frac{1}{4}(\mathrm{x} - 8)\)
• Expand: \(\mathrm{h(x)} + 1 = \frac{1}{4}\mathrm{x} - 2\)
• Solve for \(\mathrm{h(x)}\): \(\mathrm{h(x)} = \frac{1}{4}\mathrm{x} - 2 - 1 = \frac{1}{4}\mathrm{x} - 3\)

5. SIMPLIFY to find the y-intercept

• Evaluate at \(\mathrm{x} = 0\): \(\mathrm{h(0)} = \frac{1}{4}(0) - 3 = 0 - 3 = -3\)
• The y-intercept is the point \(\mathrm{(0, -3)}\)

Answer: A. \(\mathrm{(0, -3)}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students may misinterpret "increases by \(0.25\)" as meaning the y-intercept is related to \(0.25\), or they might not recognize this describes the slope. Some students confuse this with the y-value itself changing by \(0.25\) from the given point.

This leads to confusion and guessing among the answer choices.

Second Most Common Error:

Inadequate SIMPLIFY execution: Students correctly identify the slope and set up point-slope form, but make algebraic errors when expanding \(\frac{1}{4}(\mathrm{x} - 8)\) or when combining \(-2 - 1\). Common mistakes include getting \(\mathrm{h(x)} = \frac{1}{4}\mathrm{x} - 2\) instead of \(\mathrm{h(x)} = \frac{1}{4}\mathrm{x} - 3\).

This may lead them to select Choice B: \(\mathrm{(0, -2)}\).

The Bottom Line:

This problem tests whether students can connect verbal descriptions of linear behavior to mathematical formulation. The key insight is recognizing that "increases by \(0.25\) for each increase of 1" directly gives you the slope, then systematically building the equation to find where it crosses the y-axis.