In the xy-plane, a linear function h satisfies the following: For every increase of 8 in x, the value of...

1. TRANSLATE the problem information

• Given information:
  • \(\mathrm{h}\) is a linear function
  • For every increase of 8 in \(\mathrm{x}\), \(\mathrm{h(x)}\) increases by 20
  • \(\mathrm{h(-2) = 5}\)
• What this tells us: We need to find the equation \(\mathrm{y = h(x)}\) in slope-intercept form

2. INFER the approach

• Since \(\mathrm{h}\) is linear, it has the form \(\mathrm{y = mx + b}\)
• The rate of change gives us the slope directly
• We'll use the given point to find the y-intercept

3. TRANSLATE the rate information to find slope

• "For every increase of 8 in x, h(x) increases by 20"
• This means: when \(\mathrm{Δx = 8}\), \(\mathrm{Δy = 20}\)
• Slope \(\mathrm{m = \frac{Δy}{Δx} = \frac{20}{8} = \frac{5}{2}}\)

4. SIMPLIFY to find the y-intercept

• Use the point \(\mathrm{(-2, 5)}\) in \(\mathrm{y = mx + b}\):
• \(\mathrm{5 = \frac{5}{2}(-2) + b}\)
• \(\mathrm{5 = -5 + b}\)
• \(\mathrm{b = 10}\)

5. INFER the final equation

• With \(\mathrm{m = \frac{5}{2}}\) and \(\mathrm{b = 10}\): \(\mathrm{y = \frac{5}{2}x + 10}\)

Answer: B. \(\mathrm{y = \frac{5}{2}x + 10}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Misinterpreting "for every increase of 8 in x, h(x) increases by 20" as meaning the slope is \(\mathrm{\frac{8}{20}}\) instead of \(\mathrm{\frac{20}{8}}\).

Students might think "8 in x" comes first, so they put 8 in the numerator, getting slope = \(\mathrm{\frac{8}{20} = \frac{2}{5}}\). Using this wrong slope with point \(\mathrm{(-2, 5)}\):

\(\mathrm{5 = \frac{2}{5}(-2) + b}\)

\(\mathrm{5 = -\frac{4}{5} + b}\)

\(\mathrm{b = 5 + \frac{4}{5} = \frac{29}{5}}\)

This doesn't lead to any of the given answers, causing confusion and guessing. However, if they somehow get \(\mathrm{b = 10}\) through calculation errors, this may lead them to select Choice A (\(\mathrm{y = \frac{2}{5}x + 10}\)).

Second Most Common Error:

Inadequate SIMPLIFY execution: Getting the slope correct as \(\mathrm{\frac{5}{2}}\) but making sign errors when finding the y-intercept.

Students might write: \(\mathrm{5 = \frac{5}{2}(-2) + b}\) → \(\mathrm{5 = 5 + b}\) → \(\mathrm{b = 0}\), forgetting that \(\mathrm{\frac{5}{2}(-2) = -5}\), not \(\mathrm{+5}\). Or they might get confused with the negative signs and calculate \(\mathrm{b = -10}\) instead of \(\mathrm{+10}\).

This may lead them to select Choice C (\(\mathrm{y = \frac{5}{2}x - 10}\)).

The Bottom Line:

This problem tests whether students can correctly translate a verbal rate description into mathematical slope, then execute the algebra accurately to find the y-intercept.