The function h is linear. The values of h are \(\mathrm{h(1) = 8}\) and \(\mathrm{h(4) = 20}\). Which equation defines...

1. TRANSLATE the problem information

• Given information:
  • \(\mathrm{h(1) = 8}\) means when \(\mathrm{x = 1}\), the output is 8
  • \(\mathrm{h(4) = 20}\) means when \(\mathrm{x = 4}\), the output is 20
• This gives us two coordinate points: \(\mathrm{(1, 8)}\) and \(\mathrm{(4, 20)}\)

2. INFER the solution approach

• Since h is linear, it has form \(\mathrm{h(x) = mx + b}\)
• We need to find both m (slope) and b (y-intercept)
• Strategy: Use the two points to find slope first, then use either point to find y-intercept

3. SIMPLIFY to find the slope

• Using slope formula: \(\mathrm{m = \frac{y_2 - y_1}{x_2 - x_1}}\)
• \(\mathrm{m = \frac{20 - 8}{4 - 1}}\)
  \(\mathrm{= \frac{12}{3}}\)
  \(\mathrm{= 4}\)

4. SIMPLIFY to find the y-intercept

• Use point \(\mathrm{(1, 8)}\) in the equation \(\mathrm{h(x) = 4x + b}\):
  • \(\mathrm{8 = 4(1) + b}\)
  • \(\mathrm{8 = 4 + b}\)
  • \(\mathrm{b = 4}\)

5. Write final equation

• \(\mathrm{h(x) = 4x + 4}\)

Answer: A

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Students make algebraic errors when solving for b, particularly with sign operations.

When solving \(\mathrm{8 = 4 + b}\), they might incorrectly subtract: \(\mathrm{b = 8 - 4 = 4}\), but then incorrectly think the y-intercept should be negative, or make a sign error during the process and get \(\mathrm{b = -4}\).

This may lead them to select Choice D \(\mathrm{(h(x) = 4x - 4)}\)

Second Most Common Error:

Poor INFER reasoning: Students confuse slope calculation and use the slope value incorrectly.

Some students calculate the slope as 12 (numerator only) instead of \(\mathrm{\frac{12}{3} = 4}\), then try to make an equation work. With slope = 12 and point \(\mathrm{(1, 8)}\): \(\mathrm{8 = 12(1) + b}\) gives \(\mathrm{b = -4}\).

This may lead them to select Choice C \(\mathrm{(h(x) = 12x - 4)}\)

The Bottom Line:

Linear function problems require systematic execution - students who rush through the algebra or don't verify their slope calculation often select tempting incorrect answers that use the right numbers in wrong places.