The function h is defined by \(\mathrm{h(x) = 4x + 28}\). The graph of \(\mathrm{y = h(x)}\) in the xy-plane...

1. TRANSLATE the problem information

• Given information:
  • Function: \(\mathrm{h(x) = 4x + 28}\)
  • Graph has x-intercept at \(\mathrm{(a, 0)}\) and y-intercept at \(\mathrm{(0, b)}\)
  • Need to find: \(\mathrm{a + b}\)

• What this tells us: We need to find where the graph crosses both axes

2. INFER the approach

• To find intercepts, we substitute specific values:
  • X-intercept: set \(\mathrm{y = 0}\) and solve for x
  • Y-intercept: set \(\mathrm{x = 0}\) and solve for y
• The function \(\mathrm{y = h(x)}\) becomes \(\mathrm{y = 4x + 28}\)

3. SIMPLIFY to find the x-intercept

• Set \(\mathrm{y = 0}\): \(\mathrm{0 = 4x + 28}\)
• Subtract 28: \(\mathrm{-28 = 4x}\)
• Divide by 4: \(\mathrm{x = -7}\)
• So \(\mathrm{a = -7}\) (the x-coordinate of the x-intercept)

4. SIMPLIFY to find the y-intercept

• Set \(\mathrm{x = 0}\): \(\mathrm{y = 4(0) + 28}\)
• Calculate: \(\mathrm{y = 28}\)
• So \(\mathrm{b = 28}\) (the y-coordinate of the y-intercept)

5. Calculate final answer

• \(\mathrm{a + b = -7 + 28 = 21}\)

Answer: A. 21

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY skill: Students make a sign error when solving \(\mathrm{-28 = 4x}\), dividing incorrectly to get \(\mathrm{x = 7}\) instead of \(\mathrm{x = -7}\).

When they calculate \(\mathrm{7 + 28 = 35}\), this leads them to select Choice D (35).

Second Most Common Error:

Poor TRANSLATE reasoning: Students confuse the coordinates, thinking the x-intercept gives them the y-value they need, or mixing up which variable to set to zero.

This leads to confusion about what a and b represent, causing them to get stuck and guess among the remaining choices.

The Bottom Line:

This problem tests whether students can correctly connect the geometric concept of intercepts with the algebraic process of substitution, while maintaining accuracy through negative number arithmetic.