The function h satisfies \(\mathrm{h(x + 2) = 5x + 30}\) for all real numbers x. The graph of \(\mathrm{y...

1. TRANSLATE the problem information

• Given: \(\mathrm{h(x + 2) = 5x + 30}\)
• Need to find: x-intercept \(\mathrm{(a, 0)}\) and y-intercept \(\mathrm{(0, b)}\), then calculate \(\mathrm{a + b}\)
• This means we need to find the actual function \(\mathrm{h(x)}\) first

2. INFER the solution strategy

• To find intercepts, we need the explicit form \(\mathrm{h(x)}\) = something
• Since we only know \(\mathrm{h(x + 2)}\), we need to use substitution to "undo" the shift
• Let's set \(\mathrm{t = x + 2}\), which means \(\mathrm{x = t - 2}\)

3. SIMPLIFY to find h(x)

• Substitute \(\mathrm{x = t - 2}\) into \(\mathrm{h(x + 2) = 5x + 30}\):
• \(\mathrm{h(t) = 5(t - 2) + 30}\)
• \(\mathrm{h(t) = 5t - 10 + 30}\)
• \(\mathrm{h(t) = 5t + 20}\)
• Therefore: \(\mathrm{h(x) = 5x + 20}\)

4. TRANSLATE intercept requirements and solve

• Y-intercept: Set \(\mathrm{x = 0}\), so \(\mathrm{b = h(0) = 5(0) + 20 = 20}\)
• X-intercept: Set \(\mathrm{h(x) = 0}\), so \(\mathrm{5a + 20 = 0}\)
• SIMPLIFY: \(\mathrm{5a = -20}\), therefore \(\mathrm{a = -4}\)

5. Calculate final answer

• \(\mathrm{a + b = -4 + 20 = 16}\)

Answer: (B) 16

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Many students don't recognize that they need to find \(\mathrm{h(x)}\) first before finding intercepts. They might try to work directly with \(\mathrm{h(x + 2) = 5x + 30}\) and attempt to find intercepts from this form, not realizing this represents a shifted function. This leads to confusion and guessing.

Second Most Common Error:

Poor TRANSLATE reasoning: Students might correctly find \(\mathrm{h(x) = 5x + 20}\) but then confuse the intercept definitions. They might think the y-intercept requires setting \(\mathrm{h(x) = 0}\) instead of finding \(\mathrm{h(0)}\), or make sign errors when solving \(\mathrm{5a + 20 = 0}\). This may lead them to select Choice (A) (14) or Choice (C) (20).

The Bottom Line:

This problem tests whether students can work backwards from a function transformation to find the original function, then apply intercept concepts correctly. The key insight is recognizing that substitution is needed to "unwrap" the shifted function.