A function g is defined by the equation \(\mathrm{g(x) = (x + 3)(x - 2)(x - 8)}\). The graph of...

1. TRANSLATE the problem information

• Given information:
  • Function: \(\mathrm{g(x) = (x + 3)(x - 2)(x - 8)}\)
  • The graph passes through point \(\mathrm{(0, b)}\)
  • Need to find the value of b

• What this tells us: If the graph passes through \(\mathrm{(0, b)}\), then when \(\mathrm{x = 0}\), the y-value equals b. In function notation: \(\mathrm{g(0) = b}\)

2. SIMPLIFY by substituting and calculating

• Substitute \(\mathrm{x = 0}\) into the function:
  \(\mathrm{g(0) = (0 + 3)(0 - 2)(0 - 8)}\)

• Simplify each factor:
  \(\mathrm{g(0) = (3)(-2)(-8)}\)

• Calculate step by step:
  • First: \(\mathrm{(3)(-2) = -6}\)
  • Then: \(\mathrm{(-6)(-8) = 48}\)

• Therefore: \(\mathrm{g(0) = 48}\), which means \(\mathrm{b = 48}\)

Answer: D) 48

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Not understanding what "passes through \(\mathrm{(0, b)}\)" means in function terms

Students may recognize they need to substitute \(\mathrm{x = 0}\), but don't connect this to finding the y-coordinate b. They might try manipulating the factored form or looking for intercepts instead of directly evaluating \(\mathrm{g(0)}\).

This leads to confusion and guessing among the answer choices.

Second Most Common Error:

Poor SIMPLIFY execution: Sign errors during multiplication

The expression \(\mathrm{(3)(-2)(-8)}\) involves multiplying negative numbers. Students often make sign errors, particularly with \(\mathrm{(-2)(-8)}\). If they incorrectly calculate \(\mathrm{(-2)(-8) = -16}\) instead of +16, they get \(\mathrm{g(0) = (3)(-16) = -48}\).

This may lead them to select Choice A (-48).

The Bottom Line:

This problem tests whether students can translate coordinate language into function evaluation and then execute arithmetic correctly with signed numbers. The key insight is recognizing that "passes through \(\mathrm{(0, b)}\)" directly translates to "evaluate the function at \(\mathrm{x = 0}\)."