\(\mathrm{f(x) = (x - 1)(x + 3)(x - 2)}\) In the xy-plane, when the graph of the function f, where...

1. TRANSLATE the transformation information

• Given information:
  • Original function: \(\mathrm{f(x) = (x - 1)(x + 3)(x - 2)}\)
  • The graph is "shifted up 6 units" to create function g
  • The graph of \(\mathrm{y = g(x)}\) "crosses through point (4, b)"

• What this tells us:
  • A vertical shift up 6 units means: \(\mathrm{g(x) = f(x) + 6}\)
  • "Crosses through point (4, b)" means when \(\mathrm{x = 4}\), the y-value is b, so \(\mathrm{g(4) = b}\)

2. TRANSLATE the question into a calculation

• We need to find the value of b
• Since \(\mathrm{g(4) = b}\), we need to calculate \(\mathrm{g(4)}\)
• Substitute the original function: \(\mathrm{g(x) = (x - 1)(x + 3)(x - 2) + 6}\)

3. SIMPLIFY by evaluating g(4)

• Substitute \(\mathrm{x = 4}\) into \(\mathrm{g(x)}\):
  \(\mathrm{g(4) = (4 - 1)(4 + 3)(4 - 2) + 6}\)

• Calculate each factor:
  • \(\mathrm{(4 - 1) = 3}\)
  • \(\mathrm{(4 + 3) = 7}\)
  • \(\mathrm{(4 - 2) = 2}\)

• Multiply the factors: \(\mathrm{(3)(7)(2) = 42}\)

• Add the vertical shift: \(\mathrm{42 + 6 = 48}\)

Answer: 48

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Misunderstanding vertical transformations and thinking "shifted up 6 units" means \(\mathrm{g(x) = f(x + 6)}\) instead of \(\mathrm{g(x) = f(x) + 6}\).

Students often confuse horizontal shifts (which affect the input) with vertical shifts (which affect the output). With this error, they would calculate \(\mathrm{f(4 + 6) = f(10)}\) instead of \(\mathrm{f(4) + 6}\), leading to a completely different and much larger result. This leads to confusion and potentially selecting a wrong answer or guessing.

Second Most Common Error:

Poor SIMPLIFY execution: Making arithmetic mistakes during the multi-step calculation.

For example, students might correctly set up \(\mathrm{g(4) = (3)(7)(2) + 6}\) but then calculate \(\mathrm{(3)(7)(2) = 48}\) instead of 42, leading to a final answer of 54 instead of 48. Or they might forget to add the +6 at the end, getting just 42. These arithmetic slips can cause them to doubt their approach or select an incorrect answer.

The Bottom Line:

This problem tests whether students truly understand the difference between horizontal and vertical function transformations, combined with careful arithmetic execution. The key insight is recognizing that "up 6 units" affects the output (y-values), not the input (x-values).