The function f is defined by \(\mathrm{f(x) = (x - 6)(x - 2)(x + 6)}\). In the xy-plane, the graph...

1. TRANSLATE the transformation information

• Given information:
  • \(\mathrm{f(x) = (x - 6)(x - 2)(x + 6)}\)
  • \(\mathrm{g(x)}\) is \(\mathrm{f(x)}\) translated up 4 units
  • Need to find \(\mathrm{g(0)}\)

• What "up 4 units" means: \(\mathrm{g(x) = f(x) + 4}\)

2. INFER the solution strategy

• To find \(\mathrm{g(0)}\), we need to substitute \(\mathrm{x = 0}\) into our expression for \(\mathrm{g(x)}\)
• We can substitute directly since we have the explicit formula

3. Set up the expression for g(x)

• Since \(\mathrm{g(x) = f(x) + 4}\):
  \(\mathrm{g(x) = (x - 6)(x - 2)(x + 6) + 4}\)

4. SIMPLIFY by substituting x = 0

• \(\mathrm{g(0) = (0 - 6)(0 - 2)(0 + 6) + 4}\)
• \(\mathrm{g(0) = (-6)(-2)(6) + 4}\)

5. SIMPLIFY the arithmetic step by step

• First multiply: \(\mathrm{(-6)(-2) = 12}\)
• Then: \(\mathrm{12 \times 6 = 72}\)
• Finally: \(\mathrm{72 + 4 = 76}\)

Answer: 76

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students might misinterpret "up 4 units" as affecting the input rather than the output, writing something like \(\mathrm{g(x) = f(x + 4)}\) instead of \(\mathrm{g(x) = f(x) + 4}\).

This fundamental misunderstanding of vertical vs horizontal translations leads to calculating \(\mathrm{f(4)}\) instead of \(\mathrm{f(0) + 4}\), which gives a completely different result and causes confusion.

Second Most Common Error:

Poor SIMPLIFY execution: Students correctly set up \(\mathrm{g(0) = (-6)(-2)(6) + 4}\) but make sign errors in the multiplication, such as getting \(\mathrm{-72}\) instead of \(\mathrm{+72}\) when multiplying the three terms.

This leads them to calculate \(\mathrm{g(0) = -72 + 4 = -68}\), resulting in a negative answer that doesn't match any reasonable expectation for this type of problem.

The Bottom Line:

This problem tests whether students understand that vertical transformations affect the function's output (adding to the entire function) rather than its input, combined with careful arithmetic involving negative numbers.