Question:The function f is defined by \(\mathrm{f(x) = (x - 4)(x - 8)(x + 2)}\). In the xy-plane, the graph...

1. TRANSLATE the transformation information

• Given information:
  • Original function: \(\mathrm{f(x) = (x - 4)(x - 8)(x + 2)}\)
  • Graph of \(\mathrm{g(x)}\) results from translating \(\mathrm{f(x)}\) left by 3 units
  • Need to find: \(\mathrm{g(0)}\)
• What this tells us: When we translate a function left by 3 units, we get \(\mathrm{g(x) = f(x + 3)}\)

2. INFER what we need to calculate

• Since \(\mathrm{g(x) = f(x + 3)}\), then \(\mathrm{g(0) = f(0 + 3) = f(3)}\)
• Strategy: Calculate \(\mathrm{f(3)}\) using the original function definition

3. SIMPLIFY the function evaluation

• Substitute \(\mathrm{x = 3}\) into \(\mathrm{f(x) = (x - 4)(x - 8)(x + 2)}\):
  • \(\mathrm{f(3) = (3 - 4)(3 - 8)(3 + 2)}\)
  • \(\mathrm{f(3) = (-1)(-5)(5)}\)
• Calculate step by step:
  • First: \(\mathrm{(-1) \times (-5) = 5}\)
  • Then: \(\mathrm{5 \times 5 = 25}\)

Answer: 25

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students confuse the direction of horizontal translations and think 'left by 3 units' means \(\mathrm{g(x) = f(x - 3)}\) instead of \(\mathrm{g(x) = f(x + 3)}\).

With this misconception, they would calculate \(\mathrm{g(0) = f(0 - 3) = f(-3)}\):

\(\mathrm{f(-3) = (-3 - 4)(-3 - 8)(-3 + 2) = (-7)(-11)(-1) = -77}\)

This leads to confusion since -77 likely isn't among the answer choices, causing them to guess.

Second Most Common Error:

Poor SIMPLIFY execution: Students correctly set up \(\mathrm{g(0) = f(3)}\) but make sign errors when multiplying \(\mathrm{(-1)(-5)(5)}\).

Common mistakes include:

• Treating \(\mathrm{(-1)(-5)}\) as -5 instead of +5
• Getting confused about whether the final result should be positive or negative
• Calculating \(\mathrm{5 \times 5}\) incorrectly under pressure

This may lead to answers like -25 or other incorrect values.

The Bottom Line:

Function transformations require careful attention to the counterintuitive relationship between the transformation description and the algebraic form. 'Left by h' means '+h' inside the function, not '-h'.