\(\mathrm{f(x) = (x + 6)(x + 5)(x - 4)}\)The function f is given. Which table of values represents \(\mathrm{y =...

1. TRANSLATE the problem information

• Given information:
  • \(\mathrm{f(x) = (x + 6)(x + 5)(x - 4)}\)
  • We need the table for \(\mathrm{y = f(x) - 3}\)

• This tells us: \(\mathrm{y = (x + 6)(x + 5)(x - 4) - 3}\)

2. INFER the solution strategy

• We need to evaluate \(\mathrm{y = (x + 6)(x + 5)(x - 4) - 3}\) for each x-value in the answer tables
• The x-values to check are -6, -5, and 4
• Key insight: Look for when factors equal zero to simplify calculations

3. SIMPLIFY by evaluating each x-value

For x = -6:

\(\mathrm{y = (-6 + 6)(-6 + 5)(-6 - 4) - 3}\)

\(\mathrm{y = (0)(-1)(-10) - 3}\)

Since one factor is 0, the product is 0

\(\mathrm{y = 0 - 3 = -3}\)

For x = -5:

\(\mathrm{y = (-5 + 6)(-5 + 5)(-5 - 4) - 3}\)

\(\mathrm{y = (1)(0)(-9) - 3}\)

Since one factor is 0, the product is 0

\(\mathrm{y = 0 - 3 = -3}\)

For x = 4:

\(\mathrm{y = (4 + 6)(4 + 5)(4 - 4) - 3}\)

\(\mathrm{y = (10)(9)(0) - 3}\)

Since one factor is 0, the product is 0

\(\mathrm{y = 0 - 3 = -3}\)

4. INFER the pattern and select answer

• All three y-values equal -3
• The correct table shows (-6, -3), (-5, -3), and (4, -3)

Answer: B

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students may not properly substitute \(\mathrm{f(x)}\) into the equation \(\mathrm{y = f(x) - 3}\), instead trying to work with \(\mathrm{f(x)}\) directly or getting confused about the transformation.

They might try to find \(\mathrm{f(-6)}\), \(\mathrm{f(-5)}\), and \(\mathrm{f(4)}\) first, then subtract 3, but make calculation errors in the intermediate steps. Or they might not realize that \(\mathrm{y = f(x) - 3}\) requires them to evaluate the entire expression \(\mathrm{(x + 6)(x + 5)(x - 4) - 3}\).

This leads to confusion and guessing, or potentially selecting Choice A if they make arithmetic errors.

Second Most Common Error:

Poor SIMPLIFY execution: Students correctly set up \(\mathrm{y = (x + 6)(x + 5)(x - 4) - 3}\) but fail to recognize the zero multiplication property when evaluating.

Instead of seeing that factors like \(\mathrm{(x + 6) = 0}\) when \(\mathrm{x = -6}\), they attempt to multiply all factors fully, leading to unnecessary complex arithmetic and potential calculation errors. This can result in wrong y-values and selection of incorrect answer choices.

The Bottom Line:

This problem tests whether students can handle function transformations and recognize computational shortcuts. The key insight is that the chosen x-values (-6, -5, 4) are specifically the zeros of the individual factors, making each product equal to zero before subtracting 3.