Question:The function p is defined by \(\mathrm{p(x) = (x + 5)(x - 1)(x - 4)}\). In the xy-plane, the graph...

1. TRANSLATE the transformation description

• Given information:
  • Original function: \(\mathrm{p(x) = (x + 5)(x - 1)(x - 4)}\)
  • Translation: right 2 units and down 3 units
  • Need to find: \(\mathrm{h(2)}\)

• TRANSLATE tells us the transformation rule:
  • "Right 2 units" means replace \(\mathrm{x}\) with \(\mathrm{(x - 2)}\)
  • "Down 3 units" means subtract 3 from the entire function

2. INFER the new function form

• Since we're translating \(\mathrm{p(x)}\) right 2 and down 3:
  \(\mathrm{h(x) = p(x - 2) - 3}\)

• INFER the strategy: To find \(\mathrm{h(2)}\), substitute \(\mathrm{x = 2}\):
  \(\mathrm{h(2) = p(2 - 2) - 3}\)
  \(\mathrm{= p(0) - 3}\)

3. SIMPLIFY by evaluating \(\mathrm{p(0)}\)

• Calculate \(\mathrm{p(0)}\) by substituting \(\mathrm{x = 0}\) into the original function:
  \(\mathrm{p(0) = (0 + 5)(0 - 1)(0 - 4)}\)
  \(\mathrm{p(0) = (5)(-1)(-4)}\)
  \(\mathrm{p(0) = 20}\)

4. SIMPLIFY the final calculation

• \(\mathrm{h(2) = p(0) - 3}\)
  \(\mathrm{= 20 - 3}\)
  \(\mathrm{= 17}\)

Answer: 17

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students often confuse the direction of horizontal transformations, writing \(\mathrm{h(x) = p(x + 2) - 3}\) instead of \(\mathrm{h(x) = p(x - 2) - 3}\). They think "right 2 units" means adding 2, when it actually means replacing \(\mathrm{x}\) with \(\mathrm{(x - 2)}\).

This leads them to calculate \(\mathrm{h(2) = p(4) - 3}\) instead of \(\mathrm{p(0) - 3}\). Since \(\mathrm{p(4) = (4 + 5)(4 - 1)(4 - 4)}\)
\(\mathrm{= (9)(3)(0)}\)
\(\mathrm{= 0}\), they get \(\mathrm{h(2) = 0 - 3 = -3}\), leading to confusion and potentially guessing.

Second Most Common Error:

Poor SIMPLIFY execution: Students make sign errors when calculating \(\mathrm{p(0) = (5)(-1)(-4)}\). They might compute this as \(\mathrm{-20}\) instead of \(\mathrm{+20}\), forgetting that \(\mathrm{(-1) \times (-4) = +4}\). This gives them \(\mathrm{h(2) = -20 - 3 = -23}\), which seems unreasonable and causes them to second-guess their work.

The Bottom Line:

Function transformations require careful attention to the counterintuitive nature of horizontal shifts - moving right means subtracting from the input. Combined with multi-step evaluation involving negative numbers, this creates multiple opportunities for errors.