\(\mathrm{f(x) = \frac{a-19}{x} + 5}\)In the given function f, a is a constant. The graph of function f in the...

1. TRANSLATE the transformation information

• Given information:
  • Original function: \(\mathrm{f(x) = \frac{(a-19)}{x} + 5}\)
  • Translation: 3 units down and 4 units right
  • Need to find: equation for \(\mathrm{g(x)}\)

2. INFER the transformation rules

• For function transformations:
  • Moving right h units: replace x with \(\mathrm{(x - h)}\)
  • Moving down k units: subtract k from the entire function
• So the general form is: \(\mathrm{g(x) = f(x - 4) - 3}\)

3. SIMPLIFY by applying the transformations

• First, find \(\mathrm{f(x - 4)}\):
  \(\mathrm{f(x - 4) = \frac{(a-19)}{(x-4)} + 5}\)

• Then subtract 3 for the downward shift:
  \(\mathrm{g(x) = f(x - 4) - 3 = \frac{(a-19)}{(x-4)} + 5 - 3}\)

• Combine constants:
  \(\mathrm{g(x) = \frac{(a-19)}{(x-4)} + 2}\)

Answer: B. \(\mathrm{g(x) = \frac{(a-19)}{(x-4)} + 2}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Confusing the direction of horizontal transformations

Students often think "4 units to the right" means adding 4 to x, writing \(\mathrm{f(x + 4)}\) instead of \(\mathrm{f(x - 4)}\). This counterintuitive rule (right means subtract) is frequently forgotten.

This may lead them to select Choice A (\(\mathrm{g(x) = \frac{(a-19)}{(x+4)} + 2}\))

Second Most Common Error:

Incomplete SIMPLIFY execution: Forgetting to combine the vertical shift

Students correctly handle the horizontal transformation but forget that "3 units down" means subtracting 3 from the entire function, not just the constant term. They might write \(\mathrm{g(x) = \frac{(a-19)}{(x-4)} + 5}\) instead of properly combining \(\mathrm{5 - 3 = 2}\).

This may lead them to select Choice D (\(\mathrm{g(x) = \frac{(a-22)}{(x-4)} + 5}\)) or cause confusion leading to guessing.

The Bottom Line:

Function transformations require careful attention to direction conventions, especially that horizontal movements work opposite to intuition (right means subtract from x, left means add to x).