Question

the graph of ( xy = 12 ) is translated up 6 units and to the right 4 units. which are possible equations for the translated graph? select all that apply.
a. ( y = 6 + \frac{12}{x - 4} )
b. ( y = -6 + \frac{12}{x + 4} )
c. ( y = \frac{-6x - 12}{x + 4} )
d. ( y = \frac{6x - 12}{x - 4} )

Explanation:

Step1: Rewrite original equation

Start with $xy=12$, solve for $y$:
$y = \frac{12}{x}$

Step2: Apply horizontal translation

Shift right 4 units: replace $x$ with $x-4$:
$y = \frac{12}{x - 4}$

Step3: Apply vertical translation

Shift up 6 units: add 6 to the function:
$y = 6 + \frac{12}{x - 4}$ (matches option A)

Step4: Combine terms for option D

Combine the terms into a single fraction:
$y = \frac{6(x - 4)}{x - 4} + \frac{12}{x - 4} = \frac{6x - 24 + 12}{x - 4} = \frac{6x - 12}{x - 4}$ (matches option D)

Answer:

A. $y = 6 + \frac{12}{x - 4}$, D. $y = \frac{6x - 12}{x - 4}$