Question

describe the transformations needed to translate the graph of $y = \frac{1}{x}$ to the graph of $y = \frac{1}{x+3} - 4$.
a. to the right 3 and up 4
b. to the left 3 and up 4
c. to the right 3 and down 4
d. to the left 3 and down 4

Explanation:

Step1: Analyze horizontal translation

For a function $y = \frac{1}{x - h}$, $h>0$ shifts right, $h<0$ shifts left. Here, $y=\frac{1}{x+3}=\frac{1}{x-(-3)}$, so $h=-3$, shift left 3.

Step2: Analyze vertical translation

For a function $y = f(x) + k$, $k>0$ shifts up, $k<0$ shifts down. Here, $y=\frac{1}{x+3}-4$, so $k=-4$, shift down 4.

Answer:

D. to the left 3 and down 4