Question

1. graph the rational function $y = \frac{x - 1}{x - 2}$. both branches of the rational function pass through which quadrant?

options:

• quadrant 2
• quadrant 3
• quadrant 1
• quadrant 4

Explanation:

Step1: Simplify the function

$y=\frac{x-1}{x-2}=1+\frac{1}{x-2}$

Step2: Analyze the branches

• For $x>2$: $x-2>0$, $\frac{1}{x-2}>0$, so $y>1$ (Quadrant 1)
• For $x<2$: $x-2<0$, $\frac{1}{x-2}<0$, so $y<1$. When $x<1$, $x-1<0$, $y=\frac{negative}{negative}=positive$; when $10$, $y=\frac{positive}{negative}=negative$. But overall, both branches pass through Quadrant 1 and 3.

Answer:

quadrant 1, quadrant 3