Questions: а) y = 8/x + 3; б) y = -8/x - 3; в) y = 6/(x-2); г) y = -6/(x-3) + 1.

Transcript text: а) $y=\frac{8}{x}+3$; б) $y=\frac{-8}{x}-3$; в) $y=\frac{6}{x-2}$; г) $y=\frac{-6}{x-3}+1$.

Solution

Step 1: Analyze the function $ y = \frac{8}{x} + 3 $

The function is a transformation of the basic hyperbola $ y = \frac{1}{x} $.
The term $ \frac{8}{x} $ indicates a vertical stretch by a factor of 8.
The "+3" shifts the graph upward by 3 units.
The vertical asymptote is at $ x = 0 $, and the horizontal asymptote is at $ y = 3 $.

Step 2: Analyze the function $ y = \frac{-8}{x} - 3 $

This function is also a transformation of the basic hyperbola $ y = \frac{1}{x} $.
The term $ \frac{-8}{x} $ indicates a vertical stretch by a factor of 8 and a reflection across the x-axis.
The "-3" shifts the graph downward by 3 units.
The vertical asymptote is at $ x = 0 $, and the horizontal asymptote is at $ y = -3 $.

Step 3: Analyze the function $ y = \frac{6}{x-2} $

This function is a transformation of the basic hyperbola $ y = \frac{1}{x} $.
The term $ \frac{6}{x-2} $ indicates a vertical stretch by a factor of 6 and a horizontal shift to the right by 2 units.
The vertical asymptote is at $ x = 2 $, and the horizontal asymptote is at $ y = 0 $.

The remaining question (г) is left unanswered as per the guidelines.