I'm trying to help a student with following rational equation question:
Describe the transformations of $$g(x) = \frac{-4x - 2}{7x +1}$$ from the graph of $$f(x) = \frac{1}{x}.$$
The given answers are horizontal shift to the left of $\frac{1}{7}$ units, a vertical shift down of $\frac{4}{7}$ units and vertical shrink factor of $\frac{10}{49}$ units.
The vertical and horizontal shifts make sense based on the new Asymptotes compared to the graph of $f(x) = \frac{1}{x}$, but I have no idea how this shrink factor is calculated. Any help would be appreciated!
$$y=\frac {-4x-2}{7x+1} \\ x \to X-\frac 17\\y \to Y+\frac 47 \\ Y=\frac {-10}{49X}=\frac {-10}{49} \times \frac 1X$$it means firstly transform by $-1$ then by $\frac {-10}{49}$
In general for $f(x)=\frac {ax+b}{cx+d}$ when you use transform $$x\to X-\frac dc\\y\to Y+\frac ac$$ it will be turnd to $$Y+\frac ac=\frac {a(X-\frac dc)+b}{c(X-\frac dc)+d}\\Y+\frac ac=\frac {aX-a\frac dc+b}{cX}\\ Y+\frac ac=\frac {aX+\frac {-(ad-bc)}{c}}{cX}\\Y=\frac {aX+\frac {-(ad-bc)}{c}}{cX}-+\frac ac\\Y=\frac {-(ad-bc)}{c^2X}\\Y=\frac{-det(\begin{bmatrix}a & b \\c & d \end{bmatrix})}{c^2}\times \frac 1X$$