Evaluate the following Sage code block to produce a Sage interact which creates plots of $\,f(x)\,$ and $\,c\,f(x+h)+k\,$ (on the same axes) where $\,f(x)\,$ can be modified by text entry and $h$, $c$ and $k$ can be controlled by sliders. The graph of the "original" function $f$ is blue. The shifted and scaled function is plotted in red.
Scaling (in implicit form)... an interesting example
Let's call the hyperbola $xy=1$, H1, and the related hyperbola $xy=2$, H2. The equation for H2 can be recast as $(\frac{1}{2}x)y=1$, $x(\frac{1}{2}y)=1$, or $(\frac{1}{\sqrt{2}}x)(\frac{1}{\sqrt{2}}y)=1$. The first of these three forms shows that the graph of H2 is the graph of H1 expanded horizontally by a factor of 2. The second shows that H2 is H1 expanded vertically by a factor 2. And the third shows that H2 is H1 expanded both horizontally and vertically by a common factor, namely $\sqrt{2}$, i.e. the graph of H2 is a simple dilation of the graph of H1 by factor of $\sqrt{2}$. Are these three different interpretions of the relationship between H1 and H2 compatible? Take a look at the following Interact and see!