Problem with understanding how to scale a function I'm trying to scale the following function:
$$y=\sqrt{1-\frac{x^2}{a+bx}\times c}$$
For example, I want to scale it proportionately by 2. Turns out I need to multiply a and c by 4 (2 squared), but b by just 2. I assume this is something about square root and it also has to do with whether the constant is standing alone or being multiplied by x, but I'm not sure of what's actually going on and how to explain this. Can someone pls help?
Edit: for clarification, what I am trying to achieve is to keep the shape of the original function but scale the major axis of the half ellipse (domain 0, 4.2), not just scaling vertically or horizontally. Essentially trying to get the same shape with bigger size. As the screenshot shows, in order to scale it proportionately by 2, I had to multiply b by 2, but a and c had to be multiplied by 4.


In order to scale the graph of an equation in two variables $x,y$ by a constant factor $k$ one replaces each occurence of $x$ with $\frac{x}{k}$ and each occurence of $y$ with $\frac{y}{k}$. In your example to double the scale for both $x$ and $y$
$$y=\sqrt{1-\frac{x^{2}}{a+bx}\times c}$$
must be replaced with
$$ \frac{y}{2}=\sqrt{1-\frac{\left(\frac{x}{2}\right)^{2}}{a+b\left(\frac{x}{2}\right)}\cdot c}$$
Here is a desmos illustration.
