4

enter image description hereI'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.

function scaled proportionately by 2

Sine of the Time
  • 4,931
Asha R
  • 341
  • 1
    In general to vertically scale a function $f$ by $s$, shouldn't you just use the function $s \cdot f(x)$? I don't see the need for those multiplications (and they, in fact, don't match up with what I envision scaling to mean, as one might see here) – PrincessEev Feb 21 '23 at 22:38
  • 1
    To enclose a number of terms inside the radical, use {}. If you click on "Edit" you'll see the exact syntax I used. – lulu Feb 21 '23 at 22:39
  • @PrincessEev I don't know if what I am doing is called scaling, but essentially 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 – Asha R Feb 21 '23 at 22:42
  • @lulu Thank you – Asha R Feb 21 '23 at 22:42
  • Can you [edit] to post a picture of what you have and what you want? Perhaps from desmos? – Ethan Bolker Feb 21 '23 at 22:50
  • @EthanBolker Yes, now there is a screenshot. I already have what I want, I just don't know how to explain it. – Asha R Feb 21 '23 at 22:59
  • 1
    I guess the mathematical term is "Homothety":https://en.wikipedia.org/wiki/Homothety – NoChance Feb 22 '23 at 00:22

1 Answers1

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.

John Wayland Bales
  • 21,814