Transformations of the circle equation

Question

I know to translate the circle equation we just need to change the values in the brackets of the general circle equation $(x-x_1)^2+(y-y_1)^2=r^2$ , and in order to dilate a circle we need to double the value of the radius i.e times by $2^2$, however when it came to stretching and squeezing, I am confused if there is even a predictable method to showcase these transformations on a circle especially since the circle graph is not actually a function .

This further gets confusing when I noticed that the up and down translation behaved in the same way as the left and right translations.

What I mean by that is - usually, in a function, if we wanted to translate a graph on the x axis to the right we do $f(x-b)$ ,as opposed to $f(x+b)$ which is what a lot of people's first notion usually is , and similarly, if we wanted to move a function up, we do $f(x)+b$.

However, for the circle, when we write $(y+y_1)^2$, the circle shifts down by a value of $\ y_1$ rather than up a value of $\ y_1$ since usually in y transformations, adding any sort of value to the y value would make the graph go up (assuming that value was positive of course).

Which makes me wonder if there is a predictable way to represent transformation AND translations of a circle like we can do with functions $f(ax+b)$or $af(x)+b$ etc., since scenario (1*) and (2*) makes it hard for me to imagine so.

Answer 1

Let $F_{x_1,y_1,r}(x,y)$ the circle centered in $(x_1,y_1)$ and radius $r$. Obviously, this is not a function, but it is just the name I am giving to the equation of the circle.

If you want to move the circle in the $y$-axis, then you want to center it in a point of the form $(x_1,y_1+y_2)$, and you can write $F_{x_1,y_1+y_2,r}(x,y)=F_{x_1,y_1,r}(x,y-y_2)$. You can do the same thing to move the circle in the $x$-axis, and combine this with the technique for the $y$-axis to move the circle to any place you want. For the radius you just can do do $$F_{x_1,y_1,\lambda r}(x,y)=\lambda^2\cdot F_{\frac{x_1}{\lambda},\frac{y_1}{\lambda},r}\left(\frac{x}{\lambda},\frac{y}{\lambda}\right),$$ for some positive real number $\lambda$, to change from radius $r$ to radius $\lambda r$.

Answer 2

The confusion here seems to be about how translation and other transformations apply to the equation of a circle, which is not a function in the sense of passing the vertical line test but rather an implicit relation.

Let's clear up the confusion:

1. Translation: For the circle's equation $ (x - x_1)^2 + (y - y_1)^2 = r^2 $, the $ x_1 $ and $ y_1 $ terms represent the coordinates of the center of the circle. To translate the circle to the right by $ a $ units, you would replace $ x $ with $ x - a $, giving the equation $ (x - a - x_1)^2 + (y - y_1)^2 = r^2 $. Similarly, to translate up by $ b $ units, you replace $ y $ with $ y - b $, giving $ (x - x_1)^2 + (y - b - y_1)^2 = r^2 $. Notice that the signs appear to be "reversed" because you're modifying the equation to keep the center of the circle at the new location.

2. Dilation: To dilate the circle (change its size while keeping the center fixed), you multiply the radius by the dilation factor. For example, if you want to double the size of the circle, you replace $ r $ with $ 2r $, which gives the new equation $ (x - x_1)^2 + (y - y_1)^2 = (2r)^2 $.

3. Squeezing and Stretching: To stretch or squeeze the circle into an ellipse, you can multiply $ x $ or $ y $ by a factor. For instance, multiplying all $ x $ values by $ k $ would give the equation $ (\frac{x - x_1}{k})^2 + (y - y_1)^2 = r^2 $, which stretches the circle into an ellipse horizontally if $ k > 1 $ or squeezes it if $ k < 1 $.

4. Consistency with Function Translations: The reason why circle translations might seem inconsistent with function translations is due to the difference in how we express equations for functions (explicit functions of $ x $ or $ y $) versus relations like a circle's equation. For functions, $ f(x - a) $ translates to the right and $ f(x) + b $ translates up, while for relations, modifying the $ (x - x_1) $ and $ (y - y_1) $ terms directly translates the center of the circle.

To represent transformations and translations of a circle predictably, you simply need to remember that the $ (x - x_1) $ and $ (y - y_1) $ terms in the circle's equation are referencing the center of the circle. When you replace $ x $ with $ x - a $ or $ y $ with $ y - b $, you're effectively moving the center of the circle by $ a $ units in the $ x $-direction or $ b $ units in the $ y $-direction, respectively. The key is to keep track of how changes in the equation affect the center and size (radius) of the circle.