An equation is any expression with an equals sign, so your example is by definition an equation. Equations appear frequently in mathematics because mathematicians love to use equal signs. A formula is a set of instructions for creating a desired result. Non-mathematical examples include such things as chemical formulas (two H and one O make H2O), or the formula for Coca-Cola (which is just a ...
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: 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 ...