Question

1. draw the circle given by $x^{2}+y^{2}=4$ in the interactive graph.

Explanation:

Step1: Recall the standard - form of a circle equation

The standard form of a circle equation is $(x - a)^2+(y - b)^2=r^2$, where $(a,b)$ is the center of the circle and $r$ is the radius. For the equation $x^{2}+y^{2}=4$, we can rewrite it as $(x - 0)^2+(y - 0)^2 = 2^2$. So the center is at the origin $(0,0)$ and the radius $r = 2$.

Step2: Plot key points

We know that for a circle centered at the origin with radius $r = 2$, the points on the circle along the $x$ - axis are $(2,0)$ and $(-2,0)$ (when $y = 0$, $x=\pm2$), and the points on the circle along the $y$ - axis are $(0,2)$ and $(0, - 2)$ (when $x = 0$, $y=\pm2$). Then we can smoothly connect these points to form a circle.

Answer:

The circle is centered at the origin $(0,0)$ with a radius of 2, passing through the points $(2,0),(-2,0),(0,2),(0, - 2)$.