Question

How do you graph `y=-sqrt(4-x^2)`?

Answer 1

Represents a semi circle, whose circumference lies below `x`-axis and which has origin `(0,0)` as its centre.

Explanation:

Plotted graph is as below.
graph{y=-sqrt(4-x^2) [-5, 5, -2.5, 2.5]}

Given expression is
`y=-sqrt(4-x^2)` ..........(1)
If we square both sides we obtain

`y^2=4-x^2`, rearranging we obtain

`x^2+y^2=2^2`.......(2)
It looks like an equation of a circle.
General equation of a circle whose center is at the point `(h,k)` and radius `r` is
`(x-h)^2+(y-k)^2=r^2`

So the equation (2) is of a circle which has radius `r=2`, and origin `(0,0)` as its center.
From the given expression we deduce that

1. Equation (1) is a curve which has a properties as above. Also it must satisfy following two conditions.

2. That `y` always has negative values due to the presence of `-ve` sign on the right hand side term.

3. As square root of any negative number is imaginary and therefore, can not be plotted on a `x,y` graph. Implies that, argument of square root term must be positive.

Mathematically it can be written as
`4-x^2>=0`

Taking `x` to the left hand side and taking square root of both sides
we obtain `x<=2`

We see that the equation (1) represents a semi circle, whose circumference lies below `x`-axis and which has origin `(0,0)` as its center and radius `r=2`