Having trouble graphing circles and semi circles

Question

I was asked to graph $y = \sqrt{1-x^2}$.

How do I do that? I tried to do a table of values but that did not turn out correct

Is their a trick to graphing circles, semi circles, hyperbolas etc

Answer 1

Hint: $$y=\sqrt{1-x^2}\iff y\geq0\wedge y^2+x^2=1$$

Answer 2

Since you're about to draw a circle do the following:

Use a string of length $1$ and tie it to a pen. Fix the string at the origin and start to draw on the left of your coordinate system...

Answer 3

Since we know it is a semi circle, we can determine the center and the radius by transforming the equation into a form of $$ (x - x_0)^2 + (y - y_0)^2 = R^2 $$

This can be done by squaring both sides of the given equation: $$ y^2 = 1 - x^2 $$ $$ x^2 + y^2 = 1 $$ $$ (x-0)^2 + (y-0)^2 = 1^2 $$

Therefore, the semi circle has a center of (0,0) and a radius of 1.

Since the result of a square root is always positive in a real domain, the positive (upper) half of the semi circle is the one you should sketch.

Ellipses and hyperbolas can be drawn using a similar transformation, only the equations are a little different

Ellipse: $$\left(\frac{x-x_0}{r_1}\right)^2 + \left(\frac{y-y_0}{r_2}\right)^2 = 1 $$

Hyperbola: $$\left(\frac{y-y_0}{r_2}\right)^2 - \left(\frac{x-x_0}{r_1}\right)^2 = 1 $$