Question

Suppose you are using a computer to draw a face on a coordinate plane. The eyes are at (–3, 3) and (3, 3). The tip of the nose is at the origin. To make a smiling mouth, you are going to enter the equation for the graph of a quadratic function, with its appropriate domain.

Post the quadratic function that you would use (in any form) with the domain. Explain how you determined your equation and why you wrote it in the form that you did.

Answer 1

x^2 + y^2 = r^2 is the standard equation for a circle.

y^2 = r^2-x^2 is the same equation, I just moved the x^2 to the other side.

In this case, r^2 = 9, because the eyes are at the x coordinates, 3 and -3 (plus and minus 3 units from the origin).

Also note that the center of the circle is at the origin (the nose).

I would write it as such: y^2=9-x^2

This would give you the whole circle, but since you want the lower half of the circle, you have to solve for the respective half. To do so, you'd do this:

y= -sqrt(9-x^2), thus getting you the smiley face you want.

***NOTE it's the negative square root, since you want only the negative values (since the origin is the center; otherwise it would be the bottom half of the circle).***

If you can use a larger domain, you can us 16 or even 25 to have a bigger smiley.

Hope that helps!