# A fun math estimation question inspired by Kate Nowak

Saw this tweet from Dan Anderson today:

The rotated graph caught my eye and I started playing around a little. Can’t say that I’d spent much time thinking about rotations of the graph of $y = x^2$ before, so I was surprised to find that any small (non-zero) rotation about (0,0) – no matter now tiny the angle – produces a point on the graph having a vertical tangent line. It also causes the rotated graph to intersect the y-axis.

Here are two fun estimation question. Don’t calculate or use the fun computer program below, just try to estimate:

(1) For a rotation of 1 degree counterclockwise, what is the x-coordinate of the point with the vertical tangent line?

(2) For a rotation of 1 degree counterclockwise, what is the y-coordinate of the point where the rotated curve crosses the y-axis?

After you think about those questions – so no cheaty cheat cheating! – play around with this graphing tool that David Griswold made:

Definitely had some fun playing around with these rotations today!