### MATH 135H Assignment #3

Due: 1 November, 2000

"Turn, turn, turn..."

You may want to brush up a little on basic trigonometry before tackling this assignment. You won't need very much, but that much you will need.

For problems 1-3, suppose that we rotate the xy-plane 30 degrees = pi/6 radians counter-clockwise about the origin (while keeping the coordinate system fixed).

1. Work out the coordinates of the points that (1,0) and (0,1) get moved to by this rotation. 

2. In general, suppose the point (x,y) is moved to the point (u,v) by the rotation. Work out formulas for x and y in terms of u and v, and then work out formulas for u and v in terms of x and y. 

3. If you are given the parabola y = x2 -2x + 3, what is the equation describing the parabola that the given one is moved to by the rotation? 

For problems 4-7, suppose that we rotate the xy-plane theta radians = 180*theta/pi degrees counter-clockwise about the origin (while keeping the coordinate system fixed).

1. Work out the coordinates of the points that (1,0) and (0,1) get moved to by this rotation. 

2. In general, suppose the point (x,y) is moved to the point (u,v) by the rotation. Work out formulas for x and y in terms of u and v, and then work out formulas for u and v in terms of x and y. 

3. If you are given the parabola y = ax2 + bx + c, what is the equation describing the parabola that the given one is moved to by the rotation? 

4. Find the equation of the parabola, if there is one, with axis of symmetry parallel to the line x + y = 0 and passing through the points (2,5), (0,1), and (3,4). 

Solutions to Assignment #3 Department of Mathematics  Trent University