### 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. [1]

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. [2]

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? [1]

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. [1]

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. [2]

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? [1]

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). [2]

Solutions to Assignment #3