In the same way that other functions can be combined, a series of transformations can be combined into a single function. For example, this statement for the function S shows one way to represent the rotation of a point 270° counterclockwise about the origin followed by a translation 3 units to the left and 1 unit up: S(270, O)/<-3,1>(x, y) = (y – 3, -x + 1). Write a function to represent each series of transformations:

rotation of 90 degrees counterclockwise about the origin, point O, then a reflection across the x-axis
reflection across the y-axis, then a translation a units to the right and b units up
translation a units to the right and b units up, then a rotation of 180 degrees counterclockwise about the origin, then a reflection across the y-axis

Irrigation channels that require regular flow monitoring are often equipped with electromagnetic flowmeters in which the magnetic field is produced by horizontal coils embedded in the bottom of the channel. a particular coil has 100 turns and a diameter of 6.0 m. when it's time for a measurement, a 4.5 a current is turned on. the large diameter of the coil means that the field in the water flowing directly above the center of the coil is approximately equal to the field in the center of the coil. the field is directed downward and the water is flowing east. the water is flowing above the center of the coil at 1.5 m/s . what is the magnitude of the field at the center of the coil? what is the direction of the force on a positive ion in the water above the center of the coil? what is the magnitude of the force on an ion with a charge +e?