Inverse Variation and Rates-of-Change Graphs

Six workers take ten days to build a wall. More workers should take fewer days, not more. The total amount of work is six times ten, which is sixty worker-days. If only four workers do it, sixty shared by four is fifteen days. With twelve workers, it is sixty divided by twelve, five days. This is inverse variation: y equals k over x, where k is sixty here. To find k, multiply a pair of values together. Then divide k by the new value. Doubling the workers halves the days.

To test a table for inverse variation, multiply each x by its y. Here two times eighteen, three times twelve, four times nine and six times six are all thirty six, so y equals thirty six over x. Notice that when x doubles from two to four, y halves from eighteen to nine, and when x doubles from three to six, y halves from twelve to six. The graph is not a straight line. It is a curve called a hyperbola that gets closer and closer to both axes but never touches them.

For a trip of fixed length, the faster you go, the less time it takes. A two hundred and forty kilometre trip at sixty kilometres per hour takes four hours. At eighty, it takes three hours, and at forty it takes six. In every case speed times time is two hundred and forty, the distance. So time varies inversely with speed, and t equals two hundred and forty over v. If you know one speed and its time, multiply them to find the distance, then divide by the new speed.

A distance time graph shows how far something has gone at each moment. Its gradient is distance divided by time, which is the speed. In the first two hours this hiker covers six kilometres, so the speed is three kilometres per hour. From two to three hours the graph is flat: the distance does not change, so the hiker has stopped. From three to five hours the hiker covers four more kilometres, at two kilometres per hour. That section is less steep, so it is slower. For the whole hike, the average speed is the total distance, ten kilometres, divided by the total time, five hours, which is two kilometres per hour.

This graph shows distance from home, so it goes up on the way out and down on the way back. The steeper side is the faster one: out, twelve kilometres in two hours, is six kilometres per hour; back, twelve kilometres in four hours, is three kilometres per hour. The average speed for the whole journey is the total distance, twenty four kilometres, divided by the total time, six hours, which is four kilometres per hour. It is not the average of six and three, which is four and a half, because more time was spent at the slower speed. Quiz and exercises now.