Inverse Variation Last week, I gave you several posts on direct variation. You can see those posts by clicking the following links: Direct Variation Q1 Q2 Q3 Today I would like to talk about inverse variation. The following are all equivalent ways of saying the same thing: (1) y varies inversely as x. (2) y is inversely proportional to x. (3) y = k/x for some constant k. (4) xy is constant. The following is a consequence of (1), (2), (3), or (4) (5) the graph of y = f(x) is a hyperbola Note: (5) is not equivalent to (1), (2), (3), or (4). Example: In the equation y = 12/x, y varies inversely as x. Here is a partial table of values for this equation. Note that we can tell that this table represents an inverse relationship between x and y because (1)(12) = (2)(6) = (3)(4) = (4)(3) = 12. Here the constant of variation is 12. Here is a graph of the equation. On the left, you can see the full graph. On the right, we have a close-up in the first quadrant. The various equivalent definitions of direct variation lead to several different ways to solve problems. Example: If y = k/x and y = 8 when x = 3, then what is y when x = 6 ? Try to solve the problem yourself before checking the solutions below. Solutions: (1) We are given that y = 8 when x = 3, so that 8 = k/3, or k = 24. Thus, y = 24/x. When x = 6, we have y = 24/6 = 4. (2) Since y varies directly as x, xy is a constant. So, we get the following equation: (3)(8) = 6y. So, 24 = 6y, and y = 24/6 = 4. (3) (8)(3)/6 = 4. Here are a few more problems for you to try. I will provide solutions to these over the next few days. 1. If y = k/x and y = 9 when x = 14, then what is y when x = 6? 2. If x varies inversely as y2, and x is 3 when y is 5, then what is x when y is 3 3. If x ≠ 0 and x is directly proportional to y, which of the following is inversely proportional to 1/y2 ? A) x2 B) x C) 1/x D) 1/x2 4. Suppose that z varies directly as x2 and inversely as y3. If z = 9 when x = 3 and y = 2, what is y when z = 4.5 and x = 6 ? More Problems with Explanations If you are preparing for the SAT, ACT, or an SAT math subject test, you may want to take a look at the Get 800 collection of test prep books. And if you liked this article, please share it with your Facebook friends: Comments comments