Variation describes how one quantity changes when another quantity changes. In direct variation, two variables grow or shrink together at a constant ratio, which creates a straight line through the origin. In inverse variation, one variable increases while the other decreases so that their product stays constant.
These patterns are important in physics, geometry, economics, and many real world situations involving rates, scale, and tradeoffs.
The constant of variation, usually called k, is the number that connects the variables in a variation equation. For direct variation, the equation is y = kx, so k is found by k = y/x. For inverse variation, the equation is y = k/x or xy = k, so k is found by multiplying x and y.
More complex models, such as joint variation, combine several variables in one equation, such as z = kxy.
Key Facts
- Direct variation has the form y = kx, where k is the constant of variation.
- In direct variation, k = y/x and the graph is a straight line through (0, 0).
- Inverse variation has the form y = k/x, where x cannot be 0.
- In inverse variation, k = xy and the graph is a hyperbola.
- Joint variation can be written as z = kxy when z varies directly with both x and y.
- To solve a variation problem, find k from given values, write the equation, then substitute the new value.
Vocabulary
- Direct variation
- A relationship where one variable is a constant multiple of another, written as y = kx.
- Inverse variation
- A relationship where the product of two variables is constant, written as y = k/x or xy = k.
- Constant of variation
- The fixed number k that connects the variables in a variation equation.
- Joint variation
- A relationship where one variable varies directly with two or more other variables, such as z = kxy.
- Hyperbola
- The curved graph of an inverse variation relationship, with branches that approach the axes but do not cross them.
Common Mistakes to Avoid
- Using y = kx for every variation problem is wrong because inverse variation uses y = k/x or xy = k.
- Finding k by multiplying in a direct variation problem is wrong because direct variation uses k = y/x.
- Assuming an inverse variation graph crosses the axes is wrong because x = 0 is not allowed and the graph only approaches the axes.
- Forgetting to write the equation after finding k is wrong because the equation is needed to solve for new values correctly.
Practice Questions
- 1 If y varies directly with x and y = 24 when x = 6, find k and write the variation equation.
- 2 If y varies inversely with x and y = 8 when x = 5, find y when x = 10.
- 3 A table shows that when x doubles, y also doubles. Explain how you can decide whether the relationship is direct variation.