Why does inverse variation not hold (as x increases y decrease) when the constant of variation is negative?
Inverse variation, also known as indirect variation, describes a relationship between two variables where an increase in one variable leads to a decrease in the other variable. It is represented mathematically as y = k/x, where k is the constant of variation.
When the constant of variation is positive, it indicates a direct relationship between the two variables, where as x increases, y decreases, and vice versa. This can be understood intuitively by considering that as the denominator (x) increases, the whole expression (k/x) decreases.
However, when the constant of variation is negative, the inverse variation does not hold. With a negative constant of variation, the equation becomes y = -k/x. This means that as x increases, the value of -k/x becomes smaller, resulting in a decrease in y. But, as x decreases, the value of -k/x becomes larger, which contradicts the expected behavior of inverse variation.
To illustrate this, consider the example x = 2 and k = -3. Plugging these values into y = -k/x, we get y = -(-3)/2, which simplifies to y = 3/2 or 1.5. Now, if we increase x to 4, plugging these values into the equation gives us y = -(-3)/4, which simplifies to y = 3/4 or 0.75. Here we see that as x increased, y actually decreased, contradicting the inverse variation pattern.
In summary, inverse variation does not hold when the constant of variation is negative because it leads to a contradiction where increasing the independent variable results in a decrease of the dependent variable.