Question

How do you determine if a function is a direct variation when given a table?

Answer 1

Hint: Suppose we have two variables x and y connected by any linear equation. Linear equation means the connection between two variables of degree one. We have direct variation, when the resulting variable of the two variables changes in the same and proportional manner as that of the other variable. If it is not direct variation, then it is an inverse variation.

Complete step by step answer:
A direct variation between y and x is typically denoted by \[y=kx\] where \[k\in \mathbb{R}\]. This means that as x goes larger, y also tends to get larger. The opposite is also true. As x goes smaller, y tends to get smaller.

As per the given question, we need to determine if a function is a direct variation when given a table. Suppose that we have a table showing different values of the variable x and its corresponding values of variable y. Take a note of the behaviour of one variable and compare with the corresponding behaviour of another.
If you increase the value of one variable, then if the value of another variable also increases by the same amount, then it has direct variation. If it decreases, then it is inverse variation.

For example, take the linear equation y=x. If we increase x by one unit, we get y value increased by the same amount. It happens the same way in reverse case also. If we decrease x by one unit, we get y value decreased by the same amount. Hence, Tables having x and y values show the direct variation between x and y.

Note: while solving such type of problem, we have to be very clear on the concept of direct variation. We have converse of direct variation also. So, we must not confuse between direct variation and inverse variation. We can also determine the relation between x and y using a sequence of values of x and y. In the sequence of x, take values which change by a common difference then, using the sequence of difference in y values. Determining the type of sequence, we can find a common relation satisfying the required conditions.