How can you tell if a table of values(t-table) represents a direct variation?

To determine if a table of values represents a direct variation, you need to examine the relationship between the two variables in the table. Here's how you can tell if a table of values represents a direct variation:

1. Look for a constant ratio: In a direct variation, there is a constant ratio between the two variables. This means that if you divide any value in one column by the corresponding value in the other column, the result should be the same for all rows. For example, if you divide the values in the second column by the values in the first column, and you consistently get the same number, then it indicates a direct variation.

Example:
Let's say you have a table with two columns representing variables x and y. If you divide the values in the y-column by the corresponding values in the x-column and get a constant ratio, such as 2, for all rows, then it suggests a direct variation.

x | y
1 | 2
2 | 4
3 | 6
4 | 8

Dividing the values in the y-column by the values in the x-column, we get a constant ratio of 2 (2/1=2, 4/2=2, 6/3=2, 8/4=2).

2. Check for linearity: In a direct variation, the relationship between the two variables should be linear, meaning the points in the table should fall on a straight line when plotted on a graph. You can plot the points from the table on a graph and see if they create a straight line. If they do, it suggests a direct variation.

Example:
Let's take the same table from the previous example and plot the points on a graph. If the points form a straight line, it indicates a direct variation.

(1, 2), (2, 4), (3, 6), (4, 8)

When these points are plotted on a graph, they form a straight line, confirming a direct variation.

By examining the constant ratio between the variables and checking for linearity on a graph, you can determine if a table of values represents a direct variation.