Which Table Represents a Linear Function?
A linear function is a function whose graph is a straight line. In other words, for any two points on the graph of a linear function, the ratio of the change in the output to the change in the input is constant.
One way to determine whether a table of values represents a linear function is to calculate the slope between each pair of points in the table. If the slope is constant for all pairs of points, then the table represents a linear function.
For example, consider the following table of values:
x | y ---|--- 1 | 2 2 | 4 3 | 6 4 | 8 The slope between the first two points is (4 – 2) / (2 – 1) = 2. The slope between the second two points is (6 – 4) / (3 – 2) = 2. The slope between the third two points is (8 – 6) / (4 – 3) = 2. And so on. Therefore, the slope is constant for all pairs of points in this table, so the table represents a linear function.
Here are some other questions that can be used to determine whether a table of values represents a linear function:
- Do the points in the table lie on a straight line? If so, then the table represents a linear function.
- Can the values in the table be represented by a linear equation? If so, then the table represents a linear function.
For example, the following table of values represents a linear function:
x | y ---|--- 0 | 1 1 | 2 2 | 3 3 | 4 The points in this table lie on a straight line, so the table represents a linear function. The values in this table can also be represented by the linear equation y = x + 1, so the table represents a linear function.
In general, any table of values that meets one of the following criteria represents a linear function:
- The slope between each pair of points is constant.
- The points in the table lie on a straight line.
- The values in the table can be represented by a linear equation.
