Many students begin working with function tables -- also known as t-tables -- in sixth grade, as part of their preparation for future algebra courses. To solve problems involving function tables, students must possess a degree of background knowledge, including understanding the configuration of a coordinate plane and how to simplify basic algebraic expressions. “Doing” function tables in sixth grade math can entail one of two tasks: constructing a function table from an equation or constructing a function table based on a graph. How to “do” the function table depends on which task has been requested, but regardless, it requires an understanding of how these tables operate.

### Function Table Layout

To solve problems pertaining to function tables, you must be familiar with their arrangement. A function table is essentially equivalent to a gridded list of ordered pairs -- that is, a list of points on the coordinate plane of the form (x, y). Function tables typically consist of two columns, with a left-hand column titled “x” and a right-hand column titled “y.” Occasionally, you may see function tables oriented horizontally in two rows, with the top row entitled “x” and the bottom row entitled “y.”

### A Relationship Between Variables

Before working with function tables, it’s also necessary to understand the crucial relationships that lie behind them. Function tables demonstrate a quantitative relationship between two variables: an independent relationship and a dependent relationship. An independent relationship is one into which numeric values are input; a dependent relationship is one in which -- after a function rule has been applied -- produces numeric outputs. As the naming convention implies, the numeric value of the dependent variable depends upon the value of the independent variable. In this relationship, “x” represents the independent variable and “y” represents the dependent variable. For instance, in the function y = x + 4, the “x” is the independent variable, while the “y” is the dependent variable. If you input the numeric value of “1” into x, the output, y, will equal 5, since 1 + 4 = 5.

### Given an Equation

Continuing with the previous example, suppose you are asked to complete a function table for y = x + 4. Start by selecting values for x. You can choose any values you like, but it is generally the best practice to select integers close to zero, because this entails relatively simpler arithmetic calculations. Write your chosen x values in the column labelled “x”, then insert each one into the function and simplify, writing your results in the “y” column. For instance, as previously determined, inputting a “1” for x results in a y-value of 5; thus, in your table, you’d write a 1 in the “x” column, with a 5 next to it in the “y” column. Now, pick another value for “x,” such as -1, which produces a y-value of 3, and write this -1 and 3 in the table. Continue in this way until you have filled in the t-table.

### Given a Graph

Because the individual rows of a function table coordinate to points on a graph, you may be asked to construct a function table from a graph. Suppose you are given the graph of a line that passes through the points (-2, -3), (0, -1) and (2, 1). Write the x-values of each point, which are -2, 0 and 2, in the x-column of the function table. Write each y-value of each point in the y-column next to the x-value to which it corresponds. For instance, write the -3 next to the -2 and so on. Later, as your studies progress, you may be asked to write an equation based on the pattern found in the function table, which in this case would be y = x – 1, since each value of “y” is 1 less than it corresponding x-value.