Patterns and Function - Input and Output Tables

Function Machine: Think of a machine where you drop a number (the Input) into a slot. Inside, a secret mathematical 'Rule' acts on it, and a new number (the Output) pops out the other side. Visually, this is often represented as a box with an arrow pointing in for the input and an arrow pointing out for the output, showing the transformation clearly.

The Input (Independent Variable): This is the starting value, typically represented by the variable $x$. In an input-output table, inputs are usually listed in the left column of a T-chart or the top row of a horizontal table. You can think of the inputs as the 'cause' in the relationship.

The Output (Dependent Variable): This is the resulting value after the rule has been applied, usually represented by the variable $y$. The output's value depends entirely on what the input was. In a visual table, these results are placed in the right column or the bottom row, aligned with their specific input.

The Function Rule: This is the specific mathematical operation—such as addition ($+$), subtraction ($-$), multiplication ($\times$), or division ($\div$)—that defines the relationship between $x$ and $y$. Visually, the rule is like the 'bridge' connecting the two columns of a table; it must work for every single pair in the set.

Identifying Patterns: To find the rule, compare each input to its output. If the outputs are getting larger than the inputs, the rule involves addition or multiplication. If the outputs are smaller, it involves subtraction or division. Visually, you can look for a constant increase or decrease as you move down the output column.

Input-Output Tables (T-Charts): A common way to organize functions is using a T-Chart, which looks like a large capital letter 'T'. The left side is labeled 'Input' and the right side 'Output'. Each row represents an ordered pair $(x, y)$ that follows the same logic, making it easy to spot missing numbers in a sequence.

Predicting Missing Values: Once you identify the pattern or rule, you can use it to find unknown numbers. For example, if the rule is $+5$ and there is an empty box in the output column next to an input of $10$, you can calculate $10 + 5$ to fill in $15$. This allows you to extend the table indefinitely.

Two-Step Rules: Some patterns are more complex and require two operations, such as multiplying then adding ($y = (x \times 2) + 1$). Visually, this can be imagined as two function machines connected in a row, where the output of the first machine becomes the input for the second.