Pattern and Function - Functional Relationships in Tables and Graphs

Understanding Functions: A function is a specific rule that relates an input to exactly one output. Think of it like a 'Function Machine'—you drop an input value ($x$) into the machine, the machine applies a mathematical rule (like adding or multiplying), and an output value ($y$) pops out the other side.

Input-Output Tables: These tables are visual grids used to organize the relationship between two variables. The first column usually lists the 'Inputs' ($x$) and the second column lists the 'Outputs' ($y$). By looking at the rows horizontally, you can see how each $x$ is transformed into a specific $y$.

Identifying the Rule: To find the functional relationship, compare the input to its corresponding output. If the $y$ value is always $5$ more than $x$, the rule is additive ($y = x + 5$). If the $y$ value is always $4$ times larger than $x$, the rule is multiplicative ($y = x \times 4$).

Ordered Pairs and Coordinate Planes: Functions can be represented visually on a coordinate plane, which is a flat grid with two perpendicular lines. The horizontal line is the $x$-axis and the vertical line is the $y$-axis. Every input-output pair from a table can be written as an ordered pair $(x, y)$ and plotted as a single dot on this grid.

Graphing Patterns: When you plot the points of a linear function on a graph, they form a straight line. You can visualize this as a 'path' or a 'staircase' where each step moves right by a certain amount and up by a consistent amount. If the points do not form a straight line, the relationship might not be linear.

Using Variables: In patterns and functions, we use letters called variables to represent numbers that can change. $x$ is typically the independent variable (the value we choose), and $y$ is the dependent variable (the result we get). Using variables allows us to write a rule that applies to any number, even very large ones.

Predicting and Extending Patterns: Once you have identified the rule in a table or graph, you can use it to find missing values. For example, if you know the rule is $y = 2 \times x$, and you are given an input of $50$, you can predict the output will be $100$ without needing to draw the entire table or graph.