## Graphed Points

The graphed points from Pattern 1 are shown below where the *\(x\)*-value represents the step and the *\(y\)**–*value represents the number of squares in that step.

What do you notice about the pattern of these points?

*The points are collinear {a set of points lying in the same straight line)**L**inear relationships can be represented in tables, graphs, situations (such as a growth pattern), and equations.**The rate of change (slope) can be identified in linear relationships.*

- Use the table from each pattern to graph the points for Patterns 2, 3, and 4.
- Are these patterns linear relationships?

How can you use the graphed points to find the slope?

*Slope is the rate of change of a linear function and is a measure of the steepness of a line using the change in \(y\) (which is the rise) and change in \(x\) (which is the run).**The change in a value is denoted by the symbol \(delta \Delta\)*

\[slope=\frac{\text {change in } y}{\text{change in }x}=\frac{\Delta y}{\Delta x} =\frac {rise}{run}\]

*The slope can be found in graphed linear relationships by determining the rise and the run between any two collinear points.**The rise and the run for Pattern 1 is shown on the graph below. Graphs are created using Desmos.com.*

\[slope=\frac{\Delta y}{\Delta x} =\frac {rise}{run}=\frac {+2}{+1}=2\]

- Compute the slope for the remaining patterns using the graphs.