Graphing y=2t-2: A Simple Visual Guide

Let's explore linear equations with Spark.E!We'll start with the equation y equals two t minus two.This equation is in slope-intercept form, which follows the pattern y equals m t plus b.In our equation, the slope m is 2, and the y-intercept b is negative two.Let's see what this means on a coordinate plane.The y-intercept of negative two means our line starts two units below the origin on the y-axis.The slope of two means that for every one unit we move right, we go up two units.This pattern of rising two units for every one unit to the right continues along the entire line.The same pattern applies when moving left from our starting point.Remember these key points about our linear equation y equals two t minus two.Now let's find some key points on our line by plugging in different t-values.Let's start with t equals zero. This will give us our y-intercept.Next, let's try t equals one. When we plug this in, y equals zero.Notice how the y-value increased by 2 units as t increased by 1.For t equals two, we get y equals two.Again, we see the same pattern - increasing by 2 units vertically for each unit to the right.Finally, let's try a negative t-value. When t equals negative one, y equals negative four.The same pattern holds true in the negative direction - decreasing by 2 units vertically for each unit to the left.Let's summarize all our points in a table. Each point follows the pattern of our equation y equals two t minus two.Now that we have our points plotted, let's connect them to create our line.When we connect these points, we can see they form a straight line that extends infinitely in both directions.Let's verify the slope between any two points. For example, between t equals zero and t equals one, we can see the rise is 2 units and the run is 1 unit.The same is true between any other pair of points. Here between t equals one and t equals two, we again see a rise of 2 and a run of 1.Let's verify that any point on this line satisfies our equation. For example, at t equals one point five...When we plug t equals one point five into our equation, y equals two times one point five minus two, which equals three minus two, giving us y equals one.And indeed, we can see this point lies exactly at y equals one on our line, confirming it satisfies our equation.Remember that this line extends infinitely in both directions, and every point on it follows the same pattern of y equals two t minus two.