Graphing a Line
Construct a linear graph from its equation, then connect the drawn line to what it really represents: every ordered pair that makes the equation true.
How does one equation become an entire geometric line?
The equation generates ordered-pair solutions. The graph is the picture of all of them at once. Slope-intercept form gives us an efficient way to construct that picture without making a giant table.
Two points determine the straight line, but the finished line represents infinitely many solutions, not just those two points.
Build y = 2x − 1 in three moves.
Set x = 0. The equation gives y = −1.
From (0, −1), run +1 and rise +2 to reach (1, 1).
The same constant rate continues in both directions, producing the full line.
Create the graph in the same order the equation describes it.
1. Where is the y-intercept?
Choose b, then plot the point (0, b).
A point belongs on the line only if it satisfies the equation.
The picture and the algebra must agree. Test a nearby point by substituting its coordinates into the rule.
Finish constructing the line, then test a point against the original equation.