Graphing from a Table
Turning a rule into points, and points into a line.
The explanation
Given a rule like y = 2x + 1, you can build a table by choosing x values and working out y.
x = 0 gives y = 1. x = 1 gives y = 3. x = 2 gives y = 5.
Plot (0,1), (1,3), (2,5) and you will see they line up. Draw the line through them and you have graphed the equation.
Choose easy x values, and include a negative one and zero. Three points is enough for a line, but a fourth is a free error check: if one point is off the line, you made an arithmetic slip.
A table of values samples the solution set of an equation in two variables. Each row is an ordered pair satisfying the equation, and the graph is the set of *all* such pairs — the drawn line is the completed picture of an infinite set, not just the points you computed.
The pattern in the table encodes the structure of the rule. For y = mx + b, equally spaced x values produce y values with a constant first difference equal to m times the spacing, which is what makes the graph straight. Constant second differences instead indicate a quadratic, and a constant ratio between successive y values indicates an exponential. Reading which of these a table shows is a standard question and a quick diagnostic.
Whether to connect the points depends on the domain: continuous quantities get a solid line, whereas a rule counting discrete objects should stay as separate points.
Worked example
Make a table for y = −3x + 4 using x = −1, 0, 1, 2 and describe the pattern.
- x = −1: y = −3(−1) + 4 = 7.
- x = 0: y = 4. x = 1: y = 1. x = 2: y = −2.
- y values fall by 3 each time x rises by 1.
Answer: (−1,7), (0,4), (1,1), (2,−2) — a constant difference of −3, so the graph is a line falling to the right.
Common mistakes
- Substituting into the wrong variable and computing x from y.
- Sign slips with negative x values in a term like −3x.