Three ways to graph a line
You only need two points to draw a straight line, but plotting a third is a good check.
Method 1: Table of values
Pick a few x-values, calculate y, plot, and join.
Example: y = 2x − 3
| x | −1 | 0 | 1 | 2 | 3 |
|---|---|---|---|---|---|
| y | −5 | −3 | −1 | 1 | 3 |
Plot the points and draw a line through them with a ruler, extending past the ends with arrows.
Tip: choose x-values that make y easy to calculate (for y = ⅓x + 1, choose multiples of 3).
Method 2: Slope and y-intercept
For y = mx + b:
- Plot (0, b).
- Use the slope (rise/run) to find another point.
- Draw the line.
Example: y = −½x + 4
- Start at (0, 4).
- Slope −½: down 1, right 2 → (2, 3), (4, 2) …
Method 3: Intercepts
Best for standard form Ax + By = C.
- x-intercept: set y = 0 and solve for x.
- y-intercept: set x = 0 and solve for y.
Example: 3x + 4y = 12
- y = 0 → x = 4 → (4, 0)
- x = 0 → y = 3 → (0, 3)
- Plot both and join.
Special lines
| Equation | Graph |
|---|---|
| y = 5 | horizontal line through (0, 5) |
| x = −2 | vertical line through (−2, 0) |
| y = x | diagonal through the origin, slope 1 |
| y = 0 | the x-axis |
Choosing a method
| Equation looks like | Best method |
|---|---|
| y = mx + b | slope and y-intercept |
| Ax + By = C | intercepts |
| unfamiliar form | table of values |
Checking a point
A point lies on the line if it makes the equation true.
Is (3, 5) on y = 2x − 1? 2(3) − 1 = 5 ✓ Yes.
Graphing with technology
Graphing calculators and free tools like Desmos or GeoGebra draw lines instantly. Use them to check hand-drawn graphs and explore what changing m and b does.
Worked examples
Example 1: Graph y = 3x + 1 by slope and intercept: start at (0, 1), up 3 right 1 → (1, 4).
Example 2: Intercepts of 2x − 5y = 10: x-int (5, 0); y-int (0, −2).
Example 3: Is (−1, 4) on y = −3x + 1? −3(−1) + 1 = 4 ✓ Yes.
Common mistakes
- Plotting (y, x) instead of (x, y).
- Uneven scales on the axes.
- Mixing up x = 3 (vertical) and y = 3 (horizontal).
- Drawing a line segment instead of a full line with arrows.
Try it yourself
- Make a table for y = −x + 2 with x = −1, 0, 1, 2.
- Find the intercepts of 5x + 2y = 10.
- Describe the graph of x = 4.
- Is (2, −3) on y = −2x + 1?
Answers: 1) 3, 2, 1, 0 2) (2, 0) and (0, 5) 3) a vertical line through (4, 0) 4) yes
Quick recap
- Graph lines with a table of values, slope and y-intercept, or intercepts.
- Find the x-intercept by setting y = 0.
- Check by substituting a point.