Step by step
- Make a table of values. Pick a few x values, including negatives and 0, and work out y for each.
- Find the y-intercept. Put x = 0. This is where the graph crosses the vertical axis.
- Find the x-intercepts (zeros). Solve y = 0. These are where the graph crosses the horizontal axis.
- Plot the points and join them with a smooth curve (or a straight line, for a linear function).
- Check the shape. Lines are straight, x² makes a U, x³ makes an S, 1/x has a gap at x = 0.
Example 1: a line, y = 2x + 1
| x | −2 | −1 | 0 | 1 | 2 |
|---|---|---|---|---|---|
| y | −3 | −1 | 1 | 3 | 5 |
The y-intercept is (0, 1). Setting 2x + 1 = 0 gives the x-intercept (−0.5, 0). The 2 in front of x is the slope: every step of 1 to the right goes 2 up.
Example 2: a parabola, y = x² − 4
| x | −3 | −2 | −1 | 0 | 1 | 2 | 3 |
|---|---|---|---|---|---|---|---|
| y | 5 | 0 | −3 | −4 | −3 | 0 | 5 |
The y-intercept is (0, −4), which is also the lowest point. Setting x² − 4 = 0 gives x = −2 and x = 2, so it crosses the x-axis at (−2, 0) and (2, 0). The table is symmetric, which is a quick check that the arithmetic is right.
Where two graphs meet
Graph both sides of an equation and the crossing points are its solutions. y = x² − 4 and y = 2x + 1 meet where x² − 4 = 2x + 1, that is x² − 2x − 5 = 0, so x = 1 ± √6 (about −1.449 and 3.449).
Use the graphing calculator
Open these two examples in Calcist's graphing calculator. It lists the intercepts for you, shows a table of values, and lets you trace any point by moving across the graph. For exact solutions with steps, use the equation solver.
Every worked example on this page was checked with Calcist's own engine and an independent reference.