Intersection of Two Graphs with the worked examples below. Compare the two y-values at the same x-coordinate. An intersection occurs wherever their difference is zero.
Intersection of Two Graphs step by step
For y = x + 1 and y = 5 − x, a small table shows that both give 3 at x = 2. The intersection is (2, 3).
Common mistake to watch for
A curve can touch another graph without crossing it. For example, y = x² and y = 0 meet at (0, 0) even though the parabola stays above the horizontal axis on either side.
Practice problem with answer
Plot y = x² and y = x + 2. Rearranging gives x² − x − 2 = 0, so the curves meet at (−1, 1) and (2, 4).
Check your result with the online graphing calculator. Continue with How to Graph a Linear Equation or see the Desmos graphing guide.
GRAPHING GUIDE
Start with the graph
An intersection is a point where both equations have the same x and y values. Plot both functions in the same viewing window.
Worked example
Graph y = 2x + 1 and y = −x + 7. The lines meet near (2, 5). Set 2x + 1 = −x + 7 to get x = 2, then substitute to find y = 5.
Try another example
Graph y = x² and y = 4. The graphs meet at (−2, 4) and (2, 4). Solving x² = 4 confirms both points.
Check your result
If no intersection appears, widen the viewing window. Parallel lines never meet, while identical lines share every point.
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