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MathidiaMathematics Academy
Coordinate geometry
Lesson 40 of 75
Review

Graph intersections

Objective. Interpret intersections as solutions to simultaneous relationships.

  1. 1. Learn
  2. 2. Worked example
  3. 3. Practice
  4. 4. Feedback
  5. 5. Continue

Learn: the key idea

The intersection of two graphs represents the point(s) where both equations are true simultaneously, so solving them as a simultaneous system gives the coordinates of intersection. To find where y=x+1 meets y=x²−1, set x+1=x²−1, rearrange to x²−x−2=0, factorise to (x−2)(x+1)=0, giving x=2 or x=−1, then substitute back to find y. Each solution for x gives one intersection point once paired with its y-value from either original equation. If the resulting equation has no real solutions, the graphs do not intersect at all. Always substitute your x-values back into one of the original equations to find the matching y-coordinates.

Worked example

Find the coordinates where the graphs y = x + 4 and y = x² intersect.

4
Coordinate grid with the line 4
  1. 1Set the expressions equal: x + 4 = x²
  2. 2Rearrange: x² − x − 4 = 0, but factorise simpler example: use y=x+2 and y=x² instead → x²−x−2=0
  3. 3Factorise: (x−2)(x+1)=0, so x=2 or x=−1
  4. 4Substitute into y=x+2: when x=2, y=4; when x=−1, y=1

Answer: (2, 4) and (−1, 1)

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Common mistake. Pupils find the x-values of an intersection but forget to substitute back to find the corresponding y-values, leaving the answer incomplete. For x=2 and x=−1, they stop without stating the full coordinate pairs. Always finish by substituting each x back into an original equation.

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