Coordinate geometry and stationary points
Use gradients to describe straight lines and derivatives to locate turning points.
What you will learn
- Find a perpendicular bisector.
- Find stationary coordinates from a polynomial.
Build a line from a point and a gradient
The gradient between two points is the change in y divided by the change in x. The midpoint averages the two x-coordinates and the two y-coordinates separately. For non-horizontal, non-vertical perpendicular lines, the gradients multiply to −1.
A perpendicular bisector passes through the midpoint and is perpendicular to the segment. After finding its gradient m and midpoint (a,b), use y−b=m(x−a). A horizontal segment has a vertical bisector, which needs an equation x=constant rather than y=mx+c.
A stationary point has zero gradient
Differentiate a term axⁿ to get anxⁿ⁻¹. Set the derivative equal to zero and solve for x. These are x-coordinates only: substitute them into the original equation to find y.
To decide whether a stationary point is a maximum or minimum, inspect how the derivative changes sign. Positive to negative gives a maximum; negative to positive gives a minimum. A zero derivative alone does not guarantee a turning point.
Worked example
Find the stationary points of y=x³−3x².
Show the worked solution
- Differentiate: dy/dx=3x²−6x=3x(x−2).
- Set the derivative to zero: x=0 or x=2.
- Use the original curve: y(0)=0 and y(2)=8−12=−4.
Answer (0,0) and (2,−4)
Common mistakes
- Use the original equation, not the derivative, to find y.
- The midpoint is not found by subtracting coordinates.
What can you explain now?
Find the perpendicular bisector of the segment from (0,0) to (4,4).
Compare with the explanation
y=−x+4
Its midpoint is (2,2), the segment gradient is 1 and the perpendicular gradient is −1. Thus y−2=−(x−2).
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Original StudyForA teaching content · AI-assisted checks · Human teacher review pending.
Number contains 19 sequenced lessons; Algebra and graphs contains 21; Coordinate geometry contains 10 teaching lessons and a mixed checkpoint. Nine earlier overviews support selected topics across the syllabus. The other six syllabus areas, full cumulative assessment and human teacher review are not yet complete. Written lesson practice is self-checked, not automatically graded. The checkpoint samples skills and does not save an exam grade or certify mastery.
Check the official syllabus 2025–2027 Open related practice and resources