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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