What is to be learned? How to use the gradient formula to calculate gradients How to use gradients to identify shapes.

The Gradient Formula m = y 2 – y 1 x 2 – x 1

The Gradient Formula m = y 2 – y 1 x 2 – x 1

The Gradient Formula m = y 2 – y 1 x 2 – x 1 Don’t let your ironing board collapse!

ExA (4, -2) and B (2, 6)m AB ? m AB = 6 – (-2) (x 1, y 1 ) (x 2, y 2 ) = 8 / -2 = – 4

Summary of Gradients Positive gradient goes uphill. Negative gradient goes downhill. Zero gradient is horizontal. Infinite gradient is vertical.

Parallel Lines Parallel lines run in the same direction so must be equally steep. Hence parallel lines have equal gradients. Example Prove that if A is (4,-3), B is (9,3) C is (11,1) & D is (2, -1) then ACBD is a parallelogram

ACBD Opposite sides are parallel, therefore ACBD is a parallelogram. A C B D Order is vital A (4,-3), C(11,1) B (9,3) D (2, -1) (4, -3)(11, 1) (9, 3) (2, -1) m AC = – 4 = 4 / 7 m DB = – 2 = 4 / 7 m AC = m DB so AC and DB are parallel m AD = – 4 = 2 / -2 m CB = 3 – 1 9 – 11 = 2 / -2 m AD = m CB so AD and CB are parallel = -1

Collinear C B A Collinear if m CB = m BA

COLLINEARITY Defn: Three or more points are said to be collinear if the gradients from any one point to all the others is always the same. _ They are in a straight line K is (5, -8), L is (-2, 6) and M is (9, -16). Prove that the three points are collinear. m KL = 6 - (-8) = = -2 m KM = (-8) = -8 4 = -2 Since KL & KM have equal gradients and a common point K then it follows that K, L & M are collinear. points in a straight line → equal gradients

Need in a straight line

Achieved if There is a Collinear It’s what I do

A Navy jet flies over two lighthouses with map coordinates (210,115) & (50,35). If it continues on the same path will it pass over a yacht at (10,15) ? m 1 = (115-35) / (210-50) = 80 / 160 = 1 / 2 m 2 = (115-15) / (210-10) = 100 / 200 = 1 / 2 Since gradients equal & (210,115) a common point then the three places are collinear so plane must fly over all three.