1 / 11

The Gradient Formula

The Gradient Formula. Gradient of AB = m AB = height / horizontal = BC / AC = (y 2 -y 1 ) (x 2 -x 1 ). B(x 2 ,y 2 ). y 2 -y 1. A(x 1 ,y 1 ). x 2 -x 1. C(x 2 ,y 1 ). The Gradient Formula ctd. Summary If A is (x 1 ,y 1 ) and B is (x 2 ,y 2 )

tallis
Download Presentation

The Gradient Formula

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. The Gradient Formula • Gradient of AB • = mAB • = height/horizontal • = BC/AC • = (y2-y1) (x2-x1) B(x2,y2) y2-y1 A(x1,y1) x2-x1 C(x2,y1)

  2. The Gradient Formula ctd. • Summary • If A is (x1,y1) and B is (x2,y2) • then the gradient of AB is given by • mAB = (y2-y1) (x2-x1)

  3. Example 4 P is (5, 9) and Q is (7,17). • mPQ = (y2-y1) (x2-x1) • = (17 - 9) • ( 7 - 5) • = 8/2 • = 4 NB: line looks like Q P

  4. Example 5 E is (15, 9) and F is (7,17). • mEF = (y2-y1) (x2-x1) • = (17 - 9) • ( 7 - 15) • = 8/-8 • = -1 NB: line looks like F E

  5. Example 6 V is (-5, -9) and W is (12,-9). • mVW = (y2-y1) (x2-x1) • = (-9 -(- 9)) • ( 12 - (-5)) • = 0/17 • = 0 • NB: A horizontal line has no steepness. NB: line looks like V W

  6. Example 7 S is (5, -9) and T is (5, 12). • mST = (y2-y1) (x2-x1) • = (12 -(- 9)) • ( 5 - 5) • = 21/0 • and this is undefined • or infinite NB: line looks like T Vertical line has infinite gradient. S

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

  8. Parallel Lines • Parallel lines run in the same direction so must be equally steep. • Hence parallel lines have equal gradients. • Example 8 • Prove that if A is (4,-3) , B is (9,3) C is (11,1) & D is (2, -1) • then ACBD is a parallelogram

  9. Ex8 ctd D B NB; The order of the letters is important. A • mAC = (1 + 3)/(11- 4) = 4/7 • mDB = (3 + 1)/(9 - 2) = 4/7 • mAC = mDB so AC is parallel to DB • mAD = (-1 + 3)/(2 - 4) = 2/-2 = -1 • mCB = (3 - 1)/(9 - 11) = 2/-2 = -1 • mAD = mCB so AC is parallel to DB • Since the opposite sides are parallel then it follows that ACBD is a parallelogram. C

  10. 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. Example 8a K is (5, -8), L is (-2, 6) and M is (9, -16). Prove that the three points are collinear. 6 - (-8) 14 = -2 mKL = = Since KL & KM have equal gradients and a common point K then it follows that K, L & M are collinear. -2 - 5 -7 -16 - (-8) -8 mKM = = = -2 9 - 5 4

  11. Ex8b 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) ? m1 = (115-35)/(210-50) = 80/160 = 1/2 m2 = (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.

More Related