Gradient - Two fixed points
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How to find the gradient of a line joining two fixed points.

This topic is part of the TCS FREE high school mathematics 'How-to Library'. It shows you how to find the gradient of a line joining two fixed points.
(See the index page for a list of all available topics in the library.) To make best use of this topic, you need to download the Maths Helper Plus software. Click here for instructions.

Theory:

Consider two fixed points: (x1,y1) and (x2,y2), joined by a straight line:

Starting at (x1,y1) on the left, then moving to the right along the line to (x2,y2), we measure the change in 'x', and the change in 'y'.

The change in 'x'  =  x2 - x1 while the change in 'y' =   y2 - y1

These quantities are shown on the following diagram:

The gradient 'm' of the line segment is defined as follows:

The Greek capital letter 'Delta' : D is often used in mathematics to mean a change in some quantity. So the gradient definition can be written like this:

To measure the change in 'x' and 'y' between the two points , we draw a right triangle beneath the graph. The length of the vertical leg is: (y2-y1), and the horizontal leg has length (x2-x1).

In the diagram below, (x1, y1) = (2, 3) and (x2, y2) = (6, 8)...

 So the gradient of this line segment is: 

Method:

Maths Helper Plus can find the gradient of a straight line between two points. It will calculate the gradient showing the working steps. It also displays a labelled diagram.  

Step 1 Download the free support file... We have created a Maths Helper Plus document containing the completed example from this topic. You can use this to practice the steps described below, and as a starting point for solving your own problems.

File name:  'Gradient between two points.mhp'   File size: 4kb
 Click here to download the file.

If you choose 'Open this file from its current location', then Maths Helper Plus should open the document immediately. If not, try the other option: 'Save this file to disk', then run Maths Helper Plus and choose the 'Open' command from the 'File' menu. Locate the saved file and open it. If you do not yet have Maths Helper Plus installed on your computer, click here for instructions.

Step 2  Display the parameters box

Press the F5 key to display the parameters box:

 

The coordinates of two (x,y) points are entered into the four edit boxes 'A', 'B', 'C' and 'D', as follows:

x1 = A,  y1 = D,  x2 = B, and y2 = X.

To find the gradient of the straight line through two points, enter the coordinates of two points into the edit boxes.

 

To enter the coordinates, click on the 'A' edit box with the mouse, then type the x1 coordinate. Now click on the 'D' edit box and type the y1 coordinate. Continue until the four coordinates are entered, then click the 'Update' button.

 

Step 3  Adjust the scale of the labelled diagram

If the points lie off the graphing area of the screen, the scale needs to be reduced. In this case, briefly press the F10 key enough times until the two plotted points are seen.

 

You can make the diagram bigger by holding down 'Ctrl' while you press F10.

 

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