PROPERTIES OF LINEAR FUNCTIONS (II) 
Analysis
 
1. ONE POINT - ONE STRAIGHT LINE
Every point on the plane, except for the origin, determines a unique linear function. y = m x
Move the red point by dragging it with the mouse, then use the arrows to move it more precisely.

1.-Find the straight line that belongs to point A.

2.-Move point A in the different quadrants until you find its corresponding straight line graph.

As we only need to know two points to draw a straight line and the straight lines associated with linear functions always pass through the origin, the line can be drawn if we know just one point (as long as it is not the origin (0,0)).
For any point (a,b), except the origin, there is only one linear function whose straight line passes through this point.
2. POSITIVE AND NEGATIVE GRADIENTS
The gradient can be either positive, zero or negative.

y = m x

3.- Is the gradient of the linear function that goes through the point (4,6) positive or negative?

4.- what about the following points:(7,6); (-3,-4); (-5,9); (4,-8); (4,-100); (10,10); (-7,-7); (0,3); (0,-5)?

5.- In your notebook explain when the gradient is positive, zero or negative.
The gradient can be positive or negative depending on whether x- and y-coordinates of the points on the line are positive or negative:
If the x- and y-coordinates are both positive or both negative the gradient is positive (in the first and third quadrants).
If the x- and y- coordinates are different (i.e. one positive and the other negative) then the gradient is negative (in the second and fourth quadrants).
3. WORKING OUT THE GRADIENT
The gradient can be worked out from the coordinates of any one point on the line.
You can cover all the points on this line by moving the red point.

6.- When m = 0.4 check that the y-coordinate divided by the x-coordinate gives this result for every point on the line. 

If you change m you change the line.

7.- Show that the gradient of any line is the ratio of its vertical distance over its horizontal distance for any point except the origin.
To find the gradient of a line choose any point and divide its y-coordinate by its x-coordinate.
4. POINTS NOT ON THE LINE
The only points that satisfy the relation defined by a linear function are those points on the line. Points which do not lie on the line do not satisfy the relation.
Move the red point.

8.- Show that for a point which is not on the line, the ratio of the y-coordinate to the x-coordinate is different to the gradient.

9.- See if this is true for other straight lines.
The ratio of the y-coordinate to the x-coordinate of any point is equal to the gradient of the line going through that point and different to the gradients of any other straight line.
       
           
  Juan Madrigal Muga
 
Spanish Ministry of Education. Year 2001
 
 

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