7: y=mx; m +ve -ve frac

Picture

Any point on a straight line is defined as two numbers and the way to look at these two numbers is 
1. the first one horizontally and 
2. the second one vertically.

First go horizontal and then go vertical

Here in the diagram you go horizontal by x, so the pt. on the x axis will be (x,0) then you go vertical to reach the line y=x. So since y is equal to x and the x coordinate in this case, we have taken to be some number x, then the y is also x, so the pt. on the line will be (x,x).

But let us not stop at (x,x) and keep going up by another vertical distance x so that now we are at (x,2x) as shown in the picture. 

Now if you join the Origin to this point you will get a line whose equation will be y = 2x. It is very simple actually. In fact, if we still go vertically up and reach (x, 3x) and connect that point to O, we will get a line y=3x and so on and so forth.

Now you know how to draw y=kx where k is a number greater than 1. It will simply be a line which will be steeper than y=x, right ? do you get that ? simple, is it not ? 


what if we change the line to less steep than y=x ?

Picture

Here what we do is go to (x,0) on x-axis (remember, we always go horizontally when we want to reach any point on the graph) then we go up vertically to reach a distance of x/2 to reach (x,x/2)


Look at the diagram and see that now we have a line y= minus x. Simply see that this line will past through Q2 & Q4 i.e. through the 2nd and the 4th Quadrants. This is because any point on this line will have one of the two numbers negative and the other positive. 

Picture

Similar to how we understood y=x with slope =45 degree and y=2x having a steeper angle while y=x/2 having a less steep angle than 45, similarly in case of y=-x, the same theme plays out and as you can see in the adjoining diagram, y=-2x is steeper than y=-x and y=-x/2 is less steep and a lesser slope than y=-x. It is quite simple and I would recommend the student to have an intutive feel for these straight lines.