8: y=mx+a; y=mx-a


Here we start with y = x line and move the line vertically upwards. Or downwards.
So, the new line is parallel and each point on y=x goes up vertically by 1 as shown in the picture and thus the new line is y=x+1. Similarly, if you push y=x down by 1, you get another parallel line y=x-1

what if we change the line to y = – x and push it up and down by 1
As you can see once again, we can very easily conceive pushing up and down y=-x and we get two parallel lines y=-x+1 when we push up and y=-x-1 when we push y=-x down by 1

Here we show that if we draw a vertical line from x=-5, as shown, or any point on the x axis, this vertical line cuts these three lines at distances of 1 each from each other which again proves that these three lines are parallel and at a distance of 1 vertically from each other. 

You can do this with any Straight Line


In the diagram we have taken the angle of the slope as 30 degrees. And the line now is y=mx with m=tan 30

And thus the line which will be 4 units up vertically, will be y=mx+4 and as shown, the line which is pushed down by 6 units will have an equation y=mx-6

This understanding will give you an intuitive feel for the general equation of the straight line y=mx+c