This is the point-slop form:
y-y1=m(x-x1)
The point slop form is led out from the definition of slop.
When separating m(the slop) on one side, we get the definition of the slop.
m=y-y1/x-x1=∆y/∆x=rise/run
But here is another possible interpretation of the point slop form.
When we know the slop m, we get the original function f(x)=mx. In the diagram below I set m = 1. It passes through the origin and all other linear functions can be seen as a transformed version of the function by adding a constant b. The distance between to point with the same x value of both graph (not the distance of the lines) will be the constant b.
Now when we look at the point slop form.
y-y1=m(x-x1)
We can separate y to one side of the equation.
y=m(x-x1)+ y1
y=mx-mx1+y1
We can compere this with the slop intercept form.
y=mx-mx1+y1
y=mx+b
Now we might think that b is going o be equal to y1-mx1
Because the function of g(x) is y1=mx1+b
So mx1 is equal to y1-b
So y1-mx1 is y1 –(y1-b) and also equals to b, as we expected.
In other words, the point slop form does the job of getting the y-intercept of the linear function using the slop and a point on the line.
This is can also be done manually like this, say a point (x-x1) is (1,5) and a m= 2
Time x1 with m to get y, y=1*2=2
Compare the difference of y and y1 to find out b, b=y1-y=3
So this is a interpretation of point slop form that I found when trying to better understand it. Finding relationships in math always make me feel satisfied. It gives me a good night sleep.
