How do I find the points of intersection between two curves?
This problem is a graphical representation of finding the solutions to a pair of simultaneous equations.
In this example we will use the curves y=2x2 , and y=x2+1. This is a very straightforward example, but demonstrates the method of finding the intersection of two curves well.
Step 1 - since the LHS of both these equations is the same (y=...) we can equate the two equations:
2x2=x2+1
This is a fairly easy equation to solve:
Lets make one side equal to zero:
-x2 +1=0
Lets move everything across to the other side to get rid of the minus signs.
x2 - 1 =0
This is the difference of two squares, so can be factorised:
(x+1)(x-1)=0
So the x-coordinates of the intersection points are +1 and -1.
Step 2 - Now we need to find the y-coordinates. We do this by plugging the x-values into the original equations. We can use either one, because the lines intersect (so they should give us the same result!)
When x= +1,
y=2x2
y=2(1)2 =2
When x= -1
y=2(-1)2 = 2
So the points of intersection have coordinates (-1,2) and (1,2)
We can see this graphically: (see how easy this example was!)
http://fooplot.com/plot/l26e9y97hd
