Originally Posted by
Illusion3 There is probably a better way than what I am about to show you, but I think I know using visualisation geometry.
y=1+2x-x^2 can be factorized as y= -(x-1)(x-1)
If we ignore the negative in this factorization, we know that it is a 'happy face' parabola that it's minimum y value is (1,0).
We also know that because the equation has a +1 (y=1+2x-x^2) it will hit the y axis at (0,1).
However, it is not a happy face parabola because it is a negative (remember we said we would first ignore it?). This therefore makes the whole graph a complete reflection of itself in the same line that it hits the y-axis. Therefore, the line of symmetry is y=1.
Therefore, we know that the highest y value of the graph is (1,2), because the original 'dip' of the graph was 1 below the line of reflection, now it is 1 above it.
So, looking at where they intersect, we can immediately figure out that the first point is (0,1) because whenever you have a power to 0 the value is 1. We can also then easily work out that the point (2,4) is also an intersection because 4 is simply a power of 2.
Kind of long, but I guess the idea is to have a rough drawing of the lines so you can work out the intersections