The red curve is f(x) (for -2≤x≤2). The blue curve is the terms up to the square term of the Taylor series. A is the vertex of this parabola, B is its focus. The orange curve is the caustic - envelope of the reflection of vertical rays reflected in a mirror aligned with f(x). Observe that the focus of the parabola traces the caustic.
You are shooting a projectile on a sloping field. Drag the red point to change the angle of the projectile and try and maximize |AB|.
Click the button to see the focus of the parabola.
What do you notice?
Create a triangle which has 3 points on the circumference of an ellipse, and two sides passing through the foci. Look at 2 curves formed by the third side: the locus of its center and its envelope. One is an ellipse, the other is not.