3 Permutations of 3 words give us his-story:

History 101

by Christopher Smith, 2012

Tree, Man, Promise (a tree not for man and a promise)

The fruit of the Tree of the Knowledge of Good and Evil was forbidden, but eaten, and a plan was made for redemption: "And I will put enmity between you and the woman, and between your offspring and hers; he will crush your head, and you will strike his heel." (Gen 3: 15)

Promise, Tree, Man (a promise on a tree for all man)

"Christ redeemed us from the curse of the law by becoming a curse for us, for it is written: 'Cursed is everyone who is hung on a tree.'" (Gal 3: 13)

Man, Promise, Tree (man and the promised tree...forever)

"For as many as received Him to them he gave power to become children of God." (John 1:12) "Blessed are those who wash their robes, that they may have the right to the tree of life and may go through the gates into the city." (Rev 22:14)

An Easier Method to Find the Vertex from 3 Points

1. from left to right, label the points A, B, C
2. put a horizontal line through A marking where vertical lines through A, B and C cross as a, b and c
3. make a point on the vertical through the midpoint of a and b that is at the same height as B and label as N
4. mark the intersection of the line through A and C with the vertical through B as d
5. make a point on the vertical through the midpoint of a and c that is at the same height as d and label as M
6. mark where the line through N and M cross the horizontal through A as S; this is the axis of symmetry; make a vertical line through it
7. label the highest vertical point (of A, B and C) as H and the lowest as L
8. on the horizontals through H and L make points H' and L' that are at the same height but on the opposite side of the vertical through S
9. mark where the line from H to L' crosses the vertical through S as O
10. label as U, the intersection of the horizontal through L and the vertical through H'
11. draw from U through O and mark the intersection with the vertical through H as U'
12. draw from U' to L; this is the tangent at L
13. the vertical through L, reflected by the tangent at L, crosses the vertical through S at F, the focus
14. locate a point on the vertical through L that is the same distance away from L as F but located on the opposite side of the horizontal through L and label this d
15. draw a horizontal through d (the directrix), marking where it crosses the vertical through S as D
16. the vertex, V, is halfway between F and D on the vertical through S
17. let H be an arbitrary point on S above D: the intersection of the horizontal through H and an arc from F, the length of HD, is an arbitrary point on the parabola.

Geometric Construction of Parabola Vertex from 3 points and a Horizontal Axis

I was wondering how easy it would be to draw the correct parabola on a whiteboard given only 3 points (given that the parabola opens in the vertical direction). It's not as easy as connecting two points...it's not even that practical in the end. But it was fun figuring it out.

This can be accomplished as follows (provided that no points are vertically aligned):

1. draw 3 vertical lines through the points and label the one whose line is between the other two as B
2. label the point whose vertical line is closer to B as A and the other one C
3. also label the point that is to the left of B as L and the one to the right as R
4. locate the midpoints of LB and RB as L' and R'
5. draw a horizontal line through B and locate the intersection between the vertical through R and this horizontal line as U
6. on the vertical through L' mark the point that is as far above/below BU as L is below/above BU and label that point D
7. draw a vertical line that is as far to the right of B as L is to the left of B and label the intersection of that line and RB as e
8. make a point on the vertical through R' that is the same height as e and label that point E
9. Mark as S the intersection between line DE and BU. The vertical through S is the axis of symmetry of the parabola.
10. make points the same height as A and B on the opposite side of the vertical through S and label them A' and B'

From here, credit to  http://mathafou.free.fr/themes_en/parabole2.html for the method of now locating the vertex.

1. mark where line AB' crosses the vertical through S as O
2. mark where line UO crosses the vertical through A as U'
3. draw a line from U' to B; this is the tangent to the parabola where it passes through B
4. reflect the vertical through B from U'B and where it cross the vertical through S is the focal point, F
5. the directrix is as far below B as the distance from B to F, mark that point below B as x
6. mark the point where the horizontal through x crosses the vertical through S as X
7. the vertex of the parabola is the midpoint between F and X, V.

It would look better in an animated GIF but here is an example with points and lines marked: