Open Middle®
Directions: Using the digits 0 to 9, at most one time each, place a digit in each box to create three equations that produce the exact same parabola.
Hint
What can you say about where this parabola must be located?
Which numbers can you rule out of some blanks?
Which numbers can you rule out of some blanks?
Answer
Source: Zack Miller
This was an interesting one to tackle. Because of the symmetry of parabolas (i.e. the x-coordinate of the vertex being exactly halfway between the x-coordinates of the x-intercepts), there are 16 possible combinations of one digit values that can go in the Factored Form equation:
1&9
1&7
2&8
3&9
1&5
2&6
3&7
4&8
5&9
1&3
2&4
3&5
4&6
5&7
6&8
7&9
Of those 16, only 6 have a b-value (from Standard Form) that’s a one-digit number. Of those 3 can be ruled out because their c-value equals one of the digits from the Factored Form (because one times anything is itself). Of the remaining 3, two would have 2-digit c-value, which means there is only one possible Vertex Form. I thought I would have to do a lot of work with the Vertex Form stuff — I’m surprised that there was no process of elimination left to do at that point — it was just solving the one I already had.