This problem was solved by Ali Abu_Hijleh, age 13, Alex Liao, age 14 and Prateek Mehrotra, age 14 from Riccarton High School, Christchurch, New Zealand and Andrei Lazanu, age: 12 from School No. 205, Bucharest, Romania. Well done all of you. This is Andrei's solution:
I observed that the equations of the parabolas given in the problem are of the form:
where positive corresponds to parabolas with the opening towards the top,and negative to parabolas with the opening towards the bottom; are the coordinates of the peak. I identified the parabolas in your pattern and I verified them with the help of a small program in Matlab:
x=-8:0.1:8
y1=-2*(x+6).*(x+6)-2
y2=-2*(x+4).*(x+4)-1
y3=-2*(x+2).*(x+2);
y4=-2*x.*x+1;
y5=-2*(x-2).*(x-2)+2;
y6=-2*(x-4).*(x-4)+3;
y7=-2*(x-6).*(x-6)+4;
y8=2*(x+6).*(x+6)+2;
y9=2*(x+4).*(x+4)+1;
y10=2*(x+2).*(x+2);
y11=2*x.*x-1;
y12=2*(x-2).*(x-2)-2;
y13=2*(x-4).*(x-4)-3;
y14=2*(x-6).*(x-6)-4;
plot(x,y1,x,y2,x,y3,x,y4,x,y5,x,y6,x,y7,x,y8,x,y9,x,y10,x,y11,x,y12,x,y13,x,y14)
hold
axis([-8 8 -4 4]);
grid
Consequently, the equations of the 14 parabolas in your pattern are:
| y | = | -2(x + 6) 2 - 2 |
| y | = | -2(x + 4) 2 - 1 |
| y | = | -2(x + 2) 2 |
| y | = | -2x 2 + 1 |
| y | = | -2(x - 2) 2 + 2 |
| y | = | -2(x - 4) 2 + 3 |
| y | = | -2(x - 6) 2 + 4 |
| y | = | 2(x + 6) 2 + 2 |
| y | = | 2(x + 4) 2 + 1 |
| y | = | 2(x + 2) 2 |
| y | = | 2x 2 - 1 |
| y | = | 2(x - 2) 2 - 2 |
| y | = | 2(x - 4) 2 - 3 |
| y | = | 2(x - 6) 2 - 4 |
I created my own pattern with a Matlab program and here are the graphs:
Andrei supplied the Matlab program but in order to challenge you to find his equations it is not reproduced here