ANSWER
[tex]y = 2 {x}^{2} + x + 2[/tex]
EXPLANATION
Let the function that represent the graph be:
[tex]y = a {x}^{2} + bx + c[/tex]
The parabola passes through the points (-2,8), (0,2), and (1,5).
These points must satisfy the function.
For (-2,8), we have
[tex]8= a{( - 2)}^{2} + b( - 2) + c[/tex]
This implies that that,
[tex]4a - 2b + c = 8...(1)[/tex]
For (0,2), we have,
[tex]2= a{( 0)}^{2} + b( 0) + c[/tex]
This implies that,
[tex]c = 2[/tex]
For (1,5), we have
[tex]5= a{( 1)}^{2} + b( 1) + c[/tex]
This implies that,
[tex]a + b + c = 5...(2)[/tex]
Put c=2 into equation (1) and (2).
[tex]4a - 2b + 2 = 8[/tex]
[tex]4a - 2b = 8 - 2[/tex]
[tex]4a - 2b = 6[/tex]
[tex]2a - b = 3...(3)[/tex]
[tex]a + b + 2=5[/tex]
[tex]a + b =5 - 2[/tex]
[tex]a + b = 3...(4)[/tex]
Add equation (3) and equation (4)
[tex]3a = 6[/tex]
[tex]a = 2[/tex]
Put a=2 into equation (4).
[tex]2 + b = 3[/tex]
[tex]b = 3 - 2 = 1[/tex]
Therefore the function is
[tex]y = 2 {x}^{2} + x + 2[/tex]
