Answer
$\Rightarrow y=2x^2-x+3$.
Work Step by Step
The given points are
$(-1,6),(1,4)(2,9)$
The quadratic function is $y=ax^2+bx+c$.
Plug $(x,y)=(-1,6)$.
$\Rightarrow 6=a(-1)^2+b(-1)+c$
Simplify.
$\Rightarrow 6=a-b+c$ ...... (1)
Plug $(x,y)=(1,4)$.
$\Rightarrow 4=a(1)^2+b(1)+c$
Simplify.
$\Rightarrow 4=a+b+c$ ...... (2)
Plug $(x,y)=(2,9)$.
$\Rightarrow 9=a(2)^2+b(2)+c$
Simplify.
$\Rightarrow 9=4a+2b+c$ ...... (3)
Subtract equation (1) from equation (3).
$\Rightarrow 9-6=4a+2b+c-(a-b+c)$
Simplify.
$\Rightarrow 3=4a+2b+c-a+b-c$
$\Rightarrow 3=3a+3b$
Divide both sides by $3$.
$\Rightarrow 1=a+b$ ...... (4)
Subtract equation (2) from equation (3).
$\Rightarrow 9-4=4a+2b+c-(a+b+c)$
Simplify.
$\Rightarrow 5=4a+2b+c-a-b-c$
$\Rightarrow 5=3a+b$ ...... (5)
Subtract equation (4) from equation (5).
$\Rightarrow 5-1=3a+b-(a+b)$
Simplify.
$\Rightarrow 4=3a+b-a-b$
$\Rightarrow 4=2a$
Divide both sides by $2$
$\Rightarrow 2=a$
Plug the value of $a$ into equation (4).
$\Rightarrow 1=2+b$
Subtract $2$ from both sides.
$\Rightarrow 1-2=2+b-2$
Simplify.
$\Rightarrow -1=b$
Substitute the values of $a$ and $b$ into equation (1).
$\Rightarrow 6=2-(-1)+c$
$\Rightarrow 6=2+1+c$
$\Rightarrow 6=3+c$
Subtract $3$ from both sides.
$\Rightarrow 6-3=3+c-3$
$\Rightarrow 3=c$
Plug all values into the quadratic euqation.
$\Rightarrow y=2x^2-1x+3$.
$\Rightarrow y=2x^2-x+3$.
