Answer
$\Rightarrow y=2x^2-x-3$.
Work Step by Step
The given points are
$(-2,7),(1,-2),(2,3)$
The quadratic function is $y=ax^2+bx+c$.
Plug $(x,y)=(-2,7)$.
$\Rightarrow 7=a(-2)^2+b(-2)+c$
Simplify.
$\Rightarrow 7=4a-2b+c$ ...... (1)
Plug $(x,y)=(1,-2)$.
$\Rightarrow -2=a(1)^2+b(1)+c$
Simplify.
$\Rightarrow -2=a+b+c$ ...... (2)
Plug $(x,y)=(2,3)$.
$\Rightarrow 3=a(2)^2+b(2)+c$
Simplify.
$\Rightarrow 3=4a+2b+c$ ...... (3)
Subtract equation (1) from equation (3).
$\Rightarrow 3-7=4a+2b+c-(4a-2b+c)$
Simplify.
$\Rightarrow -4=4a+2b+c-4a+2b-c$
$\Rightarrow -4=4b$
Divide both sides by $4$.
$\Rightarrow -1=b$ ...... (4)
Subtract equation (2) from equation (3).
$\Rightarrow 3-(-2)=4a+2b+c-(a+b+c)$
Simplify.
$\Rightarrow 3+2=4a+2b+c-a-b-c$
$\Rightarrow 5=3a+b$ ...... (5)
Substitute the value of $b$ into equation (5).
$\Rightarrow 5=3a+(-1)$
$\Rightarrow 5=3a-1$
Add $1$ to both sides.
$\Rightarrow 5+1=3a-1+1$
$\Rightarrow 6=3a$
Divide both sides by $3$
$\Rightarrow 2=a$
Substitute the values of $a$ and $b$ into equation (2)
$\Rightarrow -2=2-1+c$
$\Rightarrow -2=1+c$
Subtract $1$ from both sides.
$\Rightarrow -2-1=1+c-1$
Simplify.
$\Rightarrow -3=c$
Plug all values into the quadratic equation.
$\Rightarrow y=2x^2-1x-3$.
$\Rightarrow y=2x^2-x-3$.
