Answer
$y=2x^2+x-5$.
Work Step by Step
The given points are
$(-1,-4),(1,-2),(2,5)$
The quadratic function is $y=ax^2+bx+c$.
Plug $(x,y)=(-1,-4)$.
$\Rightarrow -4=a(-1)^2+b(-1)+c$
Simplify.
$\Rightarrow -4=a-b+c$ ...... (1)
Plug $(x,y)=(1,-2)$.
$\Rightarrow -2=a(1)^2+b(1)+c$
Simplify.
$\Rightarrow -2=a+b+c$ ...... (2)
Plug $(x,y)=(2,5)$.
$\Rightarrow 5=a(2)^2+b(2)+c$
Simplify.
$\Rightarrow 5=4a+2b+c$ ...... (3)
Subtract equation (1) from equation (3).
$\Rightarrow 5-(-4)=4a+2b+c-(a-b+c)$
Simplify.
$\Rightarrow 5+4=4a+2b+c-a+b-c$
$\Rightarrow 9=3a+3b$
Divide both sides by $3$.
$\Rightarrow 3=a+b$ ...... (4)
Subtract equation (2) from equation (3).
$\Rightarrow 5-(-2)=4a+2b+c-(a+b+c)$
Simplify.
$\Rightarrow 5+2=4a+2b+c-a-b-c$
$\Rightarrow 7=3a+b$ ...... (5)
Subtract equation (4) from equation (5).
$\Rightarrow 7-3=3a+b-(a+b)$
Simplify.
$\Rightarrow 4=3a+b-a-b$
$\Rightarrow 4=2a$
Divide both sides by $2$.
$\Rightarrow 2=a$
Substitute the value of $a$ into equation (4).
$\Rightarrow 3=2+b$
Subtract $2$ from both sides.
$\Rightarrow 3-2=2+b-2$
Simplify.
$\Rightarrow 1=b$
Substitute the values of $a$ and $b$ into equation (2)
$\Rightarrow -2=2+1+c$
Simplify.
$\Rightarrow -2=3+c$
Subtract $3$ from both sides.
$\Rightarrow -2-3=3+c-3$
Simplify.
$\Rightarrow -5=c$
Plug all values into the quadratic equation.
$\Rightarrow y=2x^2+1x-5$.
$\Rightarrow y=2x^2+x-5$.
