Answer
$y=x^2-6x+8$.
Work Step by Step
The given points are
$(1,3),(3,-1),(4,0)$
The quadratic function is $y=ax^2+bx+c$.
Plug $(x,y)=(1,3)$.
$\Rightarrow 3=a(1)^2+b(1)+c$
Simplify.
$\Rightarrow 3=a+b+c$ ...... (1)
Plug $(x,y)=(3,-1)$.
$\Rightarrow -1=a(3)^2+b(3)+c$
Simplify.
$\Rightarrow -1=9a+3b+c$ ...... (2)
Plug $(x,y)=(4,0)$.
$\Rightarrow 0=a(4)^2+b(4)+c$
Simplify.
$\Rightarrow 0=16a+4b+c$ ...... (3)
Subtract equation (1) from equation (3).
$\Rightarrow 0-3=16a+4b+c-(a+b+c)$
Simplify.
$\Rightarrow -3=16a+4b+c-a-b-c$
$\Rightarrow -3=15a+3b$
Divide both sides by $3$.
$\Rightarrow -1=5a+b$ ...... (4)
Subtract equation (2) from equation (3).
$\Rightarrow 0-(-1)=16a+4b+c-(9a+3b+c)$
Simplify.
$\Rightarrow 1=16a+4b+c-9a-3b-c$
$\Rightarrow 1=7a+b$ ...... (5)
Subtract equation (4) from equation (5).
$\Rightarrow -1-1=5a+b-(7a+b)$
Simplify.
$\Rightarrow -2=5a+b-7a-b$
$\Rightarrow -2=-2a$
Divide both sides by $-2$.
$\Rightarrow 1=a$
Substitute the value of $a$ into equation (4).
$\Rightarrow -1=5(1)+b$
$\Rightarrow -1=5+b$
Subtract $5$ from both sides.
$\Rightarrow -1-5=5+b-5$
Simplify.
$\Rightarrow -6=b$
Substitute the values of $a$ and $b$ into equation (2)
$\Rightarrow -1=9(1)+3(-6)+c$ ...... (2)
Simplify.
$\Rightarrow -1=9-18+c$
$\Rightarrow -1=-9+c$
Add $9$ to both sides.
$\Rightarrow -1+9=-9+c+9$
Simplify.
$\Rightarrow 8=c$
Plug all values into the quadratic equation.
$\Rightarrow y=1x^2+(-6)x+8$.
$\Rightarrow y=x^2-6x+8$.
