If it's a quadratic we know that it must look like [tex] y=ax^2+bx+c [/tex] for some numbers a,b and c. To find these numbers we use the coordinates.
(0,-2) lies on the curve so when x=0, y=-2
[tex] y=ax^2+bx+c \Rightarrow -2=a(0^2)+b(0)+c \Rightarrow -2 = c [/tex]
(so that solves one of them)
Next (1,0) lies on the curve so when x=1, y=0
[tex] y=ax^2+bx-2 \Rightarrow 0=a(1^2)+b(1) -2 \Rightarrow a+b-2=0 \Rightarrow a+b=2 [/tex]
Finally (3,10) lies on the curve so when x=3, y=10
[tex] y=ax^2+bx-2 \Rightarrow 10 = a(3^2)+b(3)-2 \Rightarrow 9a+3b-2=10 \Rightarrow 9a+3b=12 \Rightarrow 3a+b=4 [/tex]
So then we have to solve the simultaneous equations to get a and b
[tex] a+b=2 \\
3a+b=4
[/tex]
These equations have solutions a=1 and b=1 and so we have a=1, b=1, c=-2.
Therefore the equation of the quadratic is [tex] y=x^2+x-2 [/tex]
