The vertex form of the parabola is
[tex]y=(x-1)^2-4[/tex]the general form is
[tex]y=x^2-2x-3[/tex]and the factored form is
[tex]y=(x+1)(x-3)[/tex]To solve this, we look at the graph and see that the vertex coordinates are (1,-4)
the vertex form is
[tex]y=a(x-h)^2+k[/tex]where h is the x coordinate of the vertex and k is the y coordinate. now we have to find the value of a
the factor form is
[tex]y=a(x-x_1)(x-x_2)[/tex]where x1 and x2 are the roots of the parabola. we know the coordinates of the roots: (-1,0) (3,0)
now, we can use this to find the value of a
[tex]y=a(x+1)(x-3)[/tex]now we plug the coodinates of the vertex (1,-4) in the equation before and solve for a
[tex]-4=a(1+1)(1-3)[/tex][tex]\frac{-4}{-4}=1=a[/tex]now we just add the value of a to the factor and vertex forms. the only thing remaining is the general form. for this we need to apply the distributive property in the factor form
[tex]y=(x+1)(x-3)=x^2-2x-3[/tex]and thus, we have all three forms calculated