ANSWER
[tex]y = 2( {x - 4)}^{2} - 3[/tex]
EXPLANATION
The function equation of a parabola that opens up in vertex form is given by
[tex]y = a( {x - h)}^{2} + k[/tex]
where (h,k) is the vertex and 'a' is the leading coefficient.
The given graph is a parabola that opens up and has its vertex at (4,-3).
This implies that, h=4 and y=-3
We substitute these values into the vertex form to obtain,
[tex]y =a( {x - 4)}^{2} + - 3[/tex]
This simplifies to,
[tex]y =a( {x - 4)}^{2} - 3[/tex]
The graph also contains (3,-1). We plug x=3 and y=-1 into the equation to find the value of 'a'.
[tex] - 1=a( {3 - 4)}^{2} - 3[/tex]
[tex] - 1 + 3 = a( { - 1})^{2} [/tex]
[tex]2 = a[/tex]
We substitute this value to get:
[tex]y = 2( {x - 4)}^{2} - 3[/tex]
The last choice is correct.
