Note that a quadratic function can be written in vertex form as:.....
[tex]y=a(x-k)^2\text{ + h}[/tex]Now, its vertex will be at (k,h)... where k = -3 and h = -5
[tex]y=a(x+3)^2\text{ - 5}[/tex]Since it passes through the point (0, 13), we have .....
[tex]\begin{gathered} 13=a(0+3)^2\text{ -5} \\ 13\text{ = 9a - 5} \\ \text{13 + 5 = 9a} \\ a\text{ = }\frac{18}{9} \\ a\text{ = 2} \end{gathered}[/tex]So, the standard form of the quadratic function is....
[tex]\begin{gathered} y=2(x+3)^2\text{ - 5} \\ y\text{ = 2(x + 3)(x + 3) - 5} \\ y=2(x^2\text{ + 6x + 9) - 5} \\ y=2x^2\text{ + 12x + 18 - 5} \\ y=2x^2\text{ + 12x +13} \end{gathered}[/tex]