Answer:
[tex]f(x)=x^2+6x+7[/tex]
Step-by-step explanation:
The vertex form of a parabola is
[tex]f(x)=a(x-h)^2+k[/tex] .... (1)
where, a is a constant, (h,k) is vertex of the parabola.
From the given graph it is clear that the vertex of the parabola is (-3,-2).
Substitute h=-3 and k=-2 in equation (1).
[tex]f(x)=a(x-(-3))^2+(-2)[/tex]
[tex]f(x)=a(x+3)^2-2[/tex] .... (2)
The graph is passes through the point (-2,-1). So, the function must be satisfy by the point (-2,-1).
[tex]-1=a(-2+3)^2-2[/tex]
[tex]-1=a-2[/tex]
Add 2 on both sides.
[tex]-1+2=a-2+2[/tex]
[tex]1=a[/tex]
The value of a is 1. Substitute this value in equation (2).
[tex]f(x)=(1)(x+3)^2-2[/tex]
[tex]f(x)=(x+3)^2-2[/tex]
[tex]f(x)=x^2+6x+9-2[/tex]
[tex]f(x)=x^2+6x+7[/tex]
Therefore, the standard form of the parabola is [tex]f(x)=x^2+6x+7[/tex].
