So the equation of the red graph is already given to us: [tex]f(x) = x^2[/tex]. I assume you meant the blue graph?
We can use the vertex form: [tex]y = a(x - h)^2+ k[/tex], where [tex](h,k)[/tex] is the vertex. (We can ignore [tex]a[/tex] because [tex]a[/tex] is 1 for this graph.) Remember that the vertex is the maximum or minimum of a parabola and the point that the line of symmetry passes through. For f(x) the vertex is (0,0).
The vertex for [tex]g(x)[/tex] is at [tex](-4, -2)[/tex] on the coordinate plane. (4 left, 2 down)
Plug into the vertex form:
[tex]y = (x - (-4))^2 -2[/tex]
The double negative becomes a positive, and then you're done:
[tex]y = (x + 4)^2-2[/tex]
