Sure, let's analyze the given equation of the parabola and find its vertex step-by-step.
The given equation of the parabola is
[tex]\[ y^2 = 4(x + 1) \][/tex]
This equation is in the form of a parabolic equation with a horizontal axis of symmetry. The standard form of a parabola with a horizontal axis is:
[tex]\[ (y - k)^2 = 4p(x - h) \][/tex]
Here, [tex]\((h, k)\)[/tex] is the vertex of the parabola, and [tex]\( p \)[/tex] is the distance from the vertex to the focus.
We start by rewriting the given equation in a way that makes it easier to compare with the standard form. Notice that:
[tex]\[ y^2 = 4(x + 1) \][/tex]
can be rewritten slightly as:
[tex]\[ (y - 0)^2 = 4(x - (-1)) \][/tex]
Comparing this with the standard form [tex]\((y - k)^2 = 4p(x - h)\)[/tex], we see that:
1. The value of [tex]\( h \)[/tex] is [tex]\(-1\)[/tex].
2. The value of [tex]\( k \)[/tex] is [tex]\( 0 \)[/tex].
Therefore, the vertex of the given parabola [tex]\( y^2 = 4(x + 1) \)[/tex] is:
[tex]\[ (-1, 0) \][/tex]
So, the vertex of the parabola is [tex]\(\boxed{(-1, 0)}\)[/tex].
