To find the vertex of the quadratic equation given in the form [tex]\( y = -4(x-2)^2 -1 \)[/tex], we need to understand the vertex form of a quadratic equation, which is:
[tex]\[
y = a(x-h)^2 + k
\][/tex]
In this form:
- [tex]\( (h, k) \)[/tex] represents the vertex of the parabola.
- [tex]\( a \)[/tex] is a coefficient that indicates the direction and the width of the parabola.
Here, the given quadratic equation is:
[tex]\[
y = -4(x-2)^2 - 1
\][/tex]
We can directly compare this with the vertex form [tex]\( y = a(x-h)^2 + k \)[/tex].
From the equation, we observe the following:
- The coefficient [tex]\( a \)[/tex] is [tex]\(-4\)[/tex], which determines the direction (downward, since it is negative) and the width of the parabola.
- The value of [tex]\( h \)[/tex] is [tex]\(2\)[/tex]. This value determines the horizontal shift of the vertex from the origin.
- The value of [tex]\( k \)[/tex] is [tex]\(-1\)[/tex]. This value determines the vertical shift of the vertex from the origin.
Thus, the vertex of the given quadratic equation [tex]\( y = -4(x-2)^2 - 1 \)[/tex] is determined by the values of [tex]\( h \)[/tex] and [tex]\( k \)[/tex].
[tex]\[
h = 2
\][/tex]
[tex]\[
k = -1
\][/tex]
Therefore, the vertex is:
[tex]\[
(2, -1)
\][/tex]
So, the vertex of the quadratic equation is [tex]\( (2, -1) \)[/tex].
