Find the vertex of the quadratic equation:

[tex]\[ y = -4(x - 2)^2 - 1 \][/tex]

The vertex is [tex]\( (\square, \square) \)[/tex].



Answer :

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].