To find the coordinates of the vertex of the given parabola [tex]\( y = x^2 - 5 \)[/tex], let's follow these steps:
1. Identify the standard form of the quadratic function:
The equation given is [tex]\( y = x^2 - 5 \)[/tex]. This is a quadratic equation in the form [tex]\( y = ax^2 + bx + c \)[/tex], where [tex]\( a = 1 \)[/tex], [tex]\( b = 0 \)[/tex], and [tex]\( c = -5 \)[/tex].
2. Determine the x-coordinate of the vertex:
The x-coordinate of the vertex of a parabola given by the equation [tex]\( y = ax^2 + bx + c \)[/tex] can be found using the formula:
[tex]\[
x = -\frac{b}{2a}
\][/tex]
Since [tex]\( a = 1 \)[/tex] and [tex]\( b = 0 \)[/tex],
[tex]\[
x = -\frac{0}{2 \cdot 1} = 0
\][/tex]
3. Calculate the y-coordinate of the vertex:
Substitute the x-coordinate back into the original equation to find the y-coordinate:
[tex]\[
y = (0)^2 - 5 = -5
\][/tex]
4. Combine the coordinates:
The vertex of the parabola is at the point [tex]\((x, y)\)[/tex]. So, the vertex is:
[tex]\[
(0, -5)
\][/tex]
Therefore, the coordinates of the vertex of the parabola [tex]\( y = x^2 - 5 \)[/tex] are [tex]\( (0, -5) \)[/tex].
