To determine the vertex of the quadratic function [tex]\( f(x) = x^2 + 8x - 2 \)[/tex], we use the vertex formula. The vertex form of a quadratic function [tex]\( f(x) = ax^2 + bx + c \)[/tex] is at the point [tex]\( (h, k) \)[/tex] where:
1. The x-coordinate (h) of the vertex is given by [tex]\( h = -\frac{b}{2a} \)[/tex].
2. Once we find [tex]\( h \)[/tex], we substitute [tex]\( h \)[/tex] back into the function [tex]\( f(x) \)[/tex] to find the y-coordinate (k), i.e., [tex]\( k = f(h) \)[/tex].
For the given function [tex]\( f(x) = x^2 + 8x - 2 \)[/tex]:
- [tex]\( a = 1 \)[/tex]
- [tex]\( b = 8 \)[/tex]
- [tex]\( c = -2 \)[/tex]
### Step-by-Step Solution:
1. Calculate the x-coordinate of the vertex (h):
[tex]\[ h = -\frac{b}{2a} \][/tex]
[tex]\[ h = -\frac{8}{2 \cdot 1} \][/tex]
[tex]\[ h = -\frac{8}{2} \][/tex]
[tex]\[ h = -4 \][/tex]
2. Calculate the y-coordinate of the vertex (k):
The y-coordinate is found by substituting [tex]\( h = -4 \)[/tex] into the function [tex]\( f(x) \)[/tex]:
[tex]\[ k = f(-4) \][/tex]
[tex]\[ f(x) = x^2 + 8x - 2 \][/tex]
[tex]\[ f(-4) = (-4)^2 + 8(-4) - 2 \][/tex]
[tex]\[ f(-4) = 16 - 32 - 2 \][/tex]
[tex]\[ f(-4) = -18 \][/tex]
So, the vertex of the function [tex]\( f(x) = x^2 + 8x - 2 \)[/tex] is [tex]\((-4, -18)\)[/tex].
### Conclusion:
The correct answer is:
[tex]\[ (-4, -18) \][/tex]
