To find the vertex of the quadratic function [tex]\( f(x) = x^2 - 4x + 21 \)[/tex], we can use the vertex formula for a quadratic function in the form [tex]\( f(x) = ax^2 + bx + c \)[/tex].
The x-coordinate of the vertex is given by:
[tex]\[ x = \frac{-b}{2a} \][/tex]
For the function [tex]\( f(x) = x^2 - 4x + 21 \)[/tex]:
- The coefficient [tex]\( a \)[/tex] is 1.
- The coefficient [tex]\( b \)[/tex] is -4.
Substitute these values into the vertex formula to find the x-coordinate:
[tex]\[ x = \frac{-(-4)}{2(1)} = \frac{4}{2} = 2 \][/tex]
Now, we need to find the y-coordinate of the vertex. This is done by substituting [tex]\( x = 2 \)[/tex] back into the function [tex]\( f(x) \)[/tex]:
[tex]\[ f(2) = (2)^2 - 4(2) + 21 \][/tex]
[tex]\[ f(2) = 4 - 8 + 21 \][/tex]
[tex]\[ f(2) = 17 \][/tex]
Thus, the coordinates of the vertex are [tex]\( (2, 17) \)[/tex].
Hence, the point that gives the vertex of [tex]\( f(x) = x^2 - 4x + 21 \)[/tex] is:
[tex]\[ \boxed{(2, 17)} \][/tex]
