To find the vertex of the parabola defined by the quadratic function [tex]\( f(x) = x^2 - 6x + 13 \)[/tex], we can use the vertex formula for a quadratic equation in standard form [tex]\( ax^2 + bx + c \)[/tex].
The vertex [tex]\((h, k)\)[/tex] of a parabola given by the quadratic function [tex]\( f(x) = ax^2 + bx + c \)[/tex] can be found using the following steps:
1. Determine the x-coordinate [tex]\( h \)[/tex] of the vertex:
[tex]\[
h = -\frac{b}{2a}
\][/tex]
In our quadratic function [tex]\( f(x) = x^2 - 6x + 13 \)[/tex]:
[tex]\[
a = 1, \quad b = -6, \quad \text{and} \quad c = 13
\][/tex]
Plugging [tex]\( a \)[/tex] and [tex]\( b \)[/tex] into the formula:
[tex]\[
h = -\frac{-6}{2 \times 1} = \frac{6}{2} = 3
\][/tex]
2. Determine the y-coordinate [tex]\( k \)[/tex] of the vertex by plugging [tex]\( h \)[/tex] back into the original equation:
[tex]\[
k = f(h) = f(3)
\][/tex]
Substitute [tex]\( x = 3 \)[/tex] into [tex]\( f(x) \)[/tex]:
[tex]\[
f(3) = (3)^2 - 6(3) + 13 = 9 - 18 + 13 = 4
\][/tex]
Therefore, the vertex of the parabola is [tex]\((3, 4)\)[/tex].
Looking at the given options:
a. [tex]\((4,0)\)[/tex]
b. [tex]\((0,3)\)[/tex]
c. [tex]\((3,4)\)[/tex]
d. [tex]\((4,3)\)[/tex]
The correct answer is:
c. [tex]\((3,4)\)[/tex]
