To determine the coordinates of the vertex of the parabola represented by the equation [tex]\( y = x^2 - 4x + 3 \)[/tex], we use the standard formula for finding the vertex of a quadratic equation in the form [tex]\( y = ax^2 + bx + c \)[/tex].
Given the quadratic equation [tex]\( y = x^2 - 4x + 3 \)[/tex]:
- The coefficient [tex]\( a \)[/tex] is 1.
- The coefficient [tex]\( b \)[/tex] is -4.
- The constant term [tex]\( c \)[/tex] is 3.
The x-coordinate of the vertex of a parabola can be found using the formula:
[tex]\[ x = -\frac{b}{2a} \][/tex]
Substituting the values of [tex]\( a \)[/tex] and [tex]\( b \)[/tex]:
[tex]\[ x = -\frac{-4}{2 \cdot 1} = \frac{4}{2} = 2 \][/tex]
Next, we find the y-coordinate of the vertex by substituting [tex]\( x = 2 \)[/tex] back into the original quadratic equation:
[tex]\[ y = 1 \cdot (2)^2 - 4 \cdot 2 + 3 \][/tex]
[tex]\[ y = 4 - 8 + 3 \][/tex]
[tex]\[ y = -4 + 3 \][/tex]
[tex]\[ y = -1 \][/tex]
So, the coordinates of the vertex are:
[tex]\[ (2, -1) \][/tex]
The correct answer is:
[tex]\[ (2, -1) \][/tex]
None of the provided options match this result. It appears there may be a typo or error in the provided answer choices. The actual coordinates of the vertex are [tex]\( (2, -1) \)[/tex].
