To find the vertex of the parabola defined by the quadratic function [tex]\( f(x) = 3x^2 - 6x + 2 \)[/tex], you can follow these steps:
1. Identify the coefficients of the quadratic function: [tex]\( ax^2 + bx + c \)[/tex]
In this case, the coefficients are:
- [tex]\( a = 3 \)[/tex]
- [tex]\( b = -6 \)[/tex]
- [tex]\( c = 2 \)[/tex]
2. Find the x-coordinate of the vertex:
The x-coordinate of the vertex for a quadratic function can be determined using the formula:
[tex]\[
x_{\text{vertex}} = -\frac{b}{2a}
\][/tex]
Plugging in the values of [tex]\( a \)[/tex] and [tex]\( b \)[/tex]:
[tex]\[
x_{\text{vertex}} = -\frac{-6}{2 \cdot 3} = \frac{6}{6} = 1.0
\][/tex]
3. Calculate the y-coordinate of the vertex:
To find the y-coordinate, substitute [tex]\( x_{\text{vertex}} \)[/tex] back into the original quadratic function:
[tex]\[
y_{\text{vertex}} = f(x_{\text{vertex}}) = 3(1.0)^2 - 6(1.0) + 2
\][/tex]
Simplifying this expression:
[tex]\[
y_{\text{vertex}} = 3 \cdot 1 - 6 \cdot 1 + 2 = 3 - 6 + 2 = -1.0
\][/tex]
4. Combine the coordinates to form the vertex:
The vertex is an ordered pair [tex]\((x_{\text{vertex}}, y_{\text{vertex}})\)[/tex], which in this case is:
[tex]\[
(1.0, -1.0)
\][/tex]
Thus, the coordinates of the vertex for the parabola defined by [tex]\( f(x) = 3x^2 - 6x + 2 \)[/tex] are:
[tex]\[
(1.0, -1.0)
\][/tex]
