To find the vertex of the quadratic function \(y = 2x^2 - 4.5x + 2\), we can use the vertex formula for a quadratic equation in the form \(y = ax^2 + bx + c\). The x-coordinate of the vertex is given by:
\[ x = -\frac{b}{2a} \]
Here, \(a = 2\) and \(b = -4.5\).
Calculating the x-coordinate of the vertex:
\[ x = -\frac{-4.5}{2 \cdot 2} = \frac{4.5}{4} = 1.125 \]
Now, substituting \(x = 1.125\) back into the original equation to find the y-coordinate:
\[ y = 2(1.125)^2 - 4.5(1.125) + 2 \]
Calculating \(2(1.125)^2\):
\[ 2(1.125)^2 = 2 \cdot 1.265625 = 2.53125 \]
Calculating \(-4.5(1.125)\):
\[ -4.5(1.125) = -5.0625 \]
Now substituting these values into the equation:
\[ y = 2.53125 - 5.0625 + 2 = -0.53125 \]
Thus, the coordinates of the vertex are:
\[ (1.125, -0.53125) \]
Rounding this to three decimal places gives us the vertex as:
\[ (1.125, -0.531) \]
Therefore, the correct response is:
\[ \text{(1.125,−0.531)} \]