The given quadratic function is $f(x) = 2x^2 - 8x + 3$. To find the vertex of a parabola represented by the quadratic function in standard form $f(x) = ax^2 + bx + c$, we use the vertex formula:

$x = -\frac{b}{2a}$

In this case:

• $a = 2$
• $b = -8$
• $c = 3$

Substitute the values of $a$ and $b$ into the formula for the x-coordinate of the vertex:

$x = -\frac{-8}{2(2)} = \frac{8}{4} = 2$

Now that we have the x-coordinate of the vertex, we substitute $x = 2$ back into the original equation to find the y-coordinate:

$f(2) = 2(2)^2 - 8(2) + 3 = 2(4) - 16 + 3 = 8 - 16 + 3 = -5$

Therefore, the coordinates of the vertex are $(2, -5)$.

Would you like more details or have any questions about this? Here are some related questions:

1. What is the significance of the vertex in the graph of a quadratic function?
2. How does the value of $a$ affect the shape and direction of the parabola?
3. Can you explain how to convert a quadratic function to vertex form?
4. What is the axis of symmetry for a parabola, and how is it related to the vertex?
5. How would the graph of $f(x) = 2x^2 - 8x + 3$ look like?

Tip: The vertex form of a quadratic function can make it easier to graph and understand the transformations of the parabola.