To find the vertex of the quadratic function $f(x) = 2x^2 - 20x + 53$, we can use the vertex formula:

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

For the given quadratic function $f(x) = 2x^2 - 20x + 53$, the coefficients are:

• $a = 2$
• $b = -20$
• $c = 53$ (though $c$ is not used in the vertex formula).

Step 1: Find the x-coordinate of the vertex

$x = \frac{-(-20)}{2(2)} = \frac{20}{4} = 5$

Step 2: Find the y-coordinate of the vertex

Now that we know $x = 5$, substitute this back into the quadratic equation to find $y$:

$f(5) = 2(5)^2 - 20(5) + 53 = 2(25) - 100 + 53 = 50 - 100 + 53 = 3$

Thus, the vertex of the quadratic function is $(5, 3)$.

Final Answer:

The correct answer is A. (5, 3).

Would you like any further details or have any questions? Here are some related questions:

1. How can we derive the vertex formula from completing the square?
2. What does the vertex of a quadratic function represent geometrically?
3. How does the value of $a$ in the quadratic equation affect the graph?
4. Can you find the axis of symmetry of the quadratic function based on the vertex?
5. How would the vertex change if the quadratic equation had different coefficients?

Tip: The vertex of a parabola always lies on its axis of symmetry, which can be found using the formula $x = \frac{-b}{2a}$.