Final answer:
To convert f(x) = 4x² - 24x + 11 into vertex form a(x-b)² + c, we complete the square and rewrite it as 4(x - 3)² - 25. The coordinates of the vertex are (3, -25).
Step-by-step explanation:
The function f(x) = 4x² - 24x + 11 is to be expressed in the form a(x-b)² + c. This expression represents a quadratic equation, typically encountered in high school mathematics. To rewrite f(x) in the desired form, we need to complete the square. First, factor out the leading coefficient from the x terms:
f(x) = 4(x² - 6x) + 11
Next, we will complete the square by finding the value that makes (x² - 6x + __) a perfect square trinomial. To do this, we take half of the coefficient of x, square it, and add it inside the bracket but also subtract it outside to maintain equation balance:
f(x) = 4(x² - 6x + (6/2)²) + 11 - 4(6/2)²
f(x) = 4(x - 3)² + 11 - 36
f(x) = 4(x - 3)² - 25
Therefore, f(x) is now expressed in the form a(x-b)² + c where a = 4, b = 3, and c = -25. The coordinates of the vertex of the graph of y = f(x) are (3, -25), since the vertex form provides the vertex as (b, c).