Final answer:
The quadratic function f(x) = 4x² - 5x - 3 can be rewritten in standard form by completing the square, resulting in 4(x - 5/8)² - 73/16 with the vertex at (5/8, -73/16).
Step-by-step explanation:
To rewrite the quadratic function f(x) = 4x² - 5x - 3 in standard (vertex) form, we need to complete the square. The standard form of a quadratic function is y = a(x-h)² + k, where (h, k) is the vertex of the parabola.
First, factor out the coefficient of x² from the x-terms:
f(x) = 4(x² - (5/4)x) - 3
Next, find the value to complete the square, which is ((b/2a)²). Here, b is the coefficient of x, and a is the coefficient of x²:
((-5/4) / (2 * 1))² = (5/8)² = 25/64
Add and subtract this value inside the parentheses:
f(x) = 4((x² - (5/4)x + (25/64)) - (25/64)) - 3
Rewrite the inside of the parentheses as a perfect square:
f(x) = 4((x - (5/8))² - 25/64) - 3
Distribute the 4 and simplify:
f(x) = 4(x - (5/8))² - 4(25/64) - 3
f(x) = 4(x - (5/8))² - 25/16 - 3
Simplify the constant term:
f(x) = 4(x - (5/8))² - 73/16
Now the function is in standard form, and the vertex is (h, k) = (5/8, -73/16).