To put the quadratic function y = 4x² - 3x - 1 into vertex form by completing the square, we can follow these steps:
Factor out the coefficient of x² from the first two terms:
y = 4(x² - (3/4)x) - 1
Complete the square by adding and subtracting the square of half of the coefficient of x, which is (3/8)² = 9/64:
y = 4(x² - (3/4)x + 9/64 - 9/64) - 1
Group the first three terms and factor the perfect square trinomial:
y = 4[(x - (3/8))² - 9/64] - 1
Simplify by distributing the 4 and combining like terms:
y = 4(x - (3/8))² - (1 + 9/16)
Rewrite the constant term as a fraction with a common denominator:
y = 4(x - (3/8))² - (16/16 + 9/16)
Combine the constant terms and simplify:
y = 4(x - (3/8))² - 25/16
Therefore, the vertex form of the quadratic function y = 4x² - 3x - 1 is y = 4(x - (3/8))² - 25/16, and the vertex is at the point (3/8, -25/16).