To convert the quadratic function y = 4x² - 3x - 1 into vertex form by completing the square, we can follow these steps:
Step 1: Group the x terms together and factor out the coefficient of the x² term:
y = 4x² - 3x - 1
= 4(x² - (3/4)x) - 1
Step 2: Complete the square by adding and subtracting the square of half the coefficient of the x term inside the parentheses. The square of half of -3/4 is (-3/8)² = 9/64, so we add and subtract 9/64 inside the parentheses:
y = 4(x² - (3/4)x + 9/64 - 9/64) - 1
Step 3: Simplify the expression inside the parentheses by factoring the square trinomial and combining like terms:
y = 4((x - 3/8)² - 9/64) - 1
= 4(x - 3/8)² - 9/16 - 1
= 4(x - 3/8)² - 25/16
Therefore, the quadratic function y = 4x² - 3x - 1 in vertex form is:
y = 4(x - 3/8)² - 25/16
The vertex of the parabola is at the point (3/8, -25/16).