Final answer:
To find the vertex form of the function f(x)=-1/2x²-2x+6, complete the square to transform the quadratic function into its vertex form, resulting in f(x) = -1/2(x + 2)² + 8.
Step-by-step explanation:
To find the vertex form of the given quadratic function f(x)=-1/2x²-2x+6, we need to complete the square.
Step 1: Factor out the coefficient of the x² term from the first two terms.
f(x) = -1/2(x² + 4x) + 6
Step 2: Find the number that completes the square for the expression in parentheses.
The number needed to complete the square is found using (b/2)², where b is the coefficient of x. In this case, b is 4, so (4/2)² = 4.
However, since we factored -1/2 out, we need to divide 4 by -1/2 to keep the equation balanced, which gives us -8.
Step 3: Add and subtract this number inside the parentheses and simplify.
f(x) = -1/2(x² + 4x + 4 - 4) + 6
= -1/2((x + 2)² - 4) + 6
Step 4: Distribute the -1/2 and simplify the entire equation.
f(x) = -1/2(x + 2)² + 2 + 6
= -1/2(x + 2)² + 8
The vertex form of the quadratic function is f(x) = -1/2(x + 2)² + 8.