Final answer:
To rewrite f(x) = -4x² + 6x + 2 in vertex form, we complete the square for the x-terms after factoring out the coefficient of x². After adding and subtracting the perfect square, we distribute and simplify to obtain the vertex form f(x) = -4(x - 0.75)² + 4.25.
Step-by-step explanation:
To rewrite the quadratic function f(x) = -4x² + 6x + 2 in standard (vertex) form, we use the method of completing the square. The standard form of a quadratic function is f(x) = a(x - h)² + k, where (h,k) is the vertex of the parabola.
First, we factor out the coefficient of x² from the first two terms:
f(x) = -4(x² - 1.5x) + 2
Next, we find the value that would complete the square inside the parentheses. This value is (-b/2a)², where a is the coefficient of x² and b is the coefficient of x, which gives us (1.5/2)² = 0.5625. We add and subtract 0.5625 inside the parentheses to complete the square:
f(x) = -4(x² - 1.5x + 0.5625 - 0.5625) + 2
f(x) = -4((x - 0.75)² - 0.5625) + 2
Now we distribute the -4:
f(x) = -4(x - 0.75)² + 4 * 0.5625 + 2
f(x) = -4(x - 0.75)² + 2.25 + 2
Finally, we simplify the constants:
f(x) = -4(x - 0.75)² + 4.25
The quadratic function is now in standard form with the vertex at (0.75, 4.25).