Final answer:
The range of the function Y = -x^2 - 2x + 3 is Y ≤ 0.
Step-by-step explanation:
The range of the function Y = -x2 - 2x + 3 can be determined by finding the minimum or maximum value of the function. Since the coefficient in front of the x2 term is negative, the function opens downwards and will have a maximum value.
To find the maximum value, we can use the vertex formula: x = -b/2a. For this function, a = -1 and b = -2. Plugging these values into the formula gives us x = -(-2)/(2*(-1)) = 1. The x-coordinate of the vertex is 1.
Substituting x = 1 back into the original function, we get Y = -12 - 2(1) + 3 = -1 - 2 + 3 = 0. Therefore, the range of the function is Y ≤ 0, meaning all values of Y less than or equal to 0.