Final answer:
The range of the quadratic function y = -x^2 - 2x + 3 is Y < 4, as the parabola opens downwards with the vertex being the highest point on the graph (0,0) which dictates the maximum y-value.
Step-by-step explanation:
The subject question asks for the range of the quadratic function y = -x^2 - 2x + 3. To find the range, we should determine the vertex of the parabola represented by this function. In a quadratic function of the form y = ax^2 + bx + c, the x-coordinate of the vertex is given by the formula -b/(2a). In our case, a = -1 and b = -2, so the x-coordinate of the vertex is x = -(-2)/(2*(-1)) = 1. Substituting x = 1 back into the function, we find the y-coordinate of the vertex to be y = -(1)^2 - 2*(1) + 3 = -1 - 2 + 3 = 0. Since the coefficient of x^2 is negative, the parabola opens downwards, making the vertex the maximum point.
Therefore, the range of the function is Y \u003C 4, because the maximum y-value is at the vertex which is 0, and all other y-values are less than this. This indicates that the range of y-values for which the function exists is all real numbers less than or equal to the y-coordinate of the vertex. Hence, the correct answer is A. Y < 4.