Final answer:
For the quadratic function f(x) = -x^2 - 2x + 3, the only correct statement regarding its graph is A, the y-intercept is (0, 3). The vertex is not (-1, -4) or (-1, 4), and the axis of symmetry is not x = -1 because the actual vertex of the parabola is (1, 0) and the axis of symmetry is therefore x = 1.
Step-by-step explanation:
The function given is f(x) = -x^2 - 2x + 3 which is a quadratic function. In general, the graph of a quadratic function is a parabola. Here are the key features of the graph for this quadratic equation:
- The y-intercept of a graph is the point where the graph crosses the y-axis. To find the y-intercept of the function f(x), we set x to 0 and calculate f(0). For f(x) = -x^2 - 2x + 3, f(0) = -(0)^2 - 2(0) + 3 = 3. So, the y-intercept is indeed (0, 3), making statement A correct.
- The vertex of the graph of a quadratic function is the highest or lowest point on the graph, depending on whether the parabola opens upwards or downwards. Since the coefficient of the x^2 term is negative, this parabola opens downwards, so the vertex will be the highest point on the graph. The vertex formula for a quadratic function f(x) = ax^2 + bx + c is given by (-b/2a, f(-b/2a)). Here, a = -1 and b = -2, so the x-coordinate of the vertex is -(-2)/(2*-1) = -2/-2 = 1. Inserting it back into the function, we get f(1) = -(1)^2 - 2(1) + 3 = -1 - 2 + 3 = 0. Therefore, the vertex of this parabola is actually (1, 0), not (-1, -4) nor (-1, 4), rendering both statements B and D incorrect.
- The axis of symmetry is a vertical line that passes through the vertex of the parabola and divides it into two symmetrical halves. Since the x-coordinate of the vertex is 1, the equation of the axis of symmetry is x = 1, not x = -1, hence statement C is also incorrect.
Therefore, the only correct statement is A, the y-intercept is (0, 3).