Final answer:
To graph the quadratic equation y = -x² + 2x + 1, find the solutions to the equation -x² + 2x + 1 = 0, draw the line of symmetry, and find the intersection of the graph with the line y = -2.
Step-by-step explanation:
To graph the equation y = -x² + 2x + 1, we can use the vertex form of a quadratic function, which is y = a(x-h)² + k. Comparing it to the given equation, we can see that a = -1, h = 1, and k = 1. The vertex of the parabola is at (1, 1). From there, we can plot a few additional points and draw a smooth curve to represent the graph on the grid.
To find the solutions to the quadratic equation -x² + 2x + 1 = 0, we can set the equation equal to zero and solve for x. We can use factoring, completing the square, or the quadratic formula to find the solutions. In this case, the solutions are x = -1 and x = 3.
The line of symmetry of the graph is a vertical line that passes through the vertex of the parabola. In this case, the line of symmetry is x = 1.
To draw the line y = -2 on the grid, we can plot points that lie on the line and connect them to form a straight line. The coordinates of the intersection of the line and the graph of the quadratic function can be found by solving the simultaneous equations y = -2 and y = -x² + 2x + 1. The coordinates of the intersection are (0, -2) and (2, -2).