Y=-x²+1

The equation you provided is a quadratic equation in standard form, where y is expressed in terms of x. To prepare for your university exams, you can follow these steps to analyze and solve the equation:

1. Identify the shape: The equation y = -x² + 1 represents a downward opening parabola. The negative coefficient of x² indicates that the parabola is concave downward.

2. Find the vertex: The vertex of a parabola can be determined using the formula x = -b/2a. In this case, a = -1 and b = 0, so the x-coordinate of the vertex is x = -0/2(-1) = 0. To find the y-coordinate, substitute the value of x into the equation: y = -(0)² + 1 = 1. Hence, the vertex is at (0, 1).

3. Determine the axis of symmetry: The axis of symmetry is a vertical line passing through the vertex. In this case, the equation is y = -x² + 1, so the axis of symmetry is x = 0.

4. Calculate the y-intercept: The y-intercept is the point where the parabola intersects the y-axis. Substitute x = 0 into the equation: y = -(0)² + 1 = 1. Therefore, the y-intercept is at (0, 1).

5. Find the x-intercepts (if any): The x-intercepts represent the points where the parabola intersects the x-axis. To find the x-intercepts, set y = 0 and solve for x: 0 = -x² + 1. Rearranging the equation, we have x² = 1. Taking the square root of both sides, we get x = ± √1. Hence, there are two x-intercepts at (-1, 0) and (1, 0).

6. Sketch the graph: Based on the above information, you can now plot the vertex, the axis of symmetry, the y-intercept, and the x-intercepts on a graph. Connect these points to form a smooth curve, resulting in the graph of y = -x² + 1.

By understanding the properties and graph of the quadratic equation, you can analyze and solve other related problems on your university exams. Remember to practice solving similar equations and familiarize yourself with the concepts for a better understanding. Good luck with your preparation!