Final answer:
To find the roots and vertex of the quadratic equation y = -x^2 - 4x + 5, you can use the quadratic formula and the vertex formula. The equation represents a downward-opening parabola with roots found by the quadratic formula and the vertex at the point (2,-3).
Step-by-step explanation:
The equation you are asking about is a quadratic equation in the form y = ax^2 + bx + c, where a, b, and c are coefficients, and in this case, they are a=-1, b=-4, and c=5. To find its roots (the values of x where y=0) and the vertex (the highest or lowest point on the graph of the equation, depending on whether the parabola opens upwards or downwards), you can either factor the equation (if possible), complete the square, or use the quadratic formula.The roots of the quadratic equation are found by setting y to zero and solving the resulting equation: 0 = -x^2 - 4x + 5. You can use the quadratic formula x = (-b ± √(b^2 - 4ac)) / (2a) to find the roots.
The vertex of the parabola given by a quadratic equation can be found using the formula (-b / (2a), f(-b / (2a))), where f(x) is the original quadratic function. For your equation, the x-coordinate of the vertex will be x = -(-4) / (2*(-1)) = 2, and the y-coordinate will be y = -2^2 - 4*2 + 5 = -3. Thus, the vertex is at the point (2, -3). The graph of this equation is a parabola that opens downwards because the coefficient of x^2 is negative.