Final answer:
The graph of the equation x^2 - 2x - 3 = 0 represents a parabola that opens upward. The x-intercepts are -1 and 3. The vertex of the parabola is at the coordinates (1,-4).
Step-by-step explanation:
The graph of the equation x^2 - 2x - 3 = 0 represents a parabola. To find the graph, we need to solve the equation and determine the x-intercepts. First, we can factor the equation as (x+1)(x-3) = 0. This gives us two possible solutions: x = -1 or x = 3. These are the x-intercepts of the graph.
The graph of the equation will be a parabola that opens upward, since the coefficient of the x^2 term is positive. The x-intercepts (-1,0) and (3,0) will be points on the graph.
To determine the shape and position of the parabola, we can also find the vertex. The x-coordinate of the vertex is given by the formula x = -b/(2a), where a and b are the coefficients of the x^2 and x terms, respectively. In this case, a = 1 and b = -2, so we can calculate x = -(-2)/(2*1) = 1. The y-coordinate of the vertex can be found by substituting x = 1 into the equation: y = (1)^2 - 2(1) - 3 = -4. Therefore, the vertex of the parabola is (1,-4).