Graph the given functions, f and g, in the same rectangular coordinate system. Describe how the graph of g is related to the graph of f. f(x) = -x^2, g(x) = -x^2 + 2

Answer

To graph the functions f(x) = -x^2 and g(x) = -x^2 + 2 in the same rectangular coordinate system, we can analyze their transformations. The base function is f(x) = -x^2, which is a parabola opening downwards with its vertex at the origin (0, 0). The function g(x) = -x^2 + 2 is obtained by adding 2 to the output of f(x). In general, adding a constant c to a function, where g(x) = f(x) + c, shifts the graph vertically. Since c = 2 > 0, the graph of g is shifted vertically upward by 2 units compared to the graph of f. Therefore, the vertex of g(x) is at (0, 2), and every point on the graph of f is moved 2 units straight up to form the graph of g.