Final answer:
To find the vertex of the function f(x) = −x^2 + 2x − 6, use the formula x = -b/2a. The x-coordinate of the vertex is -2/(-2) = 1, and substituting this value into the function yields the y-coordinate of the vertex as -5. The x-intercepts can be found by solving the quadratic equation -x^2 + 2x - 6 = 0. By factoring, the x-intercepts are 3 and -2. The y-intercept is obtained by substituting x = 0 into the function, resulting in a y-coordinate of -6.
Step-by-step explanation:
To find the vertex and intercepts for the function f(x) = −x^2 + 2x − 6, we can use various methods.
To find the vertex, we can use the formula x = -b/2a. In this case, a = -1 and b = 2, so the x-coordinate of the vertex is -2/(-2) = 1. Substituting this value into the function, we find that the y-coordinate of the vertex is f(1) = -1 + 2 - 6 = -5. Therefore, the vertex is (1, -5).
To find the x-intercepts, we set f(x) = 0 and solve for x. In this case, we solve the quadratic equation -x^2 + 2x - 6 = 0 using factoring or the quadratic formula. By factoring, we get (x - 3)(x + 2) = 0, so the x-intercepts are x = 3 and x = -2.
The y-intercept is found by substituting x = 0 into the function, giving us f(0) = -0^2 + 2(0) - 6 = -6. Therefore, the y-intercept is (0, -6).