Final answer:
The vertex of the quadratic function K(x)=10-4x-x^2 is (-2, 14), the range is y≤14 for x≥1, the x-intercepts can be found by solving the equation set to zero, and the y-intercept is (0, 10).
Step-by-step explanation:
The function described is K(x) = 10 - 4x - x^2, which is a quadratic function, where x is restricted to the real numbers and x ≥1. To find the vertex, range, and intercepts, we need to analyze this quadratic function.
Vertex
The vertex form of a quadratic function is y = a(x - h)^2 + k, where (h, k) is the vertex of the parabola. For K(x), we complete the square to find its vertex form:
K(x) = -x^2 - 4x + 10
= -(x^2 + 4x - 4) + 10 + 4
= -(x + 2)^2 + 14
The vertex is at (-2, 14).
Range
Since the coefficient of the x^2 term is negative, the parabola opens downwards, making the vertex the maximum point. Therefore, the range is y ≤ 14, for x ≥ 1.
X-Intercepts
We find the x-intercepts by setting K(x) to zero and solving for x:
0 = 10 - 4x - x^2
Applying the quadratic formula, we get two x-intercepts, which are the solutions to the equation.
Y-Intercept
The y-intercept occurs when x = 0. Substituting 0 into K(x), we get:
K(0) = 10 - 4(0) - (0)^2 = 10
Thus, the y-intercept is (0, 10).