Graph the given quadratic function. Identify the vertex, axis of symmetry, intercepts, domain, range, and intervals for which the function is increasing or decreasing. $f\left(x\right)=-\left(x-5\right)^2+1$

• A.
• B.
• C.
• D.

Answer: (A)

Verified step by step guidance

1. Identify the standard form of the quadratic function, which is f(x) = a(x-h)^2 + k. In this case, f(x) = -(x-5)^2 + 1, where a = -1, h = 5, and k = 1.
2. Determine the vertex of the parabola. The vertex is given by the point (h, k). For this function, the vertex is (5, 1).
3. Identify the axis of symmetry. The axis of symmetry is the vertical line that passes through the vertex, given by x = h. Here, the axis of symmetry is x = 5.
4. Find the y-intercept by setting x = 0 in the function and solving for f(x). Substitute x = 0 into f(x) = -(x-5)^2 + 1 to find the y-intercept.
5. Determine the domain and range. The domain of any quadratic function is all real numbers. The range is determined by the vertex and the direction of the parabola. Since the parabola opens downwards (a < 0), the range is (-∞, 1].