Question

Use the vertex and intercepts to sketch the graph of each quadratic function. Give the equation of the parabola's axis of symmetry. Use the graph to determine the function's domain and range. f(x)=(x−4)2−1

Accepted Answer

Identify the vertex of the parabola from the function f(x) = (x - $$4)^2 - 1. $$The vertex form of a quadratic function is f(x) = a(x - $$h)^2 + k$$, where (h, k) is the vertex. Here, h = 4 and k = -1, so the vertex is (4, -1). Write the equation of the axis of symmetry. The axis of symmetry is a vertical line that passes through the vertex, so its equation is x = h. For this function, the axis of symmetry is x = 4. Find the y-intercept by evaluating f(0). Substitute x = 0 into the function: f(0) = (0 - $$4)^2 - 1 = 16 - 1 = 15. So$$, the y-intercept is (0, 15). Find the x-intercepts by setting f(x) = 0 and solving for x. Set (x - $$4)^2 - 1 = 0$$, then solve the equation: (x - $$4)^2 = 1$$, which gives two solutions for x. Determine the domain and range of the function. The domain of any quadratic function is all real numbers, so domain: (-∞, ∞). The range depends on the vertex and the direction of the parabola. Since the parabola opens upward and the vertex is the minimum point at y = -1, the range is [ -1, ∞ ).

Use the vertex and intercepts to sketch the graph of each quadratic function. Give the equation of the parabola's axis of symmetry. Use the graph to determine the function's domain and range. f(x)=(x−4)²−1

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Key Concepts

Vertex of a Quadratic Function

Axis of Symmetry

Domain and Range of Quadratic Functions