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

Verified step by step guidance

1. 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).
2. 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.
3. 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).
4. 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.
5. 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, ∞ ).

Key Concepts

Vertex of a Quadratic Function

The vertex of a quadratic function is the point where the parabola changes direction, representing either its maximum or minimum value. It is given by the coordinates (h, k) in the vertex form f(x) = a(x - h)^2 + k. Understanding the vertex helps in graphing the parabola and determining its key features.

Axis of Symmetry

The axis of symmetry is a vertical line that passes through the vertex of the parabola, dividing it into two mirror-image halves. Its equation is x = h, where h is the x-coordinate of the vertex. This line is crucial for graphing and analyzing the symmetry properties of quadratic functions.

Domain and Range of Quadratic Functions

The domain of any quadratic function is all real numbers since the parabola extends infinitely left and right. The range depends on the vertex; for a parabola opening upwards, the range is all y-values greater than or equal to the vertex's y-coordinate. Understanding domain and range helps describe the function's output values.