In Exercises 17–38, 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 quadratic function . The vertex form of a quadratic function is , where is the vertex. Here, the vertex is .
2. Determine the axis of symmetry. The axis of symmetry for a parabola in vertex form is the vertical line . Therefore, the axis of symmetry is .
3. Find the x-intercepts by setting and solving for . This gives . Solve to find the x-intercepts.
4. Find the y-intercept by evaluating . Substitute into the function: .
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 upwards, the range is .

Key Concepts

Vertex of a Parabola

The vertex of a parabola is the point where the curve changes direction, representing either the maximum or minimum value of the quadratic function. For the function f(x) = (x - 4)² - 1, the vertex can be found directly from the vertex form of a quadratic equation, which is (x - h)² + k, where (h, k) is the vertex. In this case, the vertex is at (4, -1).

Axis of Symmetry

The axis of symmetry of a parabola is a vertical line that divides the parabola into two mirror-image halves. For a quadratic function in vertex form, the axis of symmetry can be determined from the x-coordinate of the vertex. For the function f(x) = (x - 4)² - 1, the axis of symmetry is the line x = 4.

Domain and Range of Quadratic Functions

The domain of a quadratic function is the set of all possible input values (x-values), which is typically all real numbers for parabolas. The range, however, depends on the vertex; if the parabola opens upwards, the range starts from the y-coordinate of the vertex to positive infinity. For f(x) = (x - 4)² - 1, the domain is all real numbers, and the range is [-1, ∞) since the vertex is the minimum point.