Graph the parabola and the axis of symmetry. Label the...

Graph the parabola and the axis of symmetry. Label the coordinates of the vertex, and write the equation of the axis of symmetry. Use the graph to write the domain and range in interval notation. (See Examples $7-8.1$)
(GRAPH CANNOT COPY)
$$f(x)=-2(x+3)^{2}-1$$

Key Concepts

Vertex Form of a Quadratic Function

The vertex form of a quadratic function is written as f(x) = a(x - h)² + k, where (h, k) is the vertex of the parabola. This form allows one to quickly identify the vertex, determine the direction of opening based on the sign of coefficient a, and understand how the function has been translated from the standard form.

Axis of Symmetry

The axis of symmetry for a parabola in vertex form is a vertical line that passes through the vertex, defined by the equation x = h. This line divides the parabola into two mirror images, making it a fundamental feature for analyzing the symmetry of quadratic functions.

Graphing Parabolas

Graphing a parabola involves plotting its vertex, identifying the direction of its opening (upward or downward), and marking critical points such as the y-intercept and additional points that showcase the symmetric properties with respect to the axis of symmetry. Understanding the vertex and axis of symmetry greatly simplifies the process of sketching an accurate graph.

Domain and Range of a Quadratic Function

The domain of a quadratic function typically includes all real numbers, while the range is determined by the vertex and the direction in which the parabola opens. If the parabola opens upwards, the range is given by [k, ?), and if it opens downwards, the range is (-?, k], where (h, k) is the vertex. These interval notations concisely describe the sets of possible output values.

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