In Exercises 13–16, use the vertex and intercepts to sketch...

In Exercises 13–16, use the vertex and intercepts to sketch the graph of each quadratic function. Give the equation for the parabola’s axis of symmetry. Use the graph to determine the function’s domain and range.
$$
f(x)=(x+4)^{2}-2
$$

Key Concepts

Quadratic Function

A quadratic function is a second-degree polynomial function that can be expressed in various forms, such as the standard form (ax² + bx + c) or the vertex form (a(x - h)² + k). Its graph is a parabola, and its properties like the vertex, axis of symmetry, and intercepts help in understanding its behavior and sketching its graph.

Vertex Form

The vertex form of a quadratic function, written as a(x - h)² + k, explicitly shows the vertex of the parabola at the point (h, k). This form simplifies the process of graphing by immediately revealing the location of the vertex and how the function has been transformed from the basic quadratic y = x².

Axis of Symmetry

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

Domain and Range

The domain and range describe the set of all possible input and output values of a function, respectively. For quadratic functions, the domain is typically all real numbers. The range, however, depends on the direction in which the parabola opens: if it opens upward, the range is all real numbers greater than or equal to the y-coordinate of the vertex, and if it opens downward, it is all real numbers less than or equal to the vertex's y-coordinate.

Intercepts

Intercepts are the points where the graph of a function intersects the coordinate axes. The x-intercepts are found by setting the function equal to zero (f(x) = 0) and solving for x, while the y-intercept is determined by evaluating the function at x = 0. Intercepts serve as key reference points that aid in accurately sketching the graph of the quadratic function.

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