Use the vertex and intercepts to sketch the graph of each...

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.
$$
f(x)=2 x-x^{2}-2
$$

Key Concepts

Quadratic Functions

A quadratic function is a second-degree polynomial function of the form f(x) = ax² + bx + c. Its graph is a parabola, and understanding its structure is key to analyzing its properties such as intercepts, vertex, and symmetry, making it a fundamental concept in algebra and calculus.

Parabola

A parabola is the graphical representation of a quadratic function. It is characterized by a symmetric, curved shape and includes features such as a vertex (the maximum or minimum point) and intercepts. Analyzing the parabola’s shape helps in understanding the behavior of the quadratic function.

Vertex

The vertex of a parabola is the point where the function reaches its maximum or minimum value. It is a critical feature that not only indicates the extreme value of the quadratic function but also serves as the point through which the axis of symmetry passes.

Axis of Symmetry

The axis of symmetry is a vertical line that divides the parabola into two mirror-image halves. For a quadratic function in standard form, this line can be found using the formula x = -b/(2a). It plays a crucial role in graphing and analyzing the symmetry and balance of the quadratic function.

Intercepts

Intercepts are the points where the graph of the quadratic function crosses the coordinate axes. The x-intercepts (or roots) provide information about the solutions to the equation f(x) = 0, while the y-intercept, found by evaluating f(0), helps in plotting the graph accurately.

Domain and Range

The domain of a quadratic function is the set of all real numbers, as any real x-value can be substituted into the function. The range depends on the vertex and whether the parabola opens upward or downward; it is all values greater than or equal to the y-coordinate of the vertex when opening upward, or all values less than or equal to that y-coordinate when opening downward.

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