The graph of a quadratic function f is given. a. Find the...

The graph of a quadratic function $f$ is given.
a. Find the coordinates of the vertex and the $x$ - and $y$ -intercepts.
b. Find the maximum or minimum value of $f$
c. Find the domain and range of $f$
$$f(x)=-\frac{1}{2} x^{2}-2 x+6$$

Key Concepts

Domain and Range

The domain of a quadratic function typically includes all real numbers, as the input values can be any real number. The range, however, is determined by the vertex and whether the parabola opens upward or downward, defining the set of possible output values. Knowing the domain and range is crucial for fully describing the behavior and limitations of the function.

Intercepts

Intercepts are the points where the graph of a function crosses the coordinate axes. The x-intercepts are the points where the function equals zero and reveal the roots of the quadratic equation, while the y-intercept is found by evaluating the function at x = 0. These intercepts are important for understanding the behavior and symmetry of the quadratic graph.

Extremum (Maximum or Minimum Value)

The extremum of a quadratic function is either its maximum or minimum value, corresponding to the vertex of the parabola. In a quadratic function, the sign of the leading coefficient determines whether the parabola opens upward (resulting in a minimum value) or downward (resulting in a maximum value). This concept is central in both solving optimization problems and understanding the overall shape of the graph.

Quadratic Functions

Quadratic functions are polynomial functions of degree two, typically expressed in the form ax² + bx + c. They are fundamental in algebra and calculus, describing parabolic curves when graphed. Understanding their structure, including the coefficient signs and the terms, is essential because it influences the graph's orientation, width, and position relative to the coordinate axes.

Vertex of a Parabola

The vertex of a parabola is the point where the function reaches its maximum or minimum value, depending on the direction of the opening. It is a key feature in graphing quadratic functions and can be found using formulas derived from the standard form or by completing the square. The vertex provides insight into the optimal value of the function in optimization problems.

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