A) Find the vertex. b) Determine whether there is a maximum...

a) Find the vertex.
b) Determine whether there is a maximum or minimum value, and find that value.
c) Find the range.
d) Find the intervals on which the function is increasing and the intervals on which the function is decreasing.
$$f(x)=\frac{1}{2} x^{2}-3 x+\frac{5}{2}$$

Key Concepts

Quadratic Functions

Quadratic functions are polynomial functions of degree 2, generally written as f(x) = ax² + bx + c. They produce parabolic graphs which have a symmetric shape and a unique vertex. The parameters a, b, and c determine the opening, width, and position of the parabola.

Vertex of a Parabola

The vertex of a parabola is a critical point where the function achieves its maximum or minimum value. It is found by rewriting the quadratic equation into its vertex form via completing the square or by using the formula (-b/(2a), f(-b/(2a))). This point also acts as the axis of symmetry for the parabola.

Maximum or Minimum Value

The maximum or minimum value of a quadratic function is determined by the vertex. When the leading coefficient a is positive, the parabola opens upward so the vertex is the lowest point (minimum), and when a is negative, it opens downward making the vertex the highest point (maximum).

Range of a Quadratic Function

The range of a quadratic function depends on the direction of the parabola's opening and the vertex. For an upward opening parabola (a > 0), the range includes all y-values at or above the y-coordinate of the vertex, while for a downward opening parabola (a < 0), it includes all y-values at or below the vertex.

Intervals of Increase and Decrease

A quadratic function typically decreases until it reaches its vertex and then increases (or vice versa for downward opening parabolas). These intervals of increase and decrease are determined by the position of the vertex and can be understood either graphically or by analyzing the derivative of the function.

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