Example  1
Determine the direction of opening, the maximum or minimum value, and the domain and range of the following quadratic functions.



  1. According to the graph and equation, \(a = 2\), therefore the graph of this function opens upward (\(a > 0 \)) and has a minimum value of \(-3\).

    There are no restrictions on \(x\), therefore the domain is {\(x | x \in \thinspace \rm{R}\)}.

    From the graph, \(y\) must be greater than or equal to \(-3\), therefore the range is {\(y | y \ge -3,\thinspace y \in \thinspace \rm{R}\)}.


  2. According to the graph and equation, \(a = -4\), therefore the graph of this function opens downward \((a < 0)\) and has a maximum value of \(5\).

    Again, there are no restrictions on \(x\), therefore the domain is {\(x | x \in \thinspace \rm{R}\)}.

    From the graph, \(y\) must be less than or equal to 5, therefore the range is {\(y | y \le 5,\thinspace y \in \thinspace \rm{R}\)}.