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Completion requirements
For the following functions, determine the direction of opening,
maximum or minimum value, domain and range, and the number of \(x\)-intercepts for the related graph. Verify by graphing.
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\(f(x) = (x + 3)^2 - 2\)
\(a = 1\); \(q = -2\)
The graph opens upward because \(a\) is positive. This means the graph of the quadratic function has a minimum value of \(-2\).
The domain is {\(x | x \in \thinspace \rm{R}\)}.
The range is {\(y | y \ge -2, \thinspace y \in \thinspace \rm{R}\)}.
Because \(a\) and \(q\) are opposite signs, there are two \(x\)-intercepts.
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\(g(x) = -0.5x^2 -3\)
\(a = -0.5\); \(q = -3\)
The graph opens downward because \(a\) is negative. This means the graph of the quadratic function has a maximum of \(-3\).
The domain is {\(x | x \in \rm{R}\)}.
The range is {\(y | y \le -3, \thinspace y \in \thinspace \rm{R}\)}.
Because \(a\) and \(q\) are the same sign, there are zero \(x\)-intercepts.