Domain:
\(G\) is an absolute value function, therefore its natural domain is \((-\infty , \infty )\).
Continuity:
\(G\) is an absolute value function, therefore it is continuous on its domain and has no discontinuities.
\(G\) is an absolute value function, therefore it has no singularities.
End-Behavior:
\(\blacktriangleright \) As \(t\) becomes big, then the inside becomes big positively or negatively. Either way, the \(| t - 3|\) is big and positive. Since the leading coefficient is negative, we have
Behavior:
\(G\) is an absolute value function, therefore, it is a piecewise function. The pieces are linear functions.
\(-2 t + 12\) is a decreasing linear function. \(G(t)\) decreases on \((-\infty , 5)\).
\(2 t\) is an increasing linear function. \(G(t)\) increases on \((5, \infty )\).
Zeros:
We can find the zeros of each piece of the piecewise function.
\(-2 t + 12 \) has a zero at \(6\).
\(-2 t\) has a zero at \(0\).
Extema:
From the end-behavior, we know that \(G\) has no global or local minimum.
Since \(G\) is an absolute value funciton, \(G\) has a global and local maximum when the inside is \(0\), which is at \(3\).
\(G\) is a global and local maximum of \(G(3) + 6\) at \(3\).
This all agrees with the graph.