Section 1.10 : Common Graphs
7. Without using a graphing calculator sketch the graph of \(W\left( x \right) = {{\bf{e}}^{x + 2}} - 3\).
The Algebraic transformations we were using in the first few problems of this section can be combined to shift a graph up/down and right/left at the same time. If we know the graph of \(g\left( x \right)\) then the graph of \(g\left( {x + c} \right) + k\) is simply the graph of \(g\left( x \right)\) shifted right by \(c\) units if \(c < 0\) or shifted left by \(c\) units if \(c > 0\) and shifted up by \(k\) units if \(k > 0\) or shifted down by \(k\) units if \(k < 0\).
So, in our case if \(g\left( x \right) = {{\bf{e}}^x}\) we can see that,
\[W\left( x \right) = {{\bf{e}}^{x + 2}} - 3 = g\left( {x + 2} \right) - 3\]and so the graph we’re being asked to sketch is the graph of \(g\left( x \right) = {{\bf{e}}^x}\)shifted left by 2 units and down by 3 units.
Here is the graph of \(W\left( x \right) = {{\bf{e}}^{x + 2}} - 3\) and note that to help see the transformation we have also sketched in the graph of \(g\left( x \right) = {{\bf{e}}^x}\).
In this case the resulting sketch of \(W\left( x \right)\) that we get by shifting the graph of \(g\left( x \right)\) is not really the best, as it pretty much cuts off at \(x = 0\) so in this case we should probably extend the graph of \(W\left( x \right)\) a little. Here is a better sketch of the graph.
