Calculus I - Common Graphs

Section 1.10 : Common Graphs

7. Without using a graphing calculator sketch the graph of $W\left( x \right) = {{\bf{e}}^{x + 2}} - 3$.

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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 $g\left( x \right) = {{\bf{e}}^x}$.

$The graph of $W\left( x \right) = {{\bf{e}}^{x + 2}} - 3$, drawn as a solid red curve, along with the graph of $g\left( x \right) = {{\bf{e}}^x}$, drawn as a dashed gray curve, for $-4 \le x \le 2$. The graph of $g\left( x \right)$ is just above the $x$-axis on the left and increases through (0,1) to about 7.4 at $x = 2$. The graph of $W\left( x \right)$ is just above $y = -3$ on the left, crosses the $x$-axis near $x = -0.9$ and stops at about (0,4.4), so only the part of it with $x \le 0$ is shown.$

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.

$The same graphs as the previous figure except that the graph of $W\left( x \right) = {{\bf{e}}^{x + 2}} - 3$ has been extended past $x = 0$. The solid red curve for $W\left( x \right)$ now continues increasing past (0,4.4) and leaves the top of the graph at $y = 8$ near $x = 0.4$. The dashed gray curve for $g\left( x \right) = {{\bf{e}}^x}$ is unchanged.$