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### Section 1.10 : Common Graphs

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

Hint : The Algebraic transformations that we used to help us graph the first few graphs in this section can be used together to shift the graph of a function both up/down and right/left at the same time.
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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 $$g\left( x \right) = {{\bf{e}}^x}$$. 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. 