How Would I Graph This Exponential Function? How Would I Graph This Exponential Function?
$f(x) = \frac{-3}{2^{(x+2)}} - 1$,
How I do it: I know that it is basically $-3\times ({\frac{ 1}{2}})^{x+2} - 1$, which is then $-3 \times 2^{-(x+2)} - 1$, my base function is $2^{-x}$, and thus my mapping equation ends up being $(-x + 2, -3y - 1)$, and after plugging in points such as $(-2, 4), (-1, 2), (0, 1), (1, \frac{1}{2})$, etc, I end up with my graph but it looks completely different than the one I get in Wolframalpha - well, similar, but it is starting from the top left and it gradually proceeds down. What am I doing wrong?
Thanks
 A: *

*First graph $\frac{1}{2^x}$. This is easy.

*Next, slide the graph over 2 units to the left. This results in $\frac{1}{2^{x+2}}$.

*Next, scale the graph by 3, this results in $\frac{3}{2^{x+2}}$.

*Now, flip the graph, yielding $-\frac{3}{2^{x+2}}$.

*Finally, slide the graph down by $1$, to obtain $-\frac{3}{2^{x+2}}-1$.


To wit:


*

*$x=2 \implies y = \frac14$. $x=3 \implies y=\frac18$.

*Slide left by 2: $x=0 \implies y = \frac14$. $x=1 \implies y=\frac18$.

*Scale by 3: $x=0 \implies y = \frac34$. $x=1 \implies y=\frac38$.

*Flip it:  $x=0 \implies y = -\frac34$. $x=1 \implies y=-\frac38$.

*Slide down by 1: $x=0 \implies y = -\frac74$. $x=1\implies y=-\frac{11}{8}$


After working this out, it appears that maybe you had an extra negative sign in your computation of $x$. Remember, $2^{-x} = \frac{1}{2^x}$, so if you perform $x \mapsto x+2$, then make sure you either compute $x+2$ and plug it into $2^{-x}$, or compute $-x-2$ and plug it into $2^x$.
A: My goodness... so much work when we have computer plotting right at hand?!  If a question poser asked how to multiply two 10-digit numbers would we explain how to do it by hand?

Your points are in red, which is fit perfectly by $y = 2^{-x}$ (in orange).
