Explaining the transformation of curves I have the graph $y=4^x$ and $y=4^{x-h}+v$.

How can $y=4^x$ be transformed into $y=4^{x-h}+v$?
I know that $v$ would be the vertical shift. So where ever the graph started, it now starts $v$ units above. Not sure about the $4^{x-h}$ though. 
Also I have a parabola, $y=x^2$ and $y=a(x-h)^2+v$
 
So how can $y=x^2$ transform to $y=a(x-h)^2 +v$? Again, I know $v$ is the vertical shift. I know that $(h,v)$ is the vertex and $a$ tells you that the parabola opens up or down but I'm not sure exactly how $y=x^2$ transforms to  $y=a(x-h)^2 +v$.
 A: Try flipping the graph along the line $y=x$. For the equation $y = 4^{x-h}+v$ we have:
$$\begin{align}
y &= 4^{x-h}+v\\
4^{x-h}+v &= y\\
4^{x-h} &= y-v\\
x-h &= \log_4(y-v)\\
x &= \log_4(y-v)+h\\
\end{align}$$
Notice that $h$ is the vertical shift for the inverse function (i.e. $log_4$), and $v$ is the horizontal shift.

For the parabola, try to just consider it piecewise, one one-to-one piece at a time.
A: OK.
a tells you "how much" the parabola opens up or down. A higher absolute value stretchs the parabola towards the y-axis.
h tells you how the parabola "slides" in x-axis. If h is positive, the x-coordinate of your vertex will lay in x=h.
Same principle in $y=4^{x}$. $y=\frac{1}{4^{h}}.4^{x}$ which will make h works like the a in your parabola.
A: You need to push each slider one at a time repeatedly to watch and learn by observation the dynamic behavior caused by particular parameter change.
$$ y=\dfrac{4^x}{1}+ 0 \quad \rightarrow y=\dfrac{4^x}{4^h}+ v  $$
As we change first slider the curve opens or closes like a flower because the coefficient $ 4^h$ affects it that way. After that,  effect of $v$ is to lift  the entire curve up or down. It does not matter which slider control you use first.
Similarly
$$ y= \dfrac  {(x-0) ^2}{1^2} +0 \rightarrow  \dfrac  {(x-h) ^2}{4 f} +v $$
the starting parabola slides on x-axis direction by varying action of slider $h.$ The focal length of parabolic beam of light changes by varying its focal length . If used as a torch the beam is made wider or focused into a narrower beam by virtue of changing focal length $f$. The action of slider $v$  as before is to lift the entire thing en masse upwards or downwards.
