Elementary question on graph transformations For those of you who spend half an hour answering complicated integral questions, this will a little bit of breather.
Consider the graph of $f(x)=e^x$, how would the transformation of this graph, defined as $2f(3x+2)+1$ look? My logic tells me to execute the translations in $y$ first, then the scaling in $y$, then the translations in $x$ and then the scaling in $x$ (sort of like a reversed order of operations).
That being said, consider the point on the graph where $y=8$, when the translation in $y$ is applied, the new value is $y=9$, then after the scaling in $y$ is applied, $y=18$, now, the $x$ value at this point is $x \approx2.9$, and when the translation in $x$ is applied, the new value is $x\approx1.9$, and when the scaling in $x$ is applied, the new value is $x\approx0.6$ and hence the transformation is done.
The question is, is this a correct approach? And are there other approaches which might speed this process up slightly?
 A: First, rewrite it so that all the transformations can be easily identified, like when you complete the square of a quadratic:
$$y \;\; = \;\; 2 \cdot f\left(\, 3\left(x + \frac{2}{3} \right) \,\right) \; + \; 1 $$
or
$$(y \; - \; 1) \;\; = \;\; 2 \cdot f\left(\, 3\left(x + \frac{2}{3} \right) \,\right) $$
In general, for problems like this there are $4$ types of algebraic operations that can be performed, and their corresponding $4$ types of geometric transformations: add/subtract to input variable or output variable (horizontal or vertical shift), multiply/divide to input variable or output variable (horizontal or vertical stretch).
One possible ordered sequence of transformations to $y = f(x)$ that results in this is the following. (For #2 and #3, recall how trig. graphs behave.)
1. Horizontal shift left by $\frac{2}{3}$
$$ y \; = \; f(x) \;\;\;\; \text{becomes} \;\;\;\; y \; = \; f\left(x + \frac{2}{3}\right) $$
2. Horizontal compression by a factor of $3$
$$ y \; = \; f\left(x + \frac{2}{3}\right) \;\;\;\; \text{becomes} \;\;\;\; y \; = \; f\left(3\left(x + \frac{2}{3}\right)\right) $$
3. Vertical expansion by a factor of $2$
$$ y \; = \; f\left(3\left(x + \frac{2}{3}\right)\right) \;\;\;\; \text{becomes} \;\;\;\; y \; = \; 2\cdot f\left(3\left(x + \frac{2}{3}\right)\right) $$
4. Vertical shift up by $1$
$$ y \; = \; 2\cdot f\left(3\left(x + \frac{2}{3}\right)\right) \;\;\;\; \text{becomes} \;\;\;\; y \; = \; 2\cdot f\left(3\left(x + \frac{2}{3}\right)\right)\; + \; 1 $$
