In which order do I graph transformations of functions?

Question

In which order do I graph transformations of functions?

The 6 function transformations are:

1. Vertical Shifts

2. Horizontal Shifts

3. Reflection about the x-axis
4. Reflection about the y-axis
5. Vertical shifting or stretching
6. Horizontal shifting or stretching

Tell me if I'm wrong, but I believe that in any function, you have to do the stretching or the shrinking before the shifting. But where do the reflections fall in this process?

Answer 1

For $Af (Bx+C)+D$ perform the operations in order: C, B , $A $, $D $. For the reflection, say $-A $, it does not matter if you stretch or shrink by $A $ and then reflect. Try an example with a simple function like $-3x^2$.

Answer 2

$$y=Af(B(x+\frac{C}{B}))+D$$

Can be thought of taking $f(x)=y$ and performing the following substitution.

$(x,y) \mapsto (Bx+C, \frac{y-D}{A})$

In order to understand what works and what doesn't work you need to understand what's going on.

Here is what is going on:

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Let's say you have some function $y=f(x)$, it has some graph. This graph is a set $G$ consisting of points $(x,y)$ where $x$ is in the domain of the function.

If you consider $f(x,y)=y-f(x)=0$ then for every substitution you perform you'll witness an inverse mapping in the graph.

For example say we perform $x \mapsto x+1$, so now we have $y-f(x+1)=0$. You might expect the graph to be composed of points $(x+1,y)$ with respect to the old graph, but this is not true rather it is composed of points $(x-1,y)$, i.e. a shift left.

On the other hand say we perform $x \mapsto 2x$, now we have $y-f(2x)=0$. Now because the inverse of the mapping $x \mapsto 2x$ is $x \mapsto \frac{1}{2}x$ now the points become,

$$(\frac{1}{2}x,y)$$

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Sometimes a combination of shifts, dilations, etc are needed, for example $y=x^2$ to $y=(2x+1)^2+1$ requires the substitution $(x,y) \mapsto (2x+1,y-1)$ whose inverse $(x,y) \mapsto (\frac{x-1}{2},y+1)$ tells you exactly what to do to the graph.

Computing the inverse of $(x,y) \mapsto (Bx+C, \frac{y-D}{A})$ will tell you everything you want to know.

I get $(x,y) \mapsto (\frac{x-C}{B},Ay+D)$. (You can perform this on points in your graph, one step at a time, in whichever way makes sense).

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For example first shifting all $x$ coordinates to the left $C$, then scaling them by $\frac{1}{B}$, then scaling $y$ coordinates by $A$, then shifting up by $D$ makes sense.

But, doing all the same for $x$ and then shifting up $y$ by $D$ to get to $y+D$ then scaling by $A$ to get to $A(y+D)$ doesn't make sense!