# In which order do I graph transformations of functions?

In which order do I graph transformations of functions?

The 6 function transformations are:

1. Vertical Shifts

2. Horizontal Shifts

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?

• Where does a reflection on the line x=y fit in, for an inverse function? Does it come before or after the other reflections? Thanks! Jun 14 '18 at 22:35

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

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:

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,

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

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).

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!

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 $-3x^2$.

• Ok thank you for the info. One more question: should you do the shifting before the reflection or after the reflection? I ask this because the result of the graph does differ when you shift first and then reflect v.s if you reflect first, and then do the shifting. Oct 24 '16 at 23:39
• Alway reflect before shifting. Look at the order I mentioned. Oct 25 '16 at 0:45

if you want to plot an expression like:

$$y=a\cdot f[k\cdot (x-b)]+c$$

then the connection to the parent function is:

$$f[k\cdot (x-b)]=Y, where Y=f(X)$$

then the y-coordinate of the point on the transformed function becomes:

$$y=a\cdot Y+c$$

and the x-coordinate is deducted from;

$$Y=f(X)=f([k\cdot (x-b)]\Rightarrow X=k\cdot (x-b)\Rightarrow x=\frac{X}{k}+b$$

therefore starting with the point $$(X,Y)$$ on the parent function, the chain of transformation is this:

$$(X,Y)\rightarrow (\frac{X}{k}+b,a\cdot Y+c)$$

I do the horizontal transformations first:

1.$$(X,Y)\rightarrow(\frac{X}{k},Y)$$: horizontal stretch/compression and reflection in Y-axis when k<0

2.$$(\frac{X}{k},Y)\rightarrow(\frac{X}{k}+b,Y)$$: horizontal shift

then I do the vertical transformations:

3.$$(\frac{X}{k}+b,Y)\rightarrow(\frac{X}{k}+b,a\cdot Y)$$: vertical stretch/compression and reflection in x-axis when a<0

4.$$(\frac{X}{k}+b,a\cdot Y)\rightarrow(\frac{X}{k}+b,a\cdot Y+c)$$: vertical shift

1. Scale (Stretch or shrink) Vertical or horizontal order do not matter.
2. Reflect Vertical or horizontal order do not matter.
3. Shift Vertical or horizontal order do not matter.