Does the order of graph transformations matter

graphing-functions

The task was worded this way:

Suppose that $f(x) = \frac {1}{2}(x-1)^2 – 3$.

State exactly how the
graph of $y = f(x) $ will be transformed into:

$ y = (3x-1)^2 +1 $

The answer I provided was:

  • Firstly, it will be translated up for 4 units;
  • Secondly, it will be vertically stretched by a factor of 2;
  • Thirdly, it will be horizontally compressed by a factor of 3;
  • Finally, it will be shifted to the left by $ \frac {2}{3} $ units;

Teacher said that it was not done in the right order, the correct one being:

  • Firstly, it will be shifted to the left by $ \frac {2}{3} $ units;
  • Secondly, it will be vertically stretched by a factor of 2;
  • Thirdly, it will be horizontally compressed by a factor of 3;
  • Finally, it will be translated up for 4 units;

As can be seen, in the correct answer the first and the final records are swapped.

I wonder how much the correct order really matters. I tried to trace the transformations in this task firstly according to the order I provided and then according to the correct order. It seems that the final result was still the same.

Is there any situation when the difference in order would be critical?

(Some vivid example would be appreciated).

Best Answer

We have $f(x)=\frac12(x-1)^2-3$, and let $g(x)=(3x-1)^2+1$.

Let's see what we get if we follow your sequence of transformations:

  1. Translation by $\binom{0}{4}$ so add $4$ to the whole expression and get $$\frac12(x-1)^2+1$$
  2. Vertical stretch by factor $2$, so multiply the whole expression by $2$ and get $$(x-1)^2+2$$
  3. Horizontal compression by factor $3$, so replace every $x$ term with $3x$ and get $$(3x-1)^2+2$$
  4. Shift to the left by $\frac23$ units, so replace every $x$ term by $(x+\frac23)$ and get $$\left(3(x+\frac23)-1\right)^2+2=(3x+1)^2+2\neq g(x)$$

Now let's see what we get if we follow your teacher's sequence of transformations:

  1. Shift to the left by $\frac23$ units, and get $$\frac12\left((x+\frac23)-1\right)^2-3=\frac12(x-\frac13)^2-3$$
  2. Vertical stretch by factor $2$, and get $$(x-\frac13)^2-6$$
  3. Horizontal compression by factor $3$, and get $$(3x-\frac13)^2-6$$
  4. Translation by $\binom{0}{4}$, and get $$(3x-\frac13)^2-2\neq g(x)$$

The correct sequence should be:

  1. Horizontal compression by factor $3$, and get $$\frac12(3x-1)^2-3$$
  2. Vertical stretch by factor $2$, and get $$(3x-1)^2-6$$
  3. Translate vertically by $\binom{0}{7}$ and get $$(3x-1)^2+1$$ as required.

Rule of thumb: start with the innermost transformation and work outwards.

Generally order does not matter if the transformations consist only of translations or only of enlargements. But if there are translations and enlargements in the same axis direction, then order matters.

So for example if you take the graph of $y=x^2$ and first stretch by factor $3$ horizontally, and then translate by $\binom{1}{0}$ you will get firstly $(\frac13x)^2$ and then $\left(\frac13(x-1)\right)^2$

But the same two transformation in reverse order would result in firstly $(x-1)^2$ and then $(\frac13x-1)^2$ and clearly these resulting expressions are not the same.

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