Graphing for me has always been such a struggle. I used to detest having to constantly draw graphs, especially when transformations were applied to the functions. After a reasonable amount of practice, disagreements with teachers and corrections regarding this topic, here I am, writing an article to help you improve graphing transformations of functions.

I believe what really confuses students when it comes to graphing functions that have multiple transformations applied to them is the order in which these transformations are to be applied graphically.

When working with composition of transformations, the order in which the transformations are applied often changes the outcome and so, it is important to make the right steps in graphing your function. For example, given the function y=x^2, a vertical stretch/dilation of 5 followed by a vertical shift of 2 gives y=5x^2+2. However, a vertical shift of 2 followed by a vertical dilation of 5 produces y=5(x^2+2).

Recall the order in which the transformations are to be applied. Always look at what is inside the parentheses first.

Horizontal shift
Horizontal or vertical dilation (the order of this does not matter)
Reflection
Vertical shift

Let’s have a look at this with an example. Consider the function y=-2(3(x-1))^2+7.

If we were to describe this function in words, we would first start with what is inside the parentheses. When we apply the order of transformations as outlined above, we must expand the inner brackets to describe the function in terms of horizontal shift first, y=-2(3x-3)^2+7.

A horizontal shift of 3 places to the right has been applied.
A horizontal dilation of a factor of ⅓ has been applied.
A vertical dilation of a factor of 2 has been applied.
A reflection about the x-axis has been applied since the negative sign is outside the entire function.
A vertical shift of 7 units up has been applied.

The power of 2 tells us that the parent function was originally a quadratic function. Applying these transformations in the previously stated order to the parent coordinate (0,0) would give us the new coordinate:

To find the new coordinate, you must first expand what is inside the parentheses. y=-2(3x-3)^2+7.
The x coordinate moves 1 place to the right so we add one to it 0+3=3.
Now a horizontal dilation of ⅓ is applied to this new coordinate so we multiply it by ⅓ --> 3 x 1/3 = 1.
A vertical dilation of 2 is applied so we multiply the original y coordinate by 2 --> 0 x 2 = 0.
A reflection about the x axis causes the y coordinate to be multiplied by -1 --> 0 x -1 = 0.
A vertical shift of 7 units up means we have to add 7 to the y coordinate --> 0 + 7 = 7.
The new coordinates of the original point (0,0) are (1,7).

Tips and Tricks

In some cases, the order may not matter and the same graph will be produced. Be careful not to assume this for all questions.
Vertical transformations and horizontal transformations are independent and do not affect each other.
A reflection may sometimes produce the same graph, depending on the exponent. For e.g. a reflection about the y axis applied to the function y=x^4 is y=(-x)^4 which gives the same result.
A horizontal shift will be applied to the entire exponent term. For e.g. starting from inside the brackets, y=3(x^3+3) indicates no horizontal shift. Here, only a vertical shift of 3 units up and then a vertical dilation of 3 is applied. If a
Check if you have identified the transformations correctly by finding the new coordinates of the transformed function by plugging it into the function equation. Compare this with the coordinate you get when you apply the transformations as you have explained.
One key thing to note is that if you want to start by describing the horizontal dilation first, instead of the horizontal translation, you must factorise the brackets. Let me explain this using the example we used earlier, y=-2(3(x-1))^2+7. Here we are given the function in a factorised form in regards to the horizontal transformations. Therefore, we can start by describing the dilation first as such: a horizontal dilation of ⅓ has been applied, and then a horizontal translation of 1 place to the right has been applied. However, if we were given the same function as y=-2(3x-3)^2+7, we would describe this using a horizontal shift first - a horizontal shift of 3 units to the right and then a horizontal dilation of ⅓ to find the new coordinate. Now, both of those would give the same new coordinate of 1.

Summary

When combining vertical transformations in the form kf(x)+c, first vertically stretch by a factor of k and then shift vertically by c units.
When combining horizontal transformations in the form f(ax-b), first horizontally shift by b units and then stretch horizontally by a factor of 1a.
When combining horizontal transformations in the form f(a(x-b)), first horizontally stretch by a factor of 1a and then horizontally shift by b units.

Remember this so you know what steps to perform the transformations in when you are given an equation in a specific form.