Billy was struggling in math, and needed help on a math test. Billy just couldn't quite understand how the transformation notations of an equation work.

Fortunately for Billy, he was graced by the presence of MATH MAN!

Hello Billy! I sensed your anguish and stress, and I am here to save yourself from any further stress!

Awesome! Well I understand most of the concept, but I hardly understand what each variable means for the transformation of the equation: y=-af(-b(x-c))+d

Not to fear! I know just how to teach this! Let's begin with horizontal and vertical shifts.

In the equation: y=-af(-b(x-c))+d, the "d" variable represents your vertical shift, and the "c" represents the horizontal shift.

For vertical shifts, if "d" is a positive number, then that means that for the new graph would be translated "d" numbers upwards.This means that the y value of a coordinate would be added by "d" units since vertical shifts indicates a change to the y value of a coordinate. The same goes if "d" is a negative number. If "d" is negative, then the graph would be translated "d" units down. 

 Let's say I have y=f(x) and one of the coordinates for my graph is (1,2). If my new graph is transformed using the equation y=f(x)+2, then that means that my graph would move upwards by 2 units. For the coordinates, that means that you add 2 to the y value only. In this case, our new graph(I'll call it g(x)) will have the coordinates (1,4). If "d" were to be a -2 instead of a 2, then the graph would shift 2 units down, giving the new coordinate of (1,0).

Horizontal shifts are a little tricky. It has the opposite trend of a vertical shift. What I mean is that if "c" is a positive number, then the graph would shift towards the left. This means that the graph would be moving towards the negative side as opposed to the positive side. On the other hand, if "c" is a negative number, then the graph would shift "c" units to the right, giving a positive trend. In the equation y=-af(-b(x-c))+d, "c" is negative because it has the opposite trend. Horizontal trends only affect the x value of a coordinate, since horizontal shift applies to the shifts within the x-axis only.

For instance, let's say I have the equation y=f(x), and the graph contains the coordinates (1,2). The graph is then horizontally shifted using the new equation y=f(x+2). In order to find my new coordinate, I would subtract 2 to my x value. Since "c" is a positive number, the trend would move toward the left instead due to "c" having the opposite trend. As such, our new coordinates for g(x) would be (-1,2).

Understand so far?

To summarize, vertical shifts pertain to the translation of the y value, while horizontal shifts pertain to the translations of the x value. When "d" is positive, the graph would move up, and vise versa. If "c" is positive, it would move to the left, and vise versa.

Exactly! You're getting the hang of it now. Now let's move on to reflections.

In the equation:y=-af(-b(x-c))+d, the "-" next to the "a" and the "-" next to the "b" represent your vertical and horizontal reflections respectively.

Let's say we have the equation y=-f(x) to simplify my explanation. If there is "-" next to f(or sometimes "a" which will be explained later), then the graph will reflect across the x-axis. This means that the y value of your coordinate is multiplied by a factor of -1 to indicate reflection.

For example, let's say we have y=f(x) and the graph contains the coordinate (1,2). The new graph is reflected across the x-axis seen by the new equation y=-f(x). In order to find our new coordinates, simply multiple your y value by -1. This will give us a new coordinate of (1,-2).

As for horizontal reflections, it's quite similar. Let's say we have a simplified equation of y=f(-x). When there is a "-" next to the x(or a (x-c)), that means that the new graph would be reflected across the y-axis. This means that the x-coordinate is multiplied by a scale factor of -1 to indicate an horizontal reflection.

For example, let's say we have a graph of y=f(x) that has the coordinate (1,2). Let's also say that the graph is then reflected across the y-axis as shown by the equation y=f(-x). In order to find our new coordinates, simply multiply the x-value by a factor of -1. This would then give us a new coordinate for g(x) which is (-1,2). Simple enough, right?

Yep! To summarize, If there is a "-" sign next to "(a)f", then my graph is reflected across the x-axis, done by multiplying the y value of a coordinate by -1, thus a vertical reflection. If there is a "-" next to the "x", then the graph would then be reflected across the y-axis, done by multiplying the x-value of a coordinate by -1, thus the horizontal reflection.

Exactly! Now that you understand translations and reflections, let's move on to scale factors.

In the equation: y=-af(-b(x-c))+d, "a" represents your vertical stretch/compression, while "b" represents your horizontal stretch/compression.

In the equation: y=-af(-b(x-c))+d, "a" represents your vertical stretch/compression, while "b" represents your horizontal stretch/compression.

Let's start with vertical compression. Let's first simplify our equation to be y=af(x). If a>1, then your graph will be vertically "stretched" by a scale factor of "a". This would look as if I were "stretching" my graph along the y axis to. Think about folding your paper hotdog style. For the coordinates, this means that your y value will be multiplied by "a" as vertical just means it relates to the y component, thus increasing the value of y.

However, if 0<a<1, then your graph will be vertically "compressed" by a sale factor of 1/a. This would look like if I were to squeeze the graph to fit a wide box. Think about folding paper hamburger style This means that the y value decreases since "a" would be a fraction. For the coordinates, that still means that your y value is multiplied by "a". If "a" is negative, then that simply means that there is a vertical reflection on top of the stretch/compression.

For example, let's say I have my graph of y=f(x) and it contains the coordinate (1,2). Let's also say that the new graph is vertically stretched which is represented by the new equation y=2f(x). In order to find my coordinates, I would simply need to multiply my y value by 2, giving me the new coordinate (1,4)

This means that my graph has been vertically "stretched" by a scale factor of 2, which makes sense as my new coordinate has a y value that is double that of my original y value.

Now let's say that we have the same scenario, except instead of it being vertically stretched, it is now vertically compressed, which is shown by the new equation y=1/2f(x). In order to find my coordinates, I would simply multiply my y value by 1/2, giving me the new coordinate (1,1). This means that my graph has been vertically "compressed" by a scale factor of 2, since 1/1/2 = 2, which makes sense since my new y coordinate is half of the original y coordinate.

Now let's look at horizontal compressions. Horizontal compressions, like horizontal shifts, have the opposite trend as its vertical counterpart. In the equation y=f(bx), if b>1, then there would be an horizontal compression done to the graph. This would be the same as folding the paper hotdog style. Since horizontal transformations pertain to the x value of a coordinate, this means that for horizontal compressions, you would divide the x value by "b", as opposed to multiplying it. This would make the graph horizontally compress by a scale factor of "b".

Additionally, if 0<b<1, then the graph would horizontally stretch by a scale factor of "1/b". This would be the same as folding a paper hamburger style. To apply it into your new graph, simply divide the x component by "b", resulting in bigger number. This would make the graph horizontally stretch by a scale factor of "1/b". If "b" is negative, that would be the same as an horizontal reflection across the y-axis.

Let's look at another example. Let's say we have the graph of y=f(x), and it contains the coordinate (1,2). Then the graph is then horizontally compressed as shown by the equation: y=f(2x). In order to find our new coordinates, we would divide the value of x by "b". In this case, we divide 1 by 2, giving us a value of 1/2 as our new x coordinate and our new coordinate of (1/2,2). This means that the graph has been horizontally compressed by a factor of 2, as it i represented as a scale factor of "b".

Now let's say that instead of horizontally compressing the graph, we want to horizontally stretch the graph as shown by the new equation: y=f(1/2x). In order to find the new coordinates of our new graph, we would divide x by "b". In this case, we divide 1 by 1/2, giving us 2. This gives us the new coordinate, (2,2). This means that the graph has been horizontally stretched by a factor of 2, as it is represented as 1/(1/2). Get it so far?

I think so. To summarize, vertical compressions/stretches are shown by "a" in y=-af(-b(x-c))+d. When a>1, then it is a vertical stretch by a factor of "a", and when 0<a<1, then the graph is vertically compressed by a scale factor of "1/a".

As for horizontal stretch/compressions, it is the opposite trend like horizontal shifts. Horizontal stretch and compressions are shown with "b" in y=-af(-b(x-c))+d. When b>1, the graph is horizontally compressed by a factor of "b", and when 0<b<1, then it is horizontally stretched by a factor of "1/b".

Correct! Now you're getting it!

There's just one thing I don't understand. What if there are multiple transformations? Which component should I work on first?