top of page
Gray Structure



Suppose you are molding a piece of clay into human form. Starting with the head, you notice it is circle-ish. It isn't exactly a circle, but it has many of the characteristics you associate with a circle. It is curved, round, smooth. So you begin by taking your clay and rolling it to form a circle. Next, you notice the head isn't as perfect as a circle. It is more oblong, so you take your circle and mash it a bit, stretching it up to be thinner. Formations are the same! Knowing the mother order, we can move it, shift it, pull it, push it, stretch it, shrink it, and flip it to create a cacophony of new and interesting formations. Even better – these commands work with all functions!

Can you use transformations to match Trent's silly face? Test your skills and play now!

Marco the Great and the Mystery of Phaseville

You need this!

The content on this page is supplementary material for Marco the Great and the Mystery of Phaseville


You have sent your soldiers onto the field into the basic shape or formation you are looking for. But it isn't quite right. The easiest way to augment your formation is using vertical commands. You simply tell everyone to take one step (or many steps) forward or backward. If that's not enough, you could mold your shape as well. If you think of the battlefield as a piece of clay or rubber, you could grab the top and bottom of it and pull, making the field taller, or you could squish it. Finally, you could flip the whole formation. Let's give it a try. 



Let f be your basic shape, the original formation you wish to mold. Then y is the location for each of your soldiers. To tell everyone to step forward, we must increase their position. Conversely, to have everyone move backward, we decrease their position.  

Vertical shift y+k=f(x)+k

To increase or decrease their station, y, we increase or decrease the station! 

©MathBait created with GeoGebra

Smiley Stick Man

To move a function up or down, we need f(x)+k. When k is positive (k>0) the formation moves up, when k is negative (k<0) the formation will move down. 


Vertical Stretch/Shrink

Let f be your basic shape, the original formation you wish to mold. If we wish to harness our power to stretch it and make it taller and mightier, or shorter and stubbier, we must cast a proliferation spell! Multiply each position, y, to mold your formation.   


Play again, this time using your wizardry!

Multiplying the entire formation increases (or decreases) every soldier's position!

©MathBait created with GeoGebra

To stretch or shrink a function, we need af(x). When a is larger than 1, the formation will stretch, appearing taller. Just like tearing down a number, if we want to make a number smaller, we divide. To shrink a formation, a must be a fraction between 0 and 1. This essentially divides your soldiers' locations, making the formation shorter.

Smiley Stick Man
Vertical Flip


Let f be your basic shape, the original formation you wish to mold. Maybe it looks like a smile and you're looking for a frown. Or maybe your range of motion is all wrong and you are looking to cover some different stations. By casting a proliferation spell to turn every location into its evil twin, you'll flip the entire thing! Send your soldiers below ground, or give them some needed fresh air. With -f(x) you'll do just this!

Did You Know...

Flips are everywhere. Just in designing this webpage we needed to apply vertical flips to get the arrows to point the right way!

This is clock(time)

This is plane(x)

This is sandwich(bun)

This is –clock(time)

This is –sandwich(bun)

This is –plane(x)


A commander facing his soldier says "move right" the soldier is confused, should he go to his right or his commander's right?

When working with horizontal transformations you must enter Bizarro world. Left is right and right and left, big is little and little is big. 

Vertical transformations not working for you? Maybe you are interested in moving your army left or right? We couldn't do that vertically. Not to fear! We can apply horizontal transformations too – but it will require you to consider a new perspective...


Let f(x) be your basic shape, the original formation you wish to mold. Horizontal transformations impact your soldiers. You can't simply change their station, but instead tell each soldier to go to someone else's station.

If we wish to instruct our army to move to the right, we might think we should add a positive value to each soldier (as right is the positive direction). Au contraire! Our right is our soldiers' left! Play around and see what you find. 


Impacting the soldier, or x-value, will move the formation horizontally: left or right. 

©MathBait created with GeoGebra

Smiley Stick Man

To move a function left or right, we need f(x+h). When h is positive (h>0) the formation moves left, when h is negative (h<0) the formation will move right

Horizontal Stertch/Shrink


Let f(x) be your basic shape, the original formation you wish to mold. We can cast a proliferation spell on our soldiers to stretch or shrink our formation as well. But be careful! Stretching horizontally still requires a factor larger than 1, yet the effect will be a shorter formation as we are pulling from the sides and our command requires us to divide each soldier. Similarly, shrinking horizontally requires a factor between 0 and 1, with an effect of making the formation taller as we are pushing in from the sides. To achieve this horizontal shrink requires a command that again divides our soldiers...but what happens when we divide by a value between 0 and 1? Explore and find out! 


Multiplying every soldier will stretch or shrink your formation...but it might not be what you're expecting!

Play again, this time using your wizardry!

©MathBait created with GeoGebra

To stretch or shrink a function horizontally, we need f(bx). When b is larger than 1, the formation will shrink, appearing shorter and stubbier. To stretch a formation horizontally, b must a fraction between 0 and 1, making the formation taller. But here is the really important part! The factor of the transformation is 1/b. Calling out "stretch the formation by a factor of 4!" tells your soldiers to adhere to the order f(x/4).

Smiley Stick Man
Horizontal Flip


Let f(x) be your basic shape, the original formation you wish to mold. If you find you accidentally mixed up twins (this is easy to do after all), you may need for evil twins to swap their locations. This would send soldier x=1 to the location of soldier x=-1 and vice versa. The result is a formation flipped horizontally. To accomplish this, you'll need the order f(–x).

This is plane(–x)

This is arrow(x)

This is plane(x)

This is arrow(–x)



For each of the following, describe how the order (and thus the formation) has changed.













©MathBait created with GeoGebra

Confused stick figure


How do I know if it is a vertical transformation or a horizontal transformation?


Vertical transformations change the location, the y-values. To identify a vertical transformation, look at the original formation/function – if you are impacting the whole formation, that's vertical. On the contrary, if you are impacting the soldiers, the x-values inside the function, that's a horizontal transformation. 

Transformation Target

Does Order Matter?

Explore for yourself!

Yes! But only if you have multiple horizontal or multiple vertical transformations. 

©MathBait created with GeoGebra

Be careful! Notice the –3 is not impacting the vertical shift +4. This means we need to shift the formation up after the reflection!

Confused stick figure


What about something like this?


The vertical stretch/shrink, a, in this example is negative. This means there are two pieces of information hidden in the –3: a vertical stretch by a factor of 3 and a vertical reflection!

Transformation Art


Did You Know...

Transformations not only allow us to master every functional formation, they also are used extensively in art! Check out these images made only from a single formation and transformations.

geogebra-export (13)_edited.jpg
geogebra-export (10)_edited.jpg
geogebra-export (12)_edited.jpg



Think you have mastered formation commands? Trent will be the judge of that! Using his eyebrows and his winning smile, Trent will make a face. You'll need to use transformations to match it. Do you have what it takes?

©MathBait created with GeoGebra

© Copyright

Ready to start your adventure?

Grab your copy and jump into the pages with a mathematical learning experience like never before!

Want more games, explorations, and math done the fun way? Enroll in our online program, The Kryptografima, today!

Marco the Great and the Mystery of Phaseville
bottom of page