Transformations
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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!
VERTICAL
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.
TEN HUT!
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.
To increase or decrease their station, y, we increase or decrease the station!
©MathBait created with GeoGebra
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.
WIZARDRY
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.
FLIP IT
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)
HORIZONTAL
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...
TEN HUT!
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
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.
WIZARDRY
Let f(x) be your basic shape, the original formation you wish to mold. We can cast a proliferation spell on our x 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).
FLIP IT
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)