Transformation Tides:Vectors on the High Sea
Backstory
Start
Transformations Tutorial
Backstory: Invasion from the Deep
SouthFort, a thriving coastal city, has long stood as a beacon of trade, culture, and naval strength. But beneath the waves, a sinister force has been growing for decades. The Abyssal Covenant, a horde of evil, parasitic mutant sea creatures, has long desired the land above for themselves. They do not seek peace, negotiation, or diplomacy—they only seek domination. Their goal: invade, corrupt, and consume SouthFort, mutating its citizens or obliterating all resistance.
Next
Enter Lord Thalassor: Oceanic Tyrannt
At the forefront of this parasitic army stands Lord Thalassor, a towering, terrifying overlord of the deep. His humanoid body is muscular and battle-hardened, yet his head and face are entirely squid-like, with multiple writhing tentacles hanging from his jaw and mouth, glowing eyes, and an elongated cranium. His slick, dark skin bears subtle scale patterns, faintly illuminated by bioluminescent markings that glow eerily in dark waters. Fins along his back and sides allow him to move with terrifying speed and agility in the ocean, while his clawed hands can rend ships and sweep aside entire groups of opponents. Lord Thalassor is not just brute force—he is intelligent, cruel, and calculating, commanding the Abyssal Covenant with precision and patience, striking at the perfect moment to devastate SouthFort.
Next
A Call to Action
To defend the city, President Kwaku Anansi has called upon South Fort’s finest: The Tempest Vanguard, a naval fleet of elite soldiers and ships, tasked with repelling the invasion at all costs. At the helm stands Captain Leopold Stravinsky III, a grizzled, muscular, and battle-hardened officer whose stern leadership and tactical genius inspire his crew. Beneath his tough exterior lies loyalty and love for his sailors, making him both feared and respected.
Next
All Hands on Deck!
Captain Stravinsky is supported by his trusted lieutenants: Shaylee, a young female tactician who is both agile, tactical, and fiercely loyal, and Jason, a resourceful, headstrong, courageous naval soldier. Together, they assist Captain Stravinsky in commanding the core of the Tempest Vanguard, guiding the fleet through high-seas combat against increasingly dangerous foes. Players will interact with Shaley and Jason, learning combat and naval strategy while fighting to protect South Fort.
Next
Prepare for War!
The stakes are nothing less than total annihilation. The Abyssal Covenant has waited generations for the moment to strike, seeking to transform the world above into a mutated underwater domain. With Lord Thalassor leading the charge, their assault is methodical, terrifying, and nearly unstoppable. The navy must rise, swords in hand, cannons blazing, to stop the parasitic hordes before South Fort falls and the land becomes a feeding ground for the Abyssal Covenant.
Transformations Tutorial
Begin
Transformations Tutorial
Begin Mission
Captain Leopold Stravinksy
Soldiers of the Tempest Vanguard… listen well. The city of SouthFort stands on the edge of destruction. The mutant hordes rise from the depths, threatening everything we swore to protect. Victory will not come from strength alone — it will come from discipline, strategy, and sharp minds. To outmaneuver the enemy, you must master the art of Geometric Transformations— reflection, rotation, translation, and dilation. These are not just lessons… they are weapons. Apply them well, and you will cut through the chaos of battle. Fail, and SouthFort will fall. Your mission is clear: lead with precision, fight with unity, and bring victory to the Tempest Vanguard. If you are not familiar with Geometric Transformations I recommend clicking the tutorial button. Lieutenant Shaylee will train and prepare you for battle. However, if you are ready click the button on the right.
Tutorial Mode
Next
Tutorial Mode
Original Point
New Point
Next
“Translation is simple, recruit — it’s just sliding the figure across the grid without turning it or flipping it over. You only adjust its position by adding or subtracting from the x- and y-coordinates. For example, if a point is at (2,4) and the rule says move it to (x plus 3, y minus 2), that means you slide it 3 units to the right and 2 units down. That’s all translation is — a straight, controlled movement across the grid. Focus, and you’ll move your ship exactly where it needs to be.”
Tutorial Mode
Try this one for practice: Translate Point A (2,4) by the rule (x + 5, y - 1). Where is the new position for Point A? Move the GREEN dot to the new location and select an answer choice.
A) (7, 3)
B) (3, 7)
C) (-3, 3)
Correct!
Next
Tutorial Mode
Original Point
New Point
Rotation turns a figure around the origin without changing its shape. There are three common rotations you’ll need to know: 90° clockwise, which changes a point (x, y) to (y, -x). 90° counterclockwise, which changes it to (-y, x). 180°, which flips both coordinates: (x, y) → (-x, -y).
Next
Tutorial Mode
Original Point
New Point
In this example we have the following problem: “Rotate point C(−2, 5) 90° clockwise about the origin. What are the new coordinates?” “For a 90° clockwise rotation, we use the rule (x, y) → (y, -x). Take point C at (−2, 5). Swap the coordinates and change the sign of the original x: y becomes the new x, and -x becomes the new y. So (−2, 5) becomes (5, 2). That’s your rotated point — precise and controlled, just like maneuvering your ship.”
Next
Tutorial Mode
Try this one for practice: The Rule: 90° clockwise: (x, y) → (y, -x)Rotate point P(2, 7) 90° clockwise about the origin. What are the new coordinates?
A) (-7, 2)
B) (2, 7))
C) (7, -2)
Correct!
Next
Tutorial Mode
Try this one for practice: The Rule: 90° counterclockwise: (x, y) → (-y, x) Rotate point Q(−3, 4) 90° counterclockwise about the origin. What are the new coordinates?
A) (-4, -3)
B) (-4, 3)
C) (3, 4)
Correct!
Next
Tutorial Mode
Try this one for practice: The Rule: 180° rotation: (x, y) → (-y, x) Rotate point R(5, -2) 180° about the origin. What are the new coordinates?
A) (-5, 2)
B) (2, 5)
C) (-5, -2)
Correct!
Next
Tutorial Mode
New Point
Original Point
“Reflection flips a figure over a line, like holding it up to a mirror. There are three common reflections you need to know: Over the x-axis: flip the y-coordinate, so (x, y) becomes (x, -y). Over the y-axis: flip the x-coordinate, so (x, y) becomes (-x, y). Over the line y = x: swap the coordinates, so (x, y) becomes (y, x). Reflection changes the orientation but keeps the shape exactly the same. Pay attention to which line you’re reflecting over — one wrong move and your fleet is misaligned.”
Next
Tutorial Mode
New Point
Original Point
In this example we the following problem: Reflect point C(-4, 2) over the y-axis. What are the coordinates after the reflection? “We’re reflecting point C over the y-axis. That means we flip the x-coordinate while keeping the y-coordinate the same. Point C is at (-4, 2), so flipping the x-coordinate gives us (4, 2). That’s your reflected point — precise and ready for action.”
Next
Tutorial Mode
Try this one for practice: The Rule: Reflecting over the x-axis changes it to (x, -y). Reflect point D(3, -5) over the x-axis. What are the coordinates after the reflection?
A) (-6, 2)
B) (3, 5)
C) (-5, -2)
Correct!
Next
Tutorial Mode
New Shape's Point
Original Shape's Point
“Dilation changes the size of a figure without changing its shape. You’ll use a scale factor to enlarge or shrink it: Multiply both the x- and y-coordinates by the scale factor. A scale factor greater than 1 makes the figure bigger, between 0 and 1 makes it smaller. Keep the origin as the center unless the problem tells you otherwise. Precision is critical — one wrong calculation and your fleet won’t reach its target.”
Next
Tutorial Mode
Try this one for practice: Dilate point E(2, 3) by a scale factor of 2.
A) (4, 6)
B) (1, -1.5)
C) (6,4)
You are Ready!
Next
Try Again!
Retry
Try Again!
Retry
Try Again!
Retry
Try Again!
Retry
Try Again!
Retry
Try Again!
Retry
Mission Start!
(2, 6)
Locate the Target by solving the problemTranslate point A(1, 4) by (x + 3, y − 2). Where does A end up?
B) (2, 6)
C) (4, 6)
A) (4 , 2)
Click here for Instructions
Game Over: Mission Failed
The Tempest Vanguard has been destoryed! SouthFort is Lost!
Exit
Retry
Target Acquired! Fire!
(2, 6)
Click here for Instructions
(2, 6)
Locate the target by solving the problem Translate point B(−2, 5) by (x + 4, y − 3). Where is B now?
B) (−6, 2)
C) (2, 2)
A) (2, 8)
Click here for Instructions
Game Over: Mission Failed
The Tempest Vanguard has been destoryed! SouthFort is Lost!
Retry
Exit
Target Acquired! Fire!
(2, 6)
Click here for Instructions
(2, 6)
Locate the target by solving the problem Rotate point C(3, 1) 90° clockwise about the origin.Rule: 90° clockwise: (x, y) → (y, -x)
B) (-1, 3)
C) (3, -1)
A) (1, -3)
Click here for Instructions
Game Over: Mission Failed
The Tempest Vanguard has been destoryed! SouthFort is Lost!
Retry
Exit
Target Acquired! Fire!
(2, 6)
(2, 6)
Locate the target by solving the problem Rotate point D(−4, 2) 90° counterclockwise.90° counterclockwise: (x, y) → (-y, x)
B) (-2, -4)
C) (2, -4)
A) (-2, 4)
Click here for Instructions
Game Over: Mission Failed
The Tempest Vanguard has been destoryed! SouthFort is Lost!
Retry
Exit
Target Acquired! Fire!
(2, 6)
(2, 6)
Locate the target by solving the problem Rotate point E(5, -3) 180° about the origin.
B) (-5, 3)
C) (5, 3)
A) (3, -5)
Click here for Instructions
Game Over: Mission Failed
The Tempest Vanguard has been destoryed! SouthFort is Lost!
Retry
Exit
Target Acquired! Fire!
(2, 6)
(2, 6)
Locate the target by solving the problem Reflect point F(2, -6) over the x-axis. Rule: Over x-axis: (x, y) → (x, -y)
B) (-2, 6)
C) (6, 2)
A) (2, 6)
Click here for Instructions
Game Over: Mission Failed
The Tempest Vanguard has been destoryed! SouthFort is Lost!
Retry
Exit
Target Acquired! Fire!
(2, 6)
(2, 6)
Locate the target by solving the problem RReflect point G(-3, 5) over the y-axis. Rule: Over x-axis: (x, y) → (x, -y)
C) (-3, -5)
B) (3, 5)
A) (-5, 3)
Click here for Instructions
Game Over: Mission Failed
The Tempest Vanguard has been destoryed! SouthFort is Lost!
Retry
Exit
Target Acquired! Fire!
(2, 6)
“So… you dare send your pathetic fleet against me? My soldiers… my perfect warriors… you slaughtered them without mercy. Now, your city… your people… will pay the ultimate price!”
Lord Thalassor:
Next
Defeat Lord Thalassor
Final Boss Fight
"This is it—the final assault. The Tempest Vanguard stands between South Fort and total destruction. Captain Leopold, Shaylee, Jason, and the crew are ready, but the fate of the city rests in your hands." "Lord Thalassor, overlord of the deep, has unleashed his monstrous army. To defeat him, you must solve the last five transformation problems correctly. Each problem is a strike against his forces. You have 20 minutes—accuracy and speed will decide the outcome." "This is the moment where heroes are forged. Focus, apply everything you’ve learned, and bring victory to South Fort!"
Begin
20:00
Rule: (x,y)→ (x + 4, y – 7).
20:00
Rule: 90° Counterclockwise Rotation → (x, y) → (-y, x)
20:00
Rule: Reflection over y = x → (x, y) → (y, x)
20:00
Rule: Reflection over y = x → (x, y) → (y, x)
20:00
Rule: Reflection over y = x → (x, y) → (y, x)
Congratulations
You successfully help The Tempest Vanguard defeat Lord Thalassor! SouthFort is Safe Once again!
Exit
Thank you for playing!
Game Over: Mission Failed
The Tempest Vanguard has been destoryed! SouthFort is Lost!
Retry
Exit
Transformation Tides:Vectors on the High Sea
Yaw Owusu-Akyaw
Created on September 29, 2025
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Transcript
Transformation Tides:Vectors on the High Sea
Backstory
Start
Transformations Tutorial
Backstory: Invasion from the Deep
SouthFort, a thriving coastal city, has long stood as a beacon of trade, culture, and naval strength. But beneath the waves, a sinister force has been growing for decades. The Abyssal Covenant, a horde of evil, parasitic mutant sea creatures, has long desired the land above for themselves. They do not seek peace, negotiation, or diplomacy—they only seek domination. Their goal: invade, corrupt, and consume SouthFort, mutating its citizens or obliterating all resistance.
Next
Enter Lord Thalassor: Oceanic Tyrannt
At the forefront of this parasitic army stands Lord Thalassor, a towering, terrifying overlord of the deep. His humanoid body is muscular and battle-hardened, yet his head and face are entirely squid-like, with multiple writhing tentacles hanging from his jaw and mouth, glowing eyes, and an elongated cranium. His slick, dark skin bears subtle scale patterns, faintly illuminated by bioluminescent markings that glow eerily in dark waters. Fins along his back and sides allow him to move with terrifying speed and agility in the ocean, while his clawed hands can rend ships and sweep aside entire groups of opponents. Lord Thalassor is not just brute force—he is intelligent, cruel, and calculating, commanding the Abyssal Covenant with precision and patience, striking at the perfect moment to devastate SouthFort.
Next
A Call to Action
To defend the city, President Kwaku Anansi has called upon South Fort’s finest: The Tempest Vanguard, a naval fleet of elite soldiers and ships, tasked with repelling the invasion at all costs. At the helm stands Captain Leopold Stravinsky III, a grizzled, muscular, and battle-hardened officer whose stern leadership and tactical genius inspire his crew. Beneath his tough exterior lies loyalty and love for his sailors, making him both feared and respected.
Next
All Hands on Deck!
Captain Stravinsky is supported by his trusted lieutenants: Shaylee, a young female tactician who is both agile, tactical, and fiercely loyal, and Jason, a resourceful, headstrong, courageous naval soldier. Together, they assist Captain Stravinsky in commanding the core of the Tempest Vanguard, guiding the fleet through high-seas combat against increasingly dangerous foes. Players will interact with Shaley and Jason, learning combat and naval strategy while fighting to protect South Fort.
Next
Prepare for War!
The stakes are nothing less than total annihilation. The Abyssal Covenant has waited generations for the moment to strike, seeking to transform the world above into a mutated underwater domain. With Lord Thalassor leading the charge, their assault is methodical, terrifying, and nearly unstoppable. The navy must rise, swords in hand, cannons blazing, to stop the parasitic hordes before South Fort falls and the land becomes a feeding ground for the Abyssal Covenant.
Transformations Tutorial
Begin
Transformations Tutorial
Begin Mission
Captain Leopold Stravinksy
Soldiers of the Tempest Vanguard… listen well. The city of SouthFort stands on the edge of destruction. The mutant hordes rise from the depths, threatening everything we swore to protect. Victory will not come from strength alone — it will come from discipline, strategy, and sharp minds. To outmaneuver the enemy, you must master the art of Geometric Transformations— reflection, rotation, translation, and dilation. These are not just lessons… they are weapons. Apply them well, and you will cut through the chaos of battle. Fail, and SouthFort will fall. Your mission is clear: lead with precision, fight with unity, and bring victory to the Tempest Vanguard. If you are not familiar with Geometric Transformations I recommend clicking the tutorial button. Lieutenant Shaylee will train and prepare you for battle. However, if you are ready click the button on the right.
Tutorial Mode
Next
Tutorial Mode
Original Point
New Point
Next
“Translation is simple, recruit — it’s just sliding the figure across the grid without turning it or flipping it over. You only adjust its position by adding or subtracting from the x- and y-coordinates. For example, if a point is at (2,4) and the rule says move it to (x plus 3, y minus 2), that means you slide it 3 units to the right and 2 units down. That’s all translation is — a straight, controlled movement across the grid. Focus, and you’ll move your ship exactly where it needs to be.”
Tutorial Mode
Try this one for practice: Translate Point A (2,4) by the rule (x + 5, y - 1). Where is the new position for Point A? Move the GREEN dot to the new location and select an answer choice.
A) (7, 3)
B) (3, 7)
C) (-3, 3)
Correct!
Next
Tutorial Mode
Original Point
New Point
Rotation turns a figure around the origin without changing its shape. There are three common rotations you’ll need to know: 90° clockwise, which changes a point (x, y) to (y, -x). 90° counterclockwise, which changes it to (-y, x). 180°, which flips both coordinates: (x, y) → (-x, -y).
Next
Tutorial Mode
Original Point
New Point
In this example we have the following problem: “Rotate point C(−2, 5) 90° clockwise about the origin. What are the new coordinates?” “For a 90° clockwise rotation, we use the rule (x, y) → (y, -x). Take point C at (−2, 5). Swap the coordinates and change the sign of the original x: y becomes the new x, and -x becomes the new y. So (−2, 5) becomes (5, 2). That’s your rotated point — precise and controlled, just like maneuvering your ship.”
Next
Tutorial Mode
Try this one for practice: The Rule: 90° clockwise: (x, y) → (y, -x)Rotate point P(2, 7) 90° clockwise about the origin. What are the new coordinates?
A) (-7, 2)
B) (2, 7))
C) (7, -2)
Correct!
Next
Tutorial Mode
Try this one for practice: The Rule: 90° counterclockwise: (x, y) → (-y, x) Rotate point Q(−3, 4) 90° counterclockwise about the origin. What are the new coordinates?
A) (-4, -3)
B) (-4, 3)
C) (3, 4)
Correct!
Next
Tutorial Mode
Try this one for practice: The Rule: 180° rotation: (x, y) → (-y, x) Rotate point R(5, -2) 180° about the origin. What are the new coordinates?
A) (-5, 2)
B) (2, 5)
C) (-5, -2)
Correct!
Next
Tutorial Mode
New Point
Original Point
“Reflection flips a figure over a line, like holding it up to a mirror. There are three common reflections you need to know: Over the x-axis: flip the y-coordinate, so (x, y) becomes (x, -y). Over the y-axis: flip the x-coordinate, so (x, y) becomes (-x, y). Over the line y = x: swap the coordinates, so (x, y) becomes (y, x). Reflection changes the orientation but keeps the shape exactly the same. Pay attention to which line you’re reflecting over — one wrong move and your fleet is misaligned.”
Next
Tutorial Mode
New Point
Original Point
In this example we the following problem: Reflect point C(-4, 2) over the y-axis. What are the coordinates after the reflection? “We’re reflecting point C over the y-axis. That means we flip the x-coordinate while keeping the y-coordinate the same. Point C is at (-4, 2), so flipping the x-coordinate gives us (4, 2). That’s your reflected point — precise and ready for action.”
Next
Tutorial Mode
Try this one for practice: The Rule: Reflecting over the x-axis changes it to (x, -y). Reflect point D(3, -5) over the x-axis. What are the coordinates after the reflection?
A) (-6, 2)
B) (3, 5)
C) (-5, -2)
Correct!
Next
Tutorial Mode
New Shape's Point
Original Shape's Point
“Dilation changes the size of a figure without changing its shape. You’ll use a scale factor to enlarge or shrink it: Multiply both the x- and y-coordinates by the scale factor. A scale factor greater than 1 makes the figure bigger, between 0 and 1 makes it smaller. Keep the origin as the center unless the problem tells you otherwise. Precision is critical — one wrong calculation and your fleet won’t reach its target.”
Next
Tutorial Mode
Try this one for practice: Dilate point E(2, 3) by a scale factor of 2.
A) (4, 6)
B) (1, -1.5)
C) (6,4)
You are Ready!
Next
Try Again!
Retry
Try Again!
Retry
Try Again!
Retry
Try Again!
Retry
Try Again!
Retry
Try Again!
Retry
Mission Start!
(2, 6)
Locate the Target by solving the problemTranslate point A(1, 4) by (x + 3, y − 2). Where does A end up?
B) (2, 6)
C) (4, 6)
A) (4 , 2)
Click here for Instructions
Game Over: Mission Failed
The Tempest Vanguard has been destoryed! SouthFort is Lost!
Exit
Retry
Target Acquired! Fire!
(2, 6)
Click here for Instructions
(2, 6)
Locate the target by solving the problem Translate point B(−2, 5) by (x + 4, y − 3). Where is B now?
B) (−6, 2)
C) (2, 2)
A) (2, 8)
Click here for Instructions
Game Over: Mission Failed
The Tempest Vanguard has been destoryed! SouthFort is Lost!
Retry
Exit
Target Acquired! Fire!
(2, 6)
Click here for Instructions
(2, 6)
Locate the target by solving the problem Rotate point C(3, 1) 90° clockwise about the origin.Rule: 90° clockwise: (x, y) → (y, -x)
B) (-1, 3)
C) (3, -1)
A) (1, -3)
Click here for Instructions
Game Over: Mission Failed
The Tempest Vanguard has been destoryed! SouthFort is Lost!
Retry
Exit
Target Acquired! Fire!
(2, 6)
(2, 6)
Locate the target by solving the problem Rotate point D(−4, 2) 90° counterclockwise.90° counterclockwise: (x, y) → (-y, x)
B) (-2, -4)
C) (2, -4)
A) (-2, 4)
Click here for Instructions
Game Over: Mission Failed
The Tempest Vanguard has been destoryed! SouthFort is Lost!
Retry
Exit
Target Acquired! Fire!
(2, 6)
(2, 6)
Locate the target by solving the problem Rotate point E(5, -3) 180° about the origin.
B) (-5, 3)
C) (5, 3)
A) (3, -5)
Click here for Instructions
Game Over: Mission Failed
The Tempest Vanguard has been destoryed! SouthFort is Lost!
Retry
Exit
Target Acquired! Fire!
(2, 6)
(2, 6)
Locate the target by solving the problem Reflect point F(2, -6) over the x-axis. Rule: Over x-axis: (x, y) → (x, -y)
B) (-2, 6)
C) (6, 2)
A) (2, 6)
Click here for Instructions
Game Over: Mission Failed
The Tempest Vanguard has been destoryed! SouthFort is Lost!
Retry
Exit
Target Acquired! Fire!
(2, 6)
(2, 6)
Locate the target by solving the problem RReflect point G(-3, 5) over the y-axis. Rule: Over x-axis: (x, y) → (x, -y)
C) (-3, -5)
B) (3, 5)
A) (-5, 3)
Click here for Instructions
Game Over: Mission Failed
The Tempest Vanguard has been destoryed! SouthFort is Lost!
Retry
Exit
Target Acquired! Fire!
(2, 6)
“So… you dare send your pathetic fleet against me? My soldiers… my perfect warriors… you slaughtered them without mercy. Now, your city… your people… will pay the ultimate price!”
Lord Thalassor:
Next
Defeat Lord Thalassor
Final Boss Fight
"This is it—the final assault. The Tempest Vanguard stands between South Fort and total destruction. Captain Leopold, Shaylee, Jason, and the crew are ready, but the fate of the city rests in your hands." "Lord Thalassor, overlord of the deep, has unleashed his monstrous army. To defeat him, you must solve the last five transformation problems correctly. Each problem is a strike against his forces. You have 20 minutes—accuracy and speed will decide the outcome." "This is the moment where heroes are forged. Focus, apply everything you’ve learned, and bring victory to South Fort!"
Begin
20:00
Rule: (x,y)→ (x + 4, y – 7).
20:00
Rule: 90° Counterclockwise Rotation → (x, y) → (-y, x)
20:00
Rule: Reflection over y = x → (x, y) → (y, x)
20:00
Rule: Reflection over y = x → (x, y) → (y, x)
20:00
Rule: Reflection over y = x → (x, y) → (y, x)
Congratulations
You successfully help The Tempest Vanguard defeat Lord Thalassor! SouthFort is Safe Once again!
Exit
Thank you for playing!
Game Over: Mission Failed
The Tempest Vanguard has been destoryed! SouthFort is Lost!
Retry
Exit