March 17th, 20XX

Mario, Sonic, and their band of unlikely companions—Luigi, Yoshi, Tails, Knuckles, and Amy. They were not in their usual carefree states, frolicking through the fields or racing along the tracks; they were huddled together, their expressions a blend of determination and wariness. Their eyes, usually filled with joy and mischief, were now narrowed with intent.

Mario, the hero of the Mushroom Kingdom, looked over the group, his eyes lingering on each face. His mustache twitched as he took in their readiness. Sonic, the speedy hedgehog from a far-off land, leaned against a tree, his quills fluttering in the breeze. His usual smugness was replaced by a steely resolve that spoke volumes of the gravity of their situation. The air around them buzzed with unspoken words and unseen tension.

Yoshi, ever loyal to Mario, nodded solemnly. His long, sticky tongue flicked out, tasting the metallic scent of the enemy's approach. The ground rumbled beneath their feet as the sound of marching boots and whirring gears grew louder. Tails, his two tails swishing anxiously behind him, checked his gear one last time. Knuckles, stoic and strong, flexed his powerful fists, a silent promise to protect their comrades. Amy, her cheeks a rosy pink, gripped her hammer tightly, her eyes glinting with the fierce determination of a warrior.

The sun dipped low in the sky, casting long shadows across the battlefield. Above, the once-beautiful blue sky was marred with the dark silhouettes of Bowser's and Dr. Robotnik's flying machines. The alliance between the two once-foes was a stark reminder of the seriousness of their mission. The heroes knew that victory was not guaranteed, but they were fueled by the knowledge that their worlds depended on their unity.

With a deep breath, Mario raised his fist in the air. The signal to charge. The ground trembled as they sprinted into battle. Yoshi's roar echoed through the valley, a war cry that sent chills down the spines of their enemies. Sonic's figure blurred as he sped off, leaving a trail of blue light in his wake. Tails took to the skies, peppering the machines with explosives. Knuckles and Amy charged ahead, a wall of strength and speed. And behind them, Luigi, with a quiver of fear in his voice, unleashed a barrage of ghostly fireballs, lighting up the night with an eerie glow.

The clang of metal on metal, the roar of flaming breath, and the cries of battle filled the air. The heroes' hearts pounded in their chests as they faced the enemy head-on. This was no longer a game; it was a fight for their very existence. The fate of their worlds rested in their hands—and they were ready to give it their all.

Mario leapfrogged over a barrage of Bullet Bills, his eyes locked on a group of Eggbots targeting innocent bystanders. He somersaulted in mid-air and landed with a thud, ground-pounding the robots into a pile of scrap metal. His allies cheered, a brief respite in the chaos. Sonic streaked by, a blur of blue leaving a sonic boom in his wake, sending a shockwave that toppled a row of charging Koopas. The sight of their foes being defeated brought a glimmer of hope to their eyes.

Tails zipped through the sky, dodging enemy fire, as he made his way to the control center of one of Dr. Robotnik's most feared machines—the Death Egg. He knew that taking it down could change the tide of the battle. With a flick of his wrist, he sent a homing missile spiraling into the cockpit, causing the massive contraption to plummet towards the earth below, crushing a squadron of Eggbots under its weight. The explosion sent a shockwave across the battlefield, knocking back even the most steadfast of fighters.

Amidst the destruction, Amy's graceful swings of her hammer sent enemies flying in all directions. Her swiftness and precision were a stark contrast to the brute force displayed by Knuckles. Each blow from the echidna sent tremors through the ground, knocking down multiple Koopas at once. The two of them fought in perfect harmony, a testament to their friendship and shared resolve.

As the night grew darker, the heroes pushed forward, inch by inch, against the relentless tide of Bowser and Dr. Robotnik's allied forces. Their spirits remained unbroken, bolstered by the camaraderie and trust that had grown between them. They had faced insurmountable odds before, but this time, they had each other's backs.

Bowser, noticing the shift in momentum, bellowed an angry command. The ground below Mario and his allies began to shake as a new, colossal enemy emerged from the depths—a Mecha-Bowser. Its eyes glowed with malicious intent as it took its first steps towards the heroes.

Mario's heart skipped a beat as he took in the monstrous sight. But instead of fear, he felt a surge of adrenaline. This was it—the moment they had all been preparing for. He turned to Sonic, his voice steady. "We've got this, buddy."

Sonic grinned, his eyes alight with excitement. "Let's show them what we're made of!"

The two iconic individuals-turned-friends took off towards the mechanical behemoth, their hearts racing as they prepared to face the ultimate challenge. The battle was far from over, but with every step, they grew more determined to save their worlds from the tyrannical grasp of their foes.

Mecha-Bowser's first attack was swift and fierce. Its massive metal fist swiped through the air, aiming for Mario and Sonic. Sonic dashed aside at the last second, leaving a blue streak in his wake, while Mario gracefully somersaulted under the swing, his eyes never leaving the monster's glowing red eyes. The Mecha-Bowser roared, a mix of Bowser's fiery breath and the whine of Eggman's machinery, as it adjusted its aim.

Tails, seeing the danger, swooped down from the sky, firing a barrage of rockets at the beast's armored underbelly. The explosions sent sparks flying, but the Mecha-Bowser barely flinched. Knuckles took this opportunity to charge in, his fists a blur as he pummeled the robot's legs, trying to weaken its stance. Amy, not to be outdone, spun her hammer in a tornado of power, slamming into the robot's side with a resounding clang.

Yet, Mecha-Bowser remained steadfast, countering with a flurry of fireballs and missiles. The heroes dodged and weaved, their reflexes and teamwork on full display. As the battle raged on, the tension grew thicker than the smoke rising from the cratered earth.

Mario, knowing that they needed to find the weak spot, leaped onto Mecha-Bowser's back, using his agility to cling to the moving target. He called out to Yoshi, who, understanding the plan, shot a stream of sticky eggs at the robot's back, giving Mario a chance to dig in his feet and climb higher.

At the apex of the robot's head, Mario spotted a glowing red button—a beacon of hope. He took a deep breath, ignoring the roaring engines and clanging metal all around him. This was it—his moment to shine. With a mighty leap, he slammed his fist down onto the button, sending a jolt of electricity through the robot's frame.

The Mecha-Bowser stumbled, its systems momentarily disrupted. Sonic took this opening and homed in, delivering a powerful spin attack to the robot's neck joint. The head lurched back, and with a deafening screech, the robot collapsed, sending tremors through the ground.

Their victory over Mecha-Bowser was a symbol of their unity and power. The remaining enemy forces wavered, their morale shattered by the sight of their unstoppable leaders. Bowser and Dr. Robotnik, watching from their looming airship, knew that the tide of battle had turned. They had underestimated the strength of friendship and the power of unity.

The heroes gathered around the fallen robot, panting and bruised but unbroken. They exchanged glances of triumph and relief, knowing that they had just faced the most formidable enemy they had ever encountered—together.

But the battle was not over. The sky above them grew darker, as more enemy ships approached. The heroes had only a brief moment to catch their breath before the next phase of the war began. Yet, as they looked at one another, a silent understanding passed between them. They were ready. They were strong. And together, they could face anything.

The ships descended, raining down a storm of fire and steel upon the battlefield. The heroes dispersed, each taking on a section of the invaders. Yoshi shielded Mario with his eggshell armor, as they scaled the nearest ship. The plucky plumber, with a fiery resolve, tossed fireballs into the cockpit, sending it spiraling out of control. Meanwhile, Sonic and Tails weaved through the air, dodging laser blasts as they targeted the ships' engines, disabling them in a spectacle of speed and precision.

Amidst the chaos, Amy and Knuckles found themselves face-to-face with a squadron of elite Eggbots. Knuckles' fists hammered into their gleaming exteriors, sending them flying, while Amy's hammer sang through the air, delivering crushing blows that sent sparks flying. The duo fought in perfect harmony, their movements a dance of destruction that left no robot untouched.

On the ground, Luigi faced his fears, his eyes glowing with spectral energy. He conjured a whirlwind of ghosts that swarmed the enemy lines, causing confusion and terror. His usually timid demeanor was replaced by the fierce determination of a warrior.

The allied forces of Bowser and Dr. Robotnik had not anticipated this level of resistance. Their once-solid line began to waver and break as the heroes' counterattack grew fiercer. The air was thick with the smell of burning metal and the cries of the defeated, but the heroes pushed on, their hearts beating in unison with the hope of victory.

As the enemy retreated, the heroes gathered once more at the center of the battlefield, surrounded by the wreckage of their foes. They were bruised, their clothes tattered, but their spirits soared. They had proven that no matter the challenge, they could stand as one.

Bowser and Dr. Robotnik, their faces grim, knew they had underestimated the bonds that had formed between these heroes. Yet, the glint in their eyes spoke of a plot not fully realized. This was not the end, merely a setback. They would regroup, they would rebuild, and they would strike again.

The alliance of Mario, Sonic, and their friends had not just survived—they had thrived. The bond they shared, forged in the fires of battle, would be the key to facing whatever challenges the future held. With newfound respect and camaraderie, they looked to the horizon, ready to tackle the next adventure that awaited them.

Math Notes: (filled with academic musings)

Tangent Line

It is a line that just touches a graph of a single point.

Finding the Slope of a Line

How many points do you need normally to find the slope of a line?

Answer: 2

Limit

If you keep picking points that get closer and closer, it's really hard to get to see that original point that you got, that was right there.

But if you keep getting closer and closer and closer, won't you eventually get so close that it's almost the same slope?

So what comes with this is the idea of limit.

The idea of a limit is about how close you can get to a point without actually hitting that same point.

Points Q and P

If you let Q get really close to P, then the secant will be identical to the tangent.

Graphing a Function

There is a point right at the vertex, at the origin.

Points: (1, 1); (-1, 1); (2, 4); (-2, 4)

Point P is at (1, 1)

The goal is to find the equation of the tangent line of the curve, f(x) = x² at (1, 1).

Note: You might see this kind of question on your homework. You are given a graph and you will be told to find the equation of the tangent line at a point.

You do not know what Q is but call it (x, y) for now.

Can you represent y in terms of x?

Answer: Yes

Remember, f(x) can easily be replaced with y.

This works for any Q you choose.

Finding the Slope

Slope formula: m = (y2 - y1) / (x2 - x1)

Assume that Q is at (2, 4), find the slope.

Let x = 2

m = (2² - 1) / (2 - 1)

m = 3 / 1

m = 3

Let x = 1.5

m = (1.5² - 1) / (1.5 - 1)

m = 2.5

Let x = 1.1

m = (1.1² - 1) / (1.1 - 1)

m = 0.21 / 0.1

m = 2.1

Let x = 1.01

m = (1.01² - 1) / (1.01 - 1)

m = (1.0201 - 1) / 0.01)

m = 0.0201 / 0.01

m = 2.01

Let x = 1.001

m = (1.001² - 1) / (1.001 - 1)

m = (1.002001 - 1) / 0.001

m = 0.002001 / 0.001

m = 2.001

Let x = 1

What will happen?

m = (1² - 1) / (1 - 1)

m = 1 / 0

Answer: Undefined

Building

● The building is 80 meter tall

● The pumpkin will be used as an example of throwing something that smacks out the big, huge, and tall building.

○ 80 is the time equals zero. Pumpkin has not dropped yet.

● The pumpkin just hit the ground in four seconds at the 80-meter building.

Asymptotes

o continually approaches a given curve but does not meet any finite distance.

o A value that you get closer, and closer.

o A horizontal, vertical, or slanted line that approaches a graph but never touches.

· 4 Basic rules of Asymptotes

o Function of y = f(x)

· Horizontal Asymptotes y = k where x→∞ or x→-∞

· Vertical Asymptotes x = k where y→∞ or y→-∞

· Slant Asymptotes form y = mx + b where m ≠ o.