#precalculus formulas

[er-cryptid]

Horizontal Translation of Logarithmic Functions

-- Formulae: -> g(x) = logb (x + c) -> g(x) = logb (x - c)

-- g(x) = logb (x + c) shifts the graph of f(x) = logb (x) to the left c units and moves the vertical asymptote to x = -c

-- g(x) = logb (x - c) shifts the graph of f(x) = logb (x) to the right c units and moves the vertical asymptote to x = c

.

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[amalgamgooze]

"easier" math has become more difficult than "difficult" math

Had to take a math placement exam for college today.

It was all high school stuff--things I'd already done.

Now tell me,

why was it harder to do that stuff than the abstract algebra stuff I'm studying?

…

I'm serious.

Maybe it's a familiarity thing.

Like, nobody should ever have to memorize all the trig identities.

sin2x + cos2x = 1, that's an acceptable one to memorize.

Super short, simple, and widely applicable.

But the bullshit like sin(2x) = sin(x)cos(x)?

Or sin(x/2) = sqrt((1 - cos(x)) / 2 ) ?

What? The? Fuck?

Is this actually stuff I studied two years ago?

This is formula sheet bullshit.

Not stuff you test for to gauge math proficiency.

And not without a calculator either.

For fucks sake--discrete math and modern algebra without a calculator is easier than precalculus without a calculator!

…

I can see why some people hate math.

That shit's just not fair.

Even if you're using math professionally, you're not going to have that bullshit restriction of "you need to know everything off the top of your head".

…

If you couldn't tell, I'm not a fan of all of math.

There's some areas that I don't feel are as important to understand as the rest.

Like conic sections.

Or eccentricity.

…don't get me wrong. Geometry is pretty cool sometimes.

But when my job is to literally solve the equation for a specific point on an ellipse, I'm just left thinking, "why bother?".

…

Calculus was cooler than that shit.

Vector calculus was cooler than that shit.

Precalculus was hell.

I offer my condolences to you poor souls who've yet to take Precalc.

There's cooler math beyond precalc.

Yes, it's more incomprehensible, but it's enjoyably incomprehensible.

It's supposed to be abstract.

It--

Physics wasn't as infuriating as that test!

I-

I don't know how much more I can prove I love math than to write a whole monologue about me descending into insanity over my love for math!

But this fuckin' test!

It just- wow.

I'm sorry. I'm still kinda worked up about it.

I'm not writing about it in a very unbiased lens.

(I don't need to. I'm not claiming to preach universal truth in this post.)

…

Again.

I suppose I want to emphasize that precalculus and later "college algebra" classes are kinda infuriating and not very fun.

There's fascinating fields like calculus, linear algebra, and abstract algebra beyond that rut.

…

But again. Math's not for everyone. So I suppose you can also hate all of it. I won't judge.

[authorjond369]

❖ A Way to remember the Entire Unit Circle for Trigonometry ❖

🎯 Memorize the Entire Unit Circle for Trigonometry with This Easy Trick! 🎯 In this video, I share the best way to remember the unit circle for trigonometry, a skill that’s essential for anyone taking trigonometry, precalculus, or calculus. I start by explaining the formula for the circumference of a circle (C = 2πr) and how we can use it to measure angles around the circle. After filling in all…

[math-journal2]

Algebra 2 - here - Book Ed1

Precalculus 1

Book: Precalculus 2e, Abramson, Jay, OpenStax, 2021

Transformations / Trig Review + 7.1

Unit 1

Review / Algebra 2 Cheat Sheet

Parent Function: D & R

Set Builder Notation

Piecewise Function

^Tough Example I x I

Unit 2

Function Operations

Decomposing

Composites

Inverse Function

Review

Unit 3 [Book Notes]

Parabola Basics

Switching Between Formats

Quadratic Word Problem

Imaginary numbers

#imaginary review

Complex Conjugate Solving

Complex Plane / Solving i^#

Power Function (3rd degree +)

Graphing Polynomials (pg1)

*Power function is a format of polynomials

Unit 3 [Class Notes]

Imaginary / Quadratic Formula

Polynomial Functions (include end behavior and D/R)

Note: there weren’t any word problems on the quizz. Also solving for a (sub n) was a significant part. Plus, make sure to write x-intercepts as ordered pairs.

Unit 4 [Book Notes]

Dividing Polynomials

Long Division

Synthetic Division

Word Problem

Finding Zeros

Remainder Theorem

Basic poly division

Factor Theorem

Poly division, factor, solve for x

Rational Zero Theorem

Find possible x-intercepts or zeros

Finding Complex Zeros

^ same, a>0

Unit 4 [Class Notes]

Long D v Synth D

Eq —> Zero, Zero —> Eq

Possible Zeros, p/q

Narrow Possible Zeros

Graphing always option

Complete Factorization

Sq Root Zero —> Eq, Vertex Form

Terminology

Complex Zero Dump

MID TERM REVIEW

Unit 5

1/x

Rational Functions

Graphing

Edit: D > N —> y=0 h.a.

D < N —> y = slant

D = N — y= a/b

Exponent

Decay and Growth

Graphing

Extra

Application Problems

Compound Interest v Continuous Growth

Logs

Common v Natural

Algebra 2 Solutions

Graphing Logs

UNIT REVIEW

*note: no interpretting graphs on quiz

.

Unit 6

Log/Exponent Rules

Expanding Logs

Ways to Solve

Application Problems

Graph —> Eq

No review

Hard part: solving exponents of different bases / richter scale application problem / decibel / newton’s law of cooling

.

Unit 7

Basic Circle

Circle Problems

NonStandard —> Standard

Ellipses

Parabola

*my book notes are a mess, so yeah… ignore them. Only listing classnotes on MP

UNIT REVIEW

[tutoringmaphy01]

Navigating AP Precalculus Tutor: Common Student Challenges and Solutions | Tutoring Maphy

Navigating AP Precalculus: Overcoming Common Student Hurdles with Expert Guidance

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[isabelledoesmath]

Why I'm learning Blender

In mathematics, we often work in detail with shapes and lines. We analyze slopes, intercepts, and confined areas in order to create a mathematical picture. Because of this, from the second we start to teach mathematics, we use images and graphs. To show that the sum of two and three is five, we might draw a representation of the numbers as cookies or apples. To show that the slope of a parabola at its vertex is zero, we might plot some points and connect the dots.

As a senior in college studying mathematics, I have found myself drawing countless graphs and diagrams of the shapes and formulas I have been working on. Even when dealing with abstract concepts, I use a combination of colors and points to explain what is happening. When tutoring precalculus, my first method of explaining a concept is always a drawing.

Bettina Rosken and Katrin Rolka did research through the University of Duisburg-Essen, Germany, that shows the importance of visualization in mathematics regarding calculus. They state that “visualization allows for reducing complexity when dealing with a multitude of information,” (Bettina and Rolka 458). Being able to see a graph and label every important point allows students to keep their work clear and concise. The authors, however, make a point that a drawing or representation of the mathematical concept only does so much; a mathematical model cannot stand by itself without an understanding of what it is and its rules.

While math is best learned in a classroom, Blender, a 3D modeling program, can be one of those tools that can help assist in understanding concepts outside of class time. As much time as I can spend on drawing a graph of a 3D shape like a donut, more commonly known in the math world as a torus, if I can instead learn a program that allows me to hold two keys down and click through some interface to place a torus into a world, then I can push my understanding of mathematics even further in less time.

And that is what I have done. So far, I have spent time learning how to place shapes, edit the world around them, and their materials. While I will not need to necessarily be able to create a sphere that is reflective in order to understand a math problem, the more I understand the entire interface and abilities of Blender, the better I will be able to use it. Understanding both what to use and what not to will allow me in the future to focus further on the mathematical side of this program and how I could have used it in the courses I have taken. This week, the final project I made was a lighthouse on a rock with some small houses around it. I learned how to sculpt different shapes, create different light sources, and assign different colors to different faces. All of this was thanks to Grant Abbitt’s course on Udemy with GameDev, titled Complete Blender Creator: Learn 3D Modeling for Beginners.

Rösken, Bettina, and Katrin Rolka. "A picture is worth a 1000 words–the role of visualization in mathematics learning." Proceedings 30th conference of the International Group for the Psychology of mathematics education. Vol. 4. Prague: Charles University, 2006.

[rumplefuckingstiltzkin]

I have always been naturally gifted at maths. When in elementary school certain family circumstances prevented me from actually attending math class and everything that came after it on a daily basis, this resulted in me scoring a 79.5% on a placement exam for middle school math classes. I still remember being frustrated but being able to figure out the answers well enough. Almost well enough. I needed an 80% to be put into the class where I belonged. (Did I really belong there? We'll never know.)

As dominoes fall, I was not challenged enough in middle school maths, found tedious the busywork designed to ingrain math I already knew, and stopped torturing myself with it. This was the start to a disillusionment with all homework, but that's another story. I was moved from regular maths into remedial maths, which was worse. I still remember trying to teach myself from my old friends' textbooks, as they were doing fun things like trigonometry and precalculus, while the other boys at my table were preoccupied by exchanging masturbation tips. (No pun intended.)

When a meeting was called to asses why I was doing so terribly on paper in maths but testing excellently, I presented the fact that I, without help, had discovered that there is a pattern to finding the next square in a sequence. (It's that (x+1)^2=x^2+2x+1.) No one taught me this. I had been moved out of algebra.

Similar misunderstandings and treatment continued to dog me throughout high school. When I discovered the trick to finding the next cube in a series, I had just graduated. But something about it bugged me. Surely there must be a way to predict the formulae for discovering the next Nth power in a series. Spoiler alert, there is! I was a second semester college freshman taking Calculus 131 with a professor who wasn't actually a teacher by any stretch when I figured out that I was right, and I rediscovered Pascal's triangle.

Pascal's triangle is an organization of figures for which each row is the coefficients for the equations I had been pondering like an orb for a decade. It's free. You can look it up, they're there, they have many other uses. I independently brute-forced Pascal's Triangle. In another place and time I would be a genius. Instead, I realized at seventeen (before voting age!) that I had been failed by the system all those years ago in the sixth grade, when my math teacher could have just said "check out this triangle" and brought me up to speed with the rest of the learned 21st century world.

Now, are my teachers to blame for this? No. Not really. Overworked, understaffed, underpaid, they were not to blame for not having the foresight to predict exactly what information could have saved me. Hindsight, only hindsight, is 20/20. Am I to blame for not having the language as a child to express what I meant, or to correctly ask what I needed to know? Again, no. I was victimized by the system. Thousands, millions worldwide like me are managed rather than encouraged because it reflects well on the school, the state, the country.

It leads to questions like "Where would we be as a populace if we were somehow able to efficiently reach every child with what they need to excel?" and "In what other areas are we failing our children with incomplete education?" I don't have these answers. "Where would I be if I had gotten what I needed?" I don't have the answer to that, either. I still teach myself calculus between work, sleep, and chores.

I suppose I'm only telling this story because you asked, OP. The most screwed up thing isn't that it happened to me, but that it keeps happening, everywhere. Still.

I'll close with just a little something to think about:

"I am, somehow, less interested in the weight and convolutions of Einstein's brain than in the near certainty that people of equal talent have lived and died in cotton fields and sweatshops." - Stephen Jay Gould, The Panda's Thumb: More Reflections in Natural History

Everyone give me the most screwed up thing that has ever happened to you in the american public school system

[gracethestudent16]

Not sure if anyone wants or needs these, but here are the notes from things I’m using the most in Pre-Calc right now. :)

[ggsstudies]

4/16/19

my math folder has gotten HUGE because of all the papers we get in class plus all my scratch paper from homework sheets

it’s one of my thickest folders and tbh it’s kinda annoying but oh well, i’m dealing with it

do you like math??

xo- gg

[er-cryptid]

Patreon | Ko-fi

[sol1loqu1st]

i cant believe this. math lecture is really gonna spend 25 minutes talking about one (1) formula

[atlfics]

Worth the Wait- All Time Low Fanfiction

Jack sneaks out of school to take care of his long time crush, Alex. High School AU. Word Count: 4733

More notes at the end. 

“Jack, have you seen Alex? He usually gets here before you do, right?” Zack asked Jack as he took a seat on the bench in the quad they always occupied before school officially started for the day.

“I haven’t seen him, his car wasn’t in the lot, either. I hope he’s okay,” Jack replied, his sudden concern making Rian laugh.

“Dude, when are you going to man up and tell Alex that you like him? You’ve had a monster crush on him for, what, over a year now, right?” Rian asked, making Jack shove him as he held a finger in front of his mouth with wide eyes.

“Will you keep it down? We don’t need to let the whole school know about that, it’ll increase the likelihood of it getting back to Alex!” Jack exclaimed, a serious look on his face.

“I mean, wouldn’t that kind of be a good thing? Like, you’re clearly too scared to make the first move, maybe he should hear it through the grapevine and do something first,” Zack pointed out, making Jack roll his eyes.

“Zack’s got a point, dude. You’ve been carrying around this secret crush for the longest time now, you really should say something so you don’t have to keep torturing yourself about it,” Rian added.

“No, I don’t want to ruin our friendship if he doesn’t feel the same way,” Jack sternly stated.

“Well, he’s bi, and he’s always all over you, so I think him liking you back is a reasonable assumption,” Rian continued.

“It’s not worth the risk,” Jack said back.

“So, is it worth torturing yourself, then?” Rian countered.

“Why are we even talking about this? How did you guys ganging up on me even become a thing?”

“We’re not trying to gang up on you, we’re trying to help you, dude. Also, since he isn’t here, now is a great time to talk about Alex and that huge crush you have on him,” Zack smugly replied.

“You motherf-“ Jack started, before getting cut off by the five minute warning bell.

“Damn, I had a few more one-liners planned! Oh well, they’ll have to wait until lunch, assuming Alex doesn’t show up,” Rian said, laughing as Jack rolled his eyes yet again.

“Fuck you guys, I’ll see you in a few hours,” Jack said, standing up and putting his backpack on.

“Sounds good, lover boy,” Zack said, laughing with Rian as they walked off to their first shared class.

Jack went in the opposite direction to get to the precalculus class that he normally had with Alex. He really hoped that Alex would show up right on time, having to suffer through it alone was torture; Alex really did make everything better.

When he got to the classroom, most of the other students were already in their seats, talking about the previous night’s homework. Jack took his seat, noticing that Alex’s next to him was empty. After waiting a couple more minutes, Jack decided to text his friend and see what was going on.

Jack: Hey dude, you coming to school today?

Jack stared at his screen, waiting for a reply. Not much later, the teacher walked in and started class, so Jack put his phone at the top right corner of his desk, and pulled out his notebook. The lesson was on trigonometry, but Jack couldn’t have cared less about the different formulas, his mind was stuck on Alex. He of course wondered why Alex wasn’t in class, but he also thought about what Rian and Zack had said in the quad earlier.

He’d known Alex for four years now, but for the past year, he’d been head over heels in love with him. Jack wasn’t sure how his crush developed, but every time he was around Alex, he just felt happier and safe; no other friend had ever made him feel like this before. He had wanted to tell Alex about how he felt, but the possibility of ruining their friendship scared Jack, if Alex didn’t feel the same way and learned that Jack had a massive crush on him, would it scare him off? Would everything between the two of them become awkward and changed forever? Jack didn’t know, but these questions were enough to scare him into saying nothing and pretending his crush didn’t exist.

He hadn’t told Rian and Zack about his feelings, they’d figured it out on their own, and Jack had made them promise that they’d never tell Alex, which they agreed to, knowing that if Alex ever did find out, it should be from Jack. His thoughts shifted from his crush on Alex to wondering about where Alex was, he hoped that Alex was okay, wherever he was, and he hoped that he’d get a text back soon.

About halfway through first period, Jack’s phone vibrated, and he did his best to discreetly grab it and look at it without his teacher noticing.

Alex: No, I’m like really sick :/

Jack instantly felt bad, he knew that whenever Alex was under the weather, it was never something simple, it was always miserable.

Jack: Sorry to hear that man :( are you okay?

Alex: Not really, I don’t think I’ve ever been this sick. My parents are still out of town too which sucks

Jack: Maybe I can sneak out of here at lunch and come hang with you

Alex: You don’t have to commit truancy for me lol, plus I’m disgusting and look awful rn, idk if you’d really want to be around me

Jack: I bet you still look great, you motherfucker. Get some rest for now and I’ll text closer to lunch

Alex: You’re too good to me, JB

Jack smiled as he turned his phone off and put it back on the corner of his desk. He knew he was definitely going to be sneaking out at lunch, and he decided that he’d stop at the store on his way to Alex’s to get a few things that would hopefully cheer him up some.

If he could have everything his way, he’d get up and walk out of the school building right now, but he was pretty confident that his math teacher wouldn’t take too kindly to him doing that. For the remainder of first period, Jack genuinely tried his hardest to pay attention to the lesson, but he couldn’t get his mind off Alex; he was concerned that he was seemingly so sick and all alone, but he knew that obsessing over this wouldn’t make lunch time arrive any faster.

Eventually, first period finally ended and second began, leading to third and fourth, too. Throughout all of his classes, Jack’s mind wandered back to Alex, even though he was probably miserable, Jack was still beyond excited to spend the day with him. He really enjoyed doing things for others, and he thought that this would be the perfect opportunity to do just that for Alex.

The only concern he had was that Alex’s vulnerability from being sick might rub off on him and make him feel vulnerable enough to confess his feelings. Maybe Rian and Zack were right, maybe he should just tell Alex how he felt, there was theoretically a chance that Alex felt the same way. As much as Jack wanted to arrive at Alex’s and confess everything so they could spend the rest of the day snuggled up, he knew he shouldn’t do that. Of course, he wasn’t saying he wouldn’t confess his feelings, but he had to be careful and only do so if it felt right.

After three long hours, it was finally lunch time, and Jack figured that he should head over to their usual lunch table to let Rian and Zack know what he was doing before sneaking out. When he got to the table, Rian and Zack were already there, talking about their upcoming physics exam. Jack figured he had some sort of distressed look on his face, because the two of them immediately stopped talking and looked concerned once they saw Jack.

“Guys, I’m going to do something that I might need your help covering for,” Jack said, taking a seat across from his friends.




“You’re going over to Alex’s, aren’t you?” Zack asked, laughing some.

“How did you know?” Jack questioned, a tone of genuine surprise in his voice.

“Jack, what else would you be so passionate about after four classes? Surely it isn’t your homework or upcoming exams,” Rian continued, making Jack roll his eyes.

“Okay, yes, you’re right, I’m going over to Alex’s. He texted with me some during first period, apparently he’s, like, super sick and his parents are out of town, so I want to help if I can,” Jack explained.

“Well, are you finally going to confess your feelings? Obviously you’re trying to do a nice thing, but we’re dudes, we don’t, like, take care of each other like that,” Zack pointed out, laughing some.

“I mean, I want to, I want to walk in and confess how I feel so we can spend the rest of the day cuddled up on the couch or his bed or whatever, but I’m still scared to. Also, I don’t want to make him uncomfortable since he’s already not feeling great,” Jack stated, as Rian and Zack nodded along.

“I guess you have a point, man. Personally, I think if there’s a moment that feels right, you should just do it. Like, it could really make his day, you never know. Plus, I think he likes you back, you’ve seen how he looks at you, right?” Rian said, taking a bite of his lunch.

“Don’t go putting ideas in my head, if that isn’t true, I’m going to be disappointed and mad at you, dude,” Jack warned, trying to sound serious.

“I agree with Rian, I think he likes you, too. You guys just have this, like, vibe, that I’ve only seen in people who like each other. Just go through with your plan, go to his place and spend some time with him, I’m sure he’ll appreciate the company regardless,” Zack assured, making a small smile appear on Jack’s face.

“Thanks, I appreciate your all’s advice; that’s what it was supposed to be, right?” Jack joked, laughing some.

“Well, some advice and some jokes, that might be more accurate. Hey, there’s only ten minutes left of lunch, if you want to get out of here, you’d better go,” Zack pointed out, looking at the time on his phone.

“Shit, you’re right, I didn’t realize it had gotten so late. Well, I guess if anything changes, like our relationship status, I’ll let you know,” Jack said, getting up and grabbing his car keys out of his pocket.

“Sounds good, just try to have a good time, dude. Regardless of what happens with the relationship shit, he’ll appreciate that you’re spending time with him,” Rian assured, catching Jack off guard.

“Woah, Rian being supportive and not being a sarcastic asshole? Are you feeling sick, too?” Jack questioned, earning a laugh from Zack and an eye roll from Rian.

“Yes, it’s support, now go before I start cracking jokes and making you feel more nervous,” Rian said in a mock-stern tone.

Jack smiled at his friends one last time before sneaking over to the back door of the cafeteria, the one that conveniently led to the student parking lot. After assuring that no staff were able to see him, Jack opened the door, then quickly bolted out and in the direction of his car. Luckily, no one had noticed him leave, and he breathed a sigh of relief when he got in his car and shut the door.

Jack quickly started up his car then drove off campus, in the direction of the nearby Target, he hoped that it would have everything he’d need for Alex. As he drove, he listened to his favorite songs and continued thinking about Alex, and what Zack and Rian had been telling him at lunch. The more he thought about it, the more Jack wanted to confess his feelings today, he knew it was risky, but he also knew that if Alex felt the same way, it would be so worth it.

His Target trip was quick, he’d grabbed Alex a bouquet of flowers to spruce up his room, a couple cans of his favorite soup, and a cute teddy bear that he thought Alex would get a kick out of. Once he had everything, he drove in the direction of Alex’s house, which luckily wasn’t too far away from where the Target was. He felt his nerves start to kick in more, and tried taking a deep breath to alleviate them to an extent. As he drove, Jack tried to eliminate any type of expectations he had for how this afternoon would go, he knew that would likely just result in him feeling hurt or disappointed. He just had to remember his primary goal, make sure Alex was okay, and do whatever he could to help him feel better.

Eventually, Jack got to Alex’s driveway and parked behind his car. He grabbed the things he’d gotten at Target, then made his way to the front door. Jack realized he’d never told Alex he was here, or even on the way, and the thought of making Alex get up from wherever he was made him feel bad. Deciding to try his luck before calling Alex, Jack twisted the knob of the door, and was very surprised to find it unlocked. He realized that Alex had probably gotten up and unlocked it when Jack had originally texted saying he’d come over, then he wouldn’t have to get up again later.

Jack stepped inside and went into the kitchen, he set the soup cans on the counter, then found a vase under the sink, and placed the flowers in it. After a couple deep breaths, Jack made his way upstairs and over to where Alex’s bedroom was. He could hear loud coughs coming from the other side, making him feel even more sympathy for his friend.

Once the coughing stopped, Jack lightly knocked on the door before walking into Alex’s room.

“Hey, how’re you doing, Alex?” Jack asked, making his way over to Alex’s bed.

The majority of the room wasn’t too messy, but the bed and Alex’s bedside table looked awful. There were used tissues everywhere, a pot from the kitchen that Jack figured Alex had brought up for an emergency, and different cups and medicine bottles everywhere. Alex was wrapped up in about three blankets and still shivering as he watched something on his TV.

“Oh, hey, Jack. I feel really awful, to be honest,” Alex replied, his voice raspy and congested.

“I’m sorry to hear that, you look pretty miserable. I got you some stuff that’ll hopefully make you feel better, though! There are two cans of your favorite soup in the kitchen, and I grabbed some flowers to brighten up your room! Oh, I also got you this,” Jack replied, setting the flowers down before reaching into the plastic bag to get out the teddy bear.

“This is cute! Thanks, Jack, this means a lot, I-“

Alex cut himself off by starting to cough again, forcing himself to sit up as he put his elbow to his mouth. Jack quickly took a seat on the bed next to Alex and rubbed his back as he coughed. Having his hand on Alex’s back made him realize how warm Alex felt, and he knew that having all of these blankets on him would be bad news in the long run, it would just make Alex’s fever worse.

Eventually, Alex finally calmed himself down and Jack helped ease him back into the spot he’d been laying in.

“Sorry, I didn’t realize that was coming until it hit me,”

“No need to apologize, it’s okay. Hey, you’re going to hate me for this, but I think you need to ditch at least one of these three blankets,”

“What? No way, I’m already freezing as it is!”

“Alex, you’re freezing because you have a fever; if you lower that fever some, your chills will go away. At the very least, they’ll be a lot more tolerable. Come on, let’s take the top one off for a bit and see if it does any good. Also, I’m taking your temperature,” Jack insisted, pulling a blanket off of a very resistant Alex.

“You sound like a mom,” Alex muttered, coughing some.

“Well, I’d rather be like a mom and help you than ignore it and have you get worse. Trust me, you’ll thank me later,”

Jack put the blanket on Alex’s desk chair so he wouldn’t grab it again, then walked into the bathroom to grab the thermometer from the medicine cabinet. When he got back to Alex’s room, Alex was blowing his nose.

“Okay, let’s see what this says. Have you taken your temperature at all since you got sick?”

Alex just shook his head as Jack stuck the thermometer under his mouth and shivered violently. Jack felt bad, he really didn’t want to take the blanket from his friend and make him feel more miserable, but Jack knew that he’d done the right thing. After a few seconds, the thermometer finally beeped, and Alex handed it to Jack, too out of it to read the number.

“101.7, damn, that’s kind of high, Alex. I’m glad we took that blanket off of you,” Jack said, setting the thermometer down as he sat on the side of the bed again.

“Jack, I’m fucking freezing,” Alex said with a sneeze.

“I know, but I promise when your fever goes down, you’ll start to feel better. Actually, I have an idea to lower it faster, one second!”

Jack quickly stood up then went back into the bathroom and grabbed a washcloth from the linen closet. He ran it under some cold water, rung out the excess water, then brought it back to Alex’s room.

“What’s that?” Alex asked, starting to tightly hug himself some to combat his chills.

“Put this on your forehead, I think it’ll get that fever lowered,”

Jack folded it some then went to place it on Alex’s head, only for him to flinch and cover his forehead with his hands.

“That thing is fucking cold, no way it’s going on my forehead,” Alex stated, doing his best to keep his hands on his forehead as Jack pulled on them.

“Alex, will you please just try it? You’ll feel so much better, I promise,”

“On one condition,”

“What would that be?” Jack asked, rolling his eyes.

“You have to lay here with me so I can take advantage of your body heat,”

“Okay fine, whatever it takes for you to stop being so stubborn,”

Jack finally placed the cloth on Alex’s forehead, making Alex shiver even more. After taking off his shoes, he laid down next to Alex and wrapped him into a hug.

“This is better,” Alex said, looking up at Jack.

Jack couldn’t believe it, Alex was insanely sick but somehow still managed to look perfect to him.

“Hey, are you good?” Alex eventually asked, pulling Jack from his thoughts.

“Oh yeah, um, I am, why do you ask?” Jack continued, hoping Alex wouldn’t think much of what had just happened.

“I mean, you were straight up staring at me for, like, a minute. I’m sick, not stupid, so I know that was kind of weird, even for you,”

“Oh, I guess I just zoned out, sorry,”

“Are you okay, dude?”

“I’m fine, but don’t worry about me, you’re the one who’s dying right now,”

“I mean, true, but I can tell something’s up, Jack. Actually, I’ve kind of been able to tell that for a while now,” Alex continued, sounding nervous but determined to make his point.

“What do you mean?” Jack asked, trying to play dumb.

“What do you mean what do I mean? Like, I’ve caught you staring at me multiple times lately, you seem a lot more invested in what I have to say, you brought me a whole care package just now. That’s more than just a friend would do, y’know?”

“Well, what’re you trying to say, Alex?”

“Jack, do you, um, see me as more than a friend?”

Jack’s heart immediately started beating ten times faster, and he felt his palms get sweaty. Alex was right, of course, but Jack had thought he’d done a great job at hiding how he felt from Alex, but apparently he hadn’t. As Jack searched for what to say next, he couldn’t help but wonder if Alex had figured out his secret because he’d been looking for it.

“Before I answer that, I have a question for you,” Jack replied, doing his best to keep his tone even.

“What? I already asked you something, you have to answer first, that’s how this works!” Alex exclaimed, coughing some.

“Not this time! Alex, are you asking me what you’re asking, because that’s what you’ve been looking for? Like, looking for clues about what you just asked?”

“Jack, I’m not answering until you do. I’ll sit here all day and wait, it’s not like I can do much else anyway,”

Despite how sick he was, Alex had a look of determination on his face, and Jack couldn’t deny it anymore. He’d promised himself, along with Zack and Rian, that if the opportunity to finally reveal his feelings to Alex presented itself, that he’d confess how he felt; maybe it was finally time to take a leap of faith.

“Okay, um, this is a bit nerve wracking for me, so be patient. Alex, yes, I do see you as more than a friend, and I have for a while now,” Jack finally replied, doing his best to take deep breaths and stay calm.

“Can you, like, give me more detail than that?” Alex asked, laughing some as he scooted closer to Jack.

“Sure, um, where do I start? Alex, I like you, like, I like-like you, and I have for about a year now. I don’t even know what made these feelings develop, like, we’ve been friends for four years, but sometime last year, a switch just flipped in me, and I woke up and realized that I was in love with you,” Jack confessed, feeling like a fifty pound weight had just been removed from his chest.

“You’ve liked me for that long, and this is just the first I’m hearing of it?”

“Yeah, but not because I wanted to keep it from you. Like, keeping this secret has been so fucking hard, like I’ll see you get lost in a song, or be super passionate about what you’re talking about, and it’s things like that that made me fall for you. I love literally everything about you, but I kept it to myself, because I didn’t want to ruin our friendship. Like, if you didn’t feel the same, I didn’t want shit to be awkward between us; you’re my best friend, I can’t just, like, lose you, y’know? I just didn’t want to put you in an uncomfortable position, but yeah, I’ve been head over heels for you for a long time now. Hey, you didn’t answer my question, were you looking for signs that I liked you?”

“Jack, I’m speechless, but in a good way, I promise. I guess you could say that I’ve been kind of hopeful that you’ve had these kinds of feelings, too. I like you too, Jack, like, I like-like you, and have for a decent amount of time now. I didn’t tell you for the same reasons, I didn’t want things to be weird if you didn’t feel the same way, I didn’t want to lose our friendship over weird feelings. This is so great, though, we both like each other like this! Where do we go from here?” Alex asked, before bursting into a coughing fit.

Jack quickly helped Alex sit up some in his bed, then rubbed his back until he finally calmed himself down. Once Alex could breathe again, Jack handed Alex a glass of water from his nightstand, and Alex took a sip before handing it back to Jack.

“Are you okay, Alex?” Jack gently asked, still rubbing Alex’s back some.

“Yeah, I’m fine. I mean, that hurt like a bitch, but I’ll live. Back to my question, though; where do we go from here?” Alex continued, looking up at Jack with a small smile.

“I think there’s only one logical thing I can ask now. Alex, do you want to be my boyfriend?”

Even though he knew they both felt the same way, Jack was still nervous to officially ask Alex out. He had all of the assurance in the world from the conversation he’d just shared with Alex, but a year of anxiety and fear of someone finding out his secret would be hard to change.

“Yes, Jack, I’d love to be your boyfriend,”

The two of them looked at each other, huge smiles on both of their faces. Jack wrapped Alex into a tight hug, rubbing his back with one hand and stroking his hair with the other as the huge smile stayed on his face. Alex was still insanely sweaty, but Jack didn’t care, nothing would ruin this moment for the two of them.

After a few minutes, Alex pulled back to sneeze, but then looked up at Jack again, a huge smile still plastered on his face.

“I can’t believe we just became a thing while I look disgusting and feel miserable,” Alex said in a tone that Jack recognized as partly sarcastic with a hint of truth.

“You don’t look disgusting, babe. Hey, can I call you that now?” Jack asked, making their smiles grow even bigger.

“Of course, you can call me any cute nickname you want, I’ll definitely do the same, baby,”

“That sounds perfect to me. I meant what I said, though, you don’t look disgusting. You’re knocking on death’s door but still look absolutely perfect, I don’t know how you do it! As far as you feeling miserable, though, what can I do to help?”

“Honestly, you just being here helps so much. Do you think we could lay down, and I could put my head on your chest? I might fall asleep eventually, but I just want to be close to you,”

“Absolutely, lay however you’re the most comfortable,”

The two of them laid back down in Alex’s bed, and Alex put his head onto Jack’s chest, and Jack wrapped Alex into a hug.

“This is perfect,” Alex said, Jack could hear the smile on Alex’s face as he spoke.

“I’m so glad to hear that, let me know if there’s anything else I can do for you. I can’t believe this is really happening, I’ve wanted this for so, so long,” Jack breathlessly said, bringing his other hand up to start stroking Alex’s hair again.

“Me either, Jack. When I wake up, you might have to tell me about this, I’ll probably think it was a fever induced delusion,” Alex joked, making them both laugh some.

“I’ll tell you as much as you want, I’ll tell everyone all the time that the Alex Gaskarth is my boyfriend,” Jack affectionately replied, placing a kiss on Alex’s forehead.

“You’re sweet, Jack. When I’m better, we’re going on a real date to celebrate this,”

“Oh, for sure, but for now, you need to focus on resting and getting better. I’ll be here for however long you want me to be to help you, okay?”

“That sounds great to me. I think I’m going to sleep for a bit now,” Alex continued, hugging Jack tighter and closing his eyes.

“Okay, babe, I’ll be here if you need me,” Jack replied, continuing to stroke Alex’s hair.

After a few minutes, Alex was fast asleep, lightly snoring due to the congestion in his sinuses. Jack stared out Alex’s bedroom window, the huge smile still plastered on his face. He couldn’t believe it, Alex was really his boyfriend now; it almost felt too good to be true. Waiting for so long hadn’t been easy, but Jack had no doubt in his mind that this new chapter in both of their lives will have been worth the wait.  

Hi guys! I posted something similar to this forever ago, but I wrote this semi recently and liked it more than I expected to, so I decided to share it! I can't believe I'm only seven fics away from having 300 posted! It blows my mind that I have so many up, and that after all this time, people still seem to enjoy them! I plan on having a new chapter of LBUA up fairly soon, and another one shot! As always, feel free to send in requests, I'd love to write your all's ideas! -Liv

[and-then-the-trash]

so these evening i spent 88 minutes on a precalculus unit test and when i went to look up in my notes how to write an explicit formula like the question was asking, this is what i found

what the fuck past me

why

why would you do this

anyways yeah I'm v tired and answered 22/24 questions and have no clue if i did remotely well cause apparently my notes are shit 🤪

[nancypi-blog1]

MIT grad shows how to do a binomial expansion with the Binomial Theorem and/or Pascal's Triangle. Nancy of mathbff explains the steps on YouTube’s NancyPi channel:

1) HOW TO START A BINOMIAL EXPANSION: If your binomial is something like (x + 3) raised to a power like (x + 3)^5, you have two parts of your binomial: x and 3. You're going to take each of those and raise them to different powers in each term of the expansion. You can start your expansion just by writing these powers out.. For the FIRST PART of your binomial, x, it will start with a power of 5 (your power number) in the first term of the expansion (x^5), and then in each term after the power will go down by 1 (x^4, x^3, etc. all the way down to a zero power, x^0). Then you take the SECOND PART of the binomial, 3, and multiply each term by a power of 3. The 3 factor will start with a power of 0 in the first term, so 3^0, and then will increase by 1 power in each term after.

2) HOW TO FIND THE COEFFICIENTS (with the FACTORIAL or COMBINATION method): Then there's one more number to find, a number that gets multiplied in front of each term, or a binomial coefficient. There are two ways to find the coefficients (for the Pascal's Triangle way, see below). To find the coefficients using the factorial, combination ("n choose k") formula of n!/(k!(n-k)!, each term has a coefficient number you find using an n value equal to the power number, 5, and a k value that runs from 0 for the first term up through 5 for the last term. This number gets multiplied by the other factors in each term, and then simplify for your final expansion.

3) HOW TO FIND THE COEFFICIENTS (with PASCAL'S TRIANGLE): You can instead use Pascal's Triangle to find the binomial coefficients. Whichever row of Pascal's Triangle has your power number in it, as the second number, is the row that gives you all the coefficient numbers you'll need for your expansion. Each coefficient is multiplied with its term, and then you can simplify the expansion.

4) WHAT IF THE BINOMIAL HAS SUBTRACTION? It's very similar. It's easier to think of the subtracted term as "adding a negative number", and then all of the negative number (in parentheses) will be raised to the power in each term of the expansion. Note: if the binomial have something like 2x - y instead of x - y, make sure that all of (2x) is raised to each power.

For more: check out the NancyPi channel on YouTube at: https://www.youtube.com/channel/UCRGXV1QlxZ8aucmE45tRx8w

[tutoringmaphy01]

Common Challenges in AP Precalculus and How a online AP Precalculus tutor to Overcome Them | Tutoring Maphy

Overcoming Common Challenges in AP Precalculus with an Online AP Precalculus Tutor

Introduction:

Embarking on the Advanced Placement (AP) Precalculus journey can be both exhilarating and challenging for students. As the coursework becomes more rigorous, many learners find themselves grappling with complex concepts. Fortunately, the advent of online education has opened up avenues for personalized learning and assistance. In this blog post, we will explore the common challenges faced by students in AP Precalculus and discuss how an online AP Precalculus tutor can help overcome these hurdles.

Complex Concepts and Formulas:

Students often encounter intricate concepts and formulas in AP Precalculus, making it challenging to grasp the fundamentals.

An experienced AP Precalculus tutor can simplify complex topics, breaking them down into more digestible segments, and provide step-by-step explanations.

Lack of Personalized Attention in Classroom Settings:

Large class sizes can hinder individualized attention, leading to students feeling lost or falling behind.

An online AP Precalculus tutor offers personalized, one-on-one sessions, addressing specific concerns and tailoring the learning experience to the student's pace.

Inadequate Practice and Reinforcement:

Regular practice is essential for mastering AP Precalculus, and students may struggle to find sufficient resources for exercises.

An AP Precalculus tutor can design custom practice sessions, ensuring that students receive targeted reinforcement on challenging topics.

Time Management and Exam Anxiety:

Balancing the demands of AP courses with other commitments can lead to stress and anxiety.

An online AP Precalculus tutor can provide time management strategies, help create effective study plans, and offer guidance on exam preparation techniques.

Limited Availability of Additional Resources:

Some students may lack access to supplementary materials or struggle to find resources that align with their learning style.

An AP Precalculus tutor, especially one who specializes in online education, can recommend and provide a variety of resources such as videos, interactive simulations, and practice exams.

Conclusion:

In the dynamic landscape of AP Precalculus, where challenges are inevitable, an online AP Precalculus tutor emerges as a valuable ally for students. With personalized attention, targeted practice, and tailored guidance, students can navigate the complexities of online calculus tutor more effectively. Investing in the support of an online AP Precalculus tutor not only enhances academic performance but also instills confidence and a love for learning in the fascinating world of advanced mathematics.

Visit us - www.tutoringmaphy.com 

Contact us - Info@tutoringmaphy.com 

Follow - https://www.youtube.com/@tutoringmaphy

[studiesgalore]

8 studying tips for math tests //

1. create overall personalized chapter reviews (chapter concepts, formulas, and common errors) after every chapter when i review i create an overall chapter review, like @tbhstudying‘s review sheet video! it’s like a place to see all the formulas and concepts. if there’s a concept i really don’t understand i search it up and add more to my notes. it’s really personalized so i suggest also putting common errors that you make & how to avoid them so you’ll remember to not make the same mistake! 

2. print online reviews for the chapter or a certain topic so this may just be me but at times i feel that my teacher’s reviews are a lot easier than what the tests actually are or it’s not enough for me lol

searching the topic up like “precalculus vectors review” often comes a lot of reviews from different schools and just doing more problems is extremely helpful. some tips for finding good reviews is adding .pdf and your class (calculus, algebra 1, etc)

3. search the topic up for online diagrams or charts perfect if you want to add them to your notes or if you want to reference it quickly on the day of the test. for example, i often forget the 6 trig function graphs so on days of trig tests i refer back to a photo i saved of the graphs! 

if you search up .pdf in your google search as well, there’s sometimes recaps of the topic (a few pages to look at while studying and before the test)

4. find youtube videos (snazzy tricks) youtube is a great place to get motivation and watch study videos but it’s also a great source for academic vids! i just search up the topic and lots of videos come up whether it’s examples or just a lesson. some of my faves are khan academy & the organic chemistry tutor. also in youtube there’s some videos that teach some really cool tricks (hand tricks), like it seems dumb but it’s actually kinda helpful!! dont forget that if you feel like it’s a bit slow or you don’t have time just put the speed at 1.5x or whatever fits for you!

5. redo homework or review problems that you got wrong 

6. create a list of problems that keep stumbling you create a list of the problems that you get wrong (it’s like your own review) and do them! make sure you understand why you got them wrong :)

7. take pictures of reviews to save on your phone (glance at them beforehand) this is for any subject, but take pics of your short reviews, so you can glance at them before the test! though you have the paper version, according to the situation it may easier to just use your phone. i’m also not too organized so papers get lost and that’s why i like to do this! you can either scan them and keep it in your notes (iOS) or save them in a folder whether it’s a camera roll album or google drive folder! doing tip #1 and this tip is extremely helpful for midterms/finals studying

8. be well rested & eat (especially at lunch) these are just a bunch of tips and you definitely shouldn’t feel the need to do them all!! the most important of them all is to be well rested and eat well. in order to retain memory and test well you need the energy and confidence bc you know you got this! if you study at lunch, make sure you do eat!! meals and sleep are extremely important and will affect your test.

as my math final comes up i just wanted to share these 8 tips! the main this is to just personalize, study according to what you need. i think a lot of the tips apply to other subjects so i hope this helps :) feel free to share any of your tips and give this a like/reblog!