Young Writers Society

hey hey! future computer science major here :] i've been fortunate enough to have some great teachers who made learning math and programming a positive experience for me, and hopefully i can help you all have a more positive experience with it as well ^^

i can answer questions on:

math

• algebra
• geometry
• trigonometry
• calculus
• i'm currently taking AP Statistics

programming

• i'm proficient in Java
• i know some basic Python
• i can search up Stack Overflow answers for other questions =P

physics
i'm taking AP Physics C right now, and the topics we've covered so far are:

• kinematics
• energy
• circular motion
• center of mass
• linear momentum and impulse
• torque

i took AP Chemistry last year, but i am extremely rusty... you can try asking me questions, though, and i'll consult the all-knowing internet and let you know what i find (or you can ask the internet directly) ^-^

...i will also ramble on about frequency analysis and substitution ciphers if you give me a chance, so there's that too. oh! you can also ask tips on being a high schooler and going through college apps, i guess?

how this works

1. you post a question or an image of a problem
2. i reply with an answer or some hints (i'll probably reply within a day or two, and i'll try to include visuals, especially if it's geometry)
3. you reply again, either letting me know you got it or asking for clarifications
4. and so on

okay then! let the questions commence =D

Hey mint! I need help with algebra 2

I can’t wrap my head around end behavior in a graph. I’ve watched videos and listened to the teacher but I just don’t understand it. Do you think you could help?

@GengarIsBestBoy Hey Gengar! Yeah, end behavior can be tricky. I'll do my best!

So, starting from the basics...
End behavior is what a graph looks like as it goes towards the far right (as x approaches positive infinity, or +∞) or the far left (as x approaches negative infinity, or -∞).

For a line like y = 3, the graph approaches 3 at both ends. (If the graph approaches a single number, that number is called a limit.)


But for a line like y = 2x + 1, there is no one number that the graph approaches. As x increases, y increases. As x decreases, y decreases. The line stretches to infinity. So, we say the end behavior is that as x approaches positive infinity, y approaches positive infinity, and as x approaches negative infinity, y approaches negative infinity. Written in symbols, that is:
As x → +∞, y → +∞, and as x → -∞, y → -∞


Okay, now, you're probably covering polynomials in class? And that's where it gets tricky...

Let's start with a polynomial in the form f(x) = ax^2 + bx + c.
I'm going to substitute random numbers, so let's say we have f(x) = 2x^2 + 3x + 7.


The first thing to note is that only the term with the highest power of x matters in finding end behavior. Why? Well, imagine x was a million (1,000,000). A million squared would be 1,000,000,000,000, which is a lot greater than a million. So the constant coefficients a, b, and c don't really matter.
(You might wonder, what if b was equal to a million? Wouldn't that mean bx matters too, then? But we can always increase x. If x is a trillion trillion, a coefficient of a million wouldn't matter. We're looking at end behavior as x approaches infinity. It's very difficult to comprehend infinity, but we can try to imagine it as a ginormous number, one that is surely larger than any of the coefficients of the polynomial.)

Okay, so now we're just looking at the term with the highest power: 2x^2. How do we get the end behavior from that?
Once again, you can try the imagining-infinity-as-a-large-number trick. Is a 1,000,000 squared positive or negative? (Hopefully you answered positive.) What about a trillion squared? (Also positive.) So, as x increases, y increases. And it never stops: the larger x gets, the larger y gets. So, we can conclude that as x → +∞, y → +∞.
We can use similar reasoning for the other end: is -1,000,000 squared positive or negative? Well, anything squared is positive, so it must be positive. And so, as x → -∞, y → +∞.

Of course, this is different if a is negative! If we had -2x^2 instead of 2x^2, the end behavior would be flipped. Anything squared is positive, so if you multiply it by a negative number, it'll be negative. So, as x → +∞, y → -∞, and as x → -∞, y → -∞.

Now then, on to x^3...
Say f(x) = 5x^3 - 7x^2 + 2x - 9.


Following the same reasoning as above, we can consider just the x^3 term, which is 5x^3.
Is 1,000,000 cubed positive or negative? (I will imagine you gleefully shouting out, "Positive!") And yep, as x increases, x^3 also increases. Thus, we can write: as x → +∞, y → +∞.
What about -1,000,000 cubed? ("Negative!") Yes indeed! :] A negative number cubed is always negative. That means that as x → -∞, y → -∞.

Once again, if a is negative, that changes the end behavior. Now, as x → +∞, y → -∞ and as x → -∞, y → +∞.

Now, f(x) = x^4 + 3x^3 - 7x^2 + 2x - 9.


The graph might look a little funky, but the same concepts hold true, and we can just consider the x^4. And lo and behold, the end behavior is the same as that of a quadratic polynomial (or a polynomial where the highest power of x is 2). Do you get why that is?



Okay, and then f(x) = 3x^5+x^4+3x^3-7x^2+2x-9


Notice anything? Yep, the end behavior is the same as that of a polynomial with the highest power of x being 3. Interestingly enough, polynomials with even highest powers have the same end behavior, and polynomials with odd highest powers have the same end behavior. Try graphing a few examples on desmos.com or reason it out for yourself

Hope that helps! Lemme know if I made any typos or mistakes, or if I didn't answer your question xD Feel free to ask more questions too! =D

@Spearmint

I still don’t think I understand… why do all the examples have the same end behavior? [edit: oh wait maybe they dont, I just wasn’t reading it right] Also I think the example using a million made things more confusing for me.

What I was doing before was looking at the graph itself. Like, the y went up when the x went up, and the y went down when the x went down. But apparently that’s not what I was supposed to do because i got problems like that wrong. I think its the negative graphs that trip me up, or problems that dont come with a graph

@GengarIsBestBoy

Hmmm I see... In that case, maybe something you could try is graphing a lot of stuff on desmos? Like, polynomials with x^3, x^2, negative coefficients, positive coefficients, etc. Then you can find the patterns for yourself, and if you get similar polynomials in the future, you can picture their graphs. :]

Also, if you give me an example problem, I could try walking you through it?

@Spearmint okay, I’ll try that!