what do you think this will look like on the right and left ends?
look at the leading coefficient and degree of your function. You need to look at whether the degree is odd or even and whether the leading coefficient is positive or negative. There are 4 different options for end behavior based on those.
would the leading coefficient be x4? @peachpi
and i'm not sure what the graph would look like @peachpi
sorry. I stepped away for a moment. The degree is the highest exponent. That's 4. The leading coefficient is the number in front of \(x^4\). There's no number actually written, so the leading coefficient is 1.
That means you have an EVEN DEGREE polynomial with a POSITIVE LEADING COEFFICIENT.
ohhh that does make sense @peachpi
ok, so now that we have that info, what will the ends of your graph look like? If you look at that drawing there are 4 options based on degree and leading coefficient -both ends up -both ends down -left end up, right end down -left end down, right end up
i think the left and the right would be up
Exactly. So as part of your answer you'd say both ends of the graph go up because the degree is even and the leading coefficient is positive.
Now we can solve the equation to find the break-even point. \[x^4-3x^3-8x^2+12x+16=0\]
alright. do you know synthetic division?
im not the best at it though
ok. I was trying to factor, but I don't think that will work. Unfortunately it looks like we have to divide. Basically any real roots have to be factors of 16. So typically you have to try a bunch of them and see which works. I've already tried 4 and it works, so we have this.|dw:1441291472156:dw|
So now it's \[(x-4)(x^3+x^2-4x-4)=0\] The second part of that can be factored by grouping \[(x-4)[x^2(x+1)-4(x+1)]=0\] \[(x-4)(x+1)(x^2-4)=0\] Then factor the difference of squares \[(x-4)(x+1)(x+2)(x-2)=0\]
yeah sorry im writing some of this down
no problem. let me know when you're ready
ok im ready now @peachpi
ok so we know the zeros are 4, -1, -2, and 2. Plot them on the x-axis. (please pardon my sorry excuse for points)|dw:1441292680506:dw|
This is where the end behavior comes in. We know that the graph goes up on both ends, so we can do this|dw:1441292840024:dw|
ohhh wow i actually understand this now :D
:) cool. Now we just need to make a reasonable guess at the middle.
basically, (anytime you don't have repeated roots), a polynomial will switch sides of the x-axis when it passes through a zero. So since it's positive to the left of -2, it will be negative between -2 and -1. Then positive again, then negative, then finally positive where we have the right end drawn |dw:1441293088769:dw|
It's really hard to draw a smooth curve on this, so plot it on https://www.desmos.com/calculator to see what it really looks like
The only other thing I think you can do is say the y-intercept is 16|dw:1441293360697:dw|
what would the end behaviors be ?
The end behavior is both ends of the graph go up because the degree is even and the leading coefficient is positive. The y-intercept is 16 so the profit was $16 when 0 laptops were produced. Then profit increases to a local maximum between 0 and 2 produced. Then the company loses money between 2 and 4 produced before breaking even permanently at 4 produced.
thank you so much i seriously appreciate all you've helped me with