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4:07 AM
How can you tell when a graph passes the horizontal asymptote and how do you solve for that point?
Like (x+1)(2x-1)/(x+3)^2
 
Horizontal asymptote is y = 2
 
It passes the horizontal asymptote of y=1 but idk how I could figure that out
 
So solve for when y = 2
 
Oh I'm dumb but yeah
wait so I just plug in 2 into the equation or what happens
 
no, you set it equal to 2
and then solve for x
and then (x, 2) will be the point
 
4:09 AM
2=(x+1)(2x-1)/(x+3)^2
O ic
 
@PrinceNorthLæraðr what happens when a graph cuts through a horizontal line?
 
@Deusovi it stoops back down and never crosses it again
 
not necessarily
 
ez linear equation
 
I'm asking more generally
remember, the graph is just drawing the points that satisfy the equation "y=(some stuff involving x)"
so if the graph crosses y=2, then that means...?
 
4:11 AM
Um
it's greater than 2?
 
what does it mean for a point to be on the graph?
 
It's rational?
 
not necessarily
say i draw a graph of a function f. and you notice that the point (5,7) is on this graph. what can you conclude?
 
It passes through (5,7)???
 
I think he's asking what you can conclude about the function f
 
4:13 AM
the graph does, yes -- that's what "the point (5,7) is on the graph" means
what can you conclude about f?
 
The graph of f == the graph of y=f(x)
 
can you tell me what the graph of a function f is? you know that you have functions -- machines that take in a number and spit out a number -- and graphs -- curvy drawings on a piece of paper. what does the "graph of a function" mean?
 
It draws the curvy things by the numbers
 
in other words, what does the graph tell you about a function? obviously there's some rule linking them together. every function has a unique graph, and you're not just drawing random lines -- there's some meaning behind those lines
 
4:16 AM
y=f(x)?
 
can you be more precise about how it "draws the curvy things by the numbers"? if you only have a function, how do you get numbers from it? how do you know where to draw curvy things?
 
So the point (5,7) is 7=f(5)?
 
i'm not asking for an equation here, i just want you to describe in words what is happening when you draw a graph
right -- "(5,7) is on this graph" means "f(5) is 7"!
 
and more generally, if the point (a,b) is on the graph...?
 
4:19 AM
f(a) is equal b?
 
yep!
 
so, say you know the asymptote is y=2. and you know that somewhere, the graph crosses that asymptote.
 
if two curves cross, what does that mean about the point of intersection? (don't overthink this!)
 
4:22 AM
It's between two curves?
 
"between" them?
 
I mean that's where they intersect
Wait that's what intersection means
It's where the two curves are equal to each other
 
------

------
is "a point between two curves" (which happen to be straight lines in this case)
a curve is a drawing. a drawing can't be equal to a different drawing
i'm not asking you to think about the equations here, just think geometrically
the point isn't "between" the two curves -- there's a much better way to describe it
 
In the middle?
Geez this time of the day is not the right time for math
 
it's not in the middle of them, because again that implies the picture i drew (or attempted to draw) above
don't overthink it - you don't even need to think of both of the curves together
 
4:27 AM
It's where the two y or x values are equal
No i'm sorry
 
i'm not asking you to think about the equations here, just think geometrically
here is an absolutely garbage picture
 
Well, that at least made me smile
 
how does the green point relate to the red curve? how does it relate to the blue curve?
 
Oh the point of intersection is when blue is equal to red?
 
i'm not asking you to think about the equations here, just think geometrically
how does the green point relate to the red curve? ignore the blue curve for now - don't overthink this
 
4:30 AM
"the green point ____ the red curve"
 
Is on the red curve
 
and for the blue curve?
 
Is on the blue curve
The green dot is on both the red and blue curve
 
exactly - that's what intersection means
 
4:31 AM
and so if the blue and red curve are defined by equations, what does that tell you about the coordinates of the green dot?
 
if blue is y=f(x) and g is y=g(x), and the two ys are equal, f(x)=g(x)
the x and y components of both the red curve and the blue curve are equal to each other
 
hm?
right - i think it might help to break it down further though

if the green dot is at coordinates (5,7), then:
- if you substitute 5 and 7 in the equation for the red line, what happens?
- if you substitute 5 and 7 in the equation for the blue line, what happens?
 
You get the same results for both red and blue
 
what do you mean by "results" exactly?
 
@Deusovi The same output for the input?
 
4:37 AM
i'm talking about equations here, not functions
a curve doesn't necessarily have to be a function
 
take this ellipse, for instance -- it's defined by "x² + y²/5 = 1"
 
I see you use desmos as well
That was fast
 
if i tell you "(0.74, 1.504) is on this ellipse", what does that tell you about those two numbers?
 
x=0.74, y=1.504
 
4:38 AM
i never said anything about x and y
i'm asking what it tells you about those two numbers, and how they relate to each other
 
When the circle is at 0.74 it is also at 1.504
That made no sense
(I'm sorry Deus)
 
i'm not asking about the graph - just the two numbers
i'm telling you that that point is on the graph. what does that tell you about the numbers?
 
The relationship? The position? When the points are equal?
Perhaps this isn't the best time for this
 
what do you mean by "points are equal"?
there's only one point involved here
 
*internal screaming*
It tells me that the point lies on the value 0.74 and 1.504????
 
4:42 AM
hint hint use the equation
 
1.504=0.74??
(cries more)
 
when i say "this ellipse is defined by x² + y²/5 = 1", do you know what i mean by that?
(also, going to move this to another room)
 
Ohh
0.74^2+1.504^2/5=1?
 
115 messages moved from The Sphinx's Lair
 
Hey
Okay so yeah that equation above ^
 
4:45 AM
right! and if i tell you instead "the point (0.3, 2) is not on this ellipse", what does that tell you?
 
you got a personal tutoring room :)
 
that 0.3^2+2^2/5=/= 1
 
exactly
 
@bobble Yay glitter
 
the graph marks the points that satisfy the equation
 
4:46 AM
Deusovi's Perfect Math Class
 
@bobble take that, math SE
 
@Deusovi So... this review exercise was for....
@Deusovi Right
 
so you've got this asymptote here in blue
and your original curve here in red
 
and you want to find the green point
 
4:46 AM
Oh I see
Since y=2
 
as you said earlier, the green point is on both the blue curve and the red curve
 
that means the green point satisfies both the blue curve's equation and the red curve's equation
and so if you call the point (a,b), you know that it satisfies "b=2" [from blue]
and also "b = (a+1)(2a-1)/(a+3)^2" [from red]
 
ohh, I see
2=(a+1)(2a-1)/(a+3)^2"
Basically explaining what bobble said
But just way longer
 
right, but hopefully in a way that actually makes you understand why that's the case
 
4:49 AM
at least I won't forget this for my math test
Alright, ready to do this all over again but for a completely different question?
(Last one, I promise)
 
"when a graph passes the horizontal asymptote" is not a special 'type' of problem -- first, you find the horizontal asymptote, and then you find where they intersect. and you can find where two graphs intersect by finding the point where their equations are both satisfied
 
Right
Since you already have the y value, you can plug and chug
 
sure, not busy at the moment - up for explaining another thing
(and yep, this happens to be a particularly "nice" system. but you could also theoretically do it with something weird, like if i instead asked you to find where the red graph crossed the ellipse from earlier)
 
How many real roots are in x^6-3x^3+1=0? Support your answer with a mathematical argument, but without relying on a graph or finding any actual roots
Okay so here's my train of thought
Max number of time something can cross the x-axis (hence, roots) is equal to the power of the leading coefficient. So, at max, this can have 6 real roots. But wait! there's more
We can apply Descarte Rules of signs to figure out that we a total of... 2? sign changes in the positive and 2? sign changes in the negative, so a total of 4? possible real roots
I'm not sure where else to go
Since I can't really solve for anything
Oh wait hold on
 
(they do Descartes rule of sign in high school? I'm surprised)
 
4:54 AM
(yep)
 
@Ankoganit (pre-calc)
 
holding on...
 
(ah, cool)
 
My two possible roots because of a separate theorem I forgot the name of tells me that p/q where p=the number without a variable and q is the leading coefficient number
The
Factors
 
rational root theorem
 
4:55 AM
Right
 
that's only for rational roots though
good thought! but there might be some irrational roots too
 
Precalc means I can't take its derivative either :/
 
It only asks for real roots
 
irrational numbers are real numbers
 
Oh darn
You're right
 
4:56 AM
for what it's worth, my head briefly went there first too
 
Well, we know that the roots could be +1 and -1. Or has to be?
 
Ok, I think I've got the answer
 
if they're rational, then they're 1 or -1
but the roots aren't necessarily rational
 
And the roots are not rational, I can confirm
 
4:57 AM
 
@bobble Already did that
 
there's something strange about the equation "x⁶ - 3x³ + 1 = 0" - do you notice anything interesting about how it's set up?
 
well, the value for f(x) and f(-x) are exactly the same, but also it moves in powers of 3s and it's positive-negative-positive
 
it's not true that f(x)=f(-x) -- the middle term will be different
"powers of 3s" is the key thing to notice here
 
Oh crap urm I gave you the wrong equation
But let's go with this for a moment
 
4:59 AM
why don't we simplify things? let's give x³ a new name -- say, call it "t".
can you write the equation in terms of t?
 
Sure. Hold on
t=(x^6-1)3
 
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