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

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...?

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

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)

*cries*

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

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"!

Right

and more generally, if the point (a,b) is on the graph...?

f(a) is equal b?

yep!

So...

so, say you know the asymptote is y=2. and you know that somewhere, the graph crosses that asymptote.

Right

if two curves cross, what does that mean about the point of intersection? (don't overthink this!)

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

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

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

"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

Right

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?

You get the same results for both red and blue

what do you mean by "results" exactly?

@Deusovi The same output for the input?

i'm talking about equations here, not functions

a curve doesn't necessarily have to be a function

True

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

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????

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 ^

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

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

Mhm

and you want to find the green point

Oh I see

Since y=2

as you said earlier, the green point is on both the blue curve and the red curve

Right

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

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)

(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

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

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

Right

And the roots are not rational, I can confirm

useful page?

@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

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