Question

Can you guys explain poles, roots and horizontal asymptotes and how to determine them in an equation/graph?

Answer 1

Roots are anywhere such that: \[
f(x)=0
\]

Answer 2

Poles are anywhere such that: \[
\left|\lim_{x\to a}f(x) \right|= \infty
\]

Answer 3

i cant really explain it that well but i have a question are you work on something in flvs.com and maybe this video can help you
https://www.youtube.com/watch?v=K2Bku1DsIto

Answer 4

Horizontal asymptotes are anywhere such that: \[
\lim_{|x|\to\infty} f(x) = L
\]

Answer 5

poles occur when the denominator is equal to zero, in the simplified or canceled version of the rational function

Answer 6

are poles vertical asymptotes?

Answer 7

Yes.

Answer 8

so say you have a rational expression...

Answer 9

For a rational expression \[
f(x)=\frac{g(x)}{h(x)}
\]The roots of \(h(x)\) (\(h(x)=0\)), are the poles of \(f(x)\).

Answer 10

I don't know something like \[4((x+2)(x-4))/(x-2)^2\]

Answer 11

The exception being if \(g(x)\) also has a root at the same place.

Answer 12

you have to be careful to cancel common factors in the numerator and denominator, or you get a pseudo pole (or just a hole)

Answer 13

Since \(h(2)=0\) and \(g(2) = -16\neq 0\), then \(f(x)\) has a pole at \(2\).

Answer 14

The limit, from both the left and right, approaches \(-\infty\).

Answer 15

The roots of \(f(x)\) are the same as the roots of \(g(x)\). So in this case they are \(-2\) and \(4\).

Answer 16

For horizontal asymptotes, it is a bit more tricky, I suppose.

Answer 17

okay

Answer 18

First you want to know the degree of the polynomials. \(g(x)\) is degree 2, and \(h(x)\) is degree 2.

Answer 19

You'll only need to look at the coefficient of the highest degree.

Answer 20

if they aren't of the same degree then there isn't a horizontal asymptote?

Answer 21

Cancel out all but the highest degree in each factor:\[
\frac{4((x+2)(x-4))}{(x-2)^2} \to \frac{4((x+\cancel{2})(x-\cancel{4}))}{(x-\cancel{2})^2} = \frac{4(x)(x)}{(x)^2} = \frac{4x^2}{x^2}=4
\]

Answer 22

This only really works for rational functions.

Answer 23

You'll end up with something of the form\[
\frac{ax^m}{bx^n}
\]

Answer 24

When \(m=n\), then your horizontal asymptote is at \(a/b\).
When \(m>n\), then you don't have a horizontal asymptote.
When \(m<n\), then your horizontal asymptote is at \(0\).

Answer 25

okay so poles are when the denom = 0, roots are where the num = 0 and hoz asymptotes are when the degree in the num is less than or greater than the denom.

Answer 26

right?

Answer 27

for rational expressions anyways

Answer 28

Yes.

Answer 29

For a graph...