Question

Analyze the graph of the function.

R(x)= x(x-20)^2/(x+15)^3

a) what is the domain?
b) what is the equation of the vertical asymptote(s) of R(x)?
c) what is the equation of the horizontal or oblique asymptote of R(x)?
d) Graph.

Answer 1

domain defines what x values can be used.

VA defines the bad values for x

HA or OA define what function the graph is limiting towards for x to +- infinity

Answer 2

can you explain more? i don't understand any of this crazy math stuff!

Answer 3

its a rather complex question which assumes you have some practice with. im not sure how much you do know of it but im willing to work with you on it

Answer 4

can you tell me what makes a bad fraction?

Answer 5

a bad person...lol im joking a negative?

Answer 6

negative fractions are fine :) its dividing by zero that is verbotten

Answer 7

oh! yes ok lol

Answer 8

so, in this case, we ask ourself, what values of x makes the bottom go zero? we would have to exclude those values from the domain of Real numbers that can be used

Answer 9

(x+15)^3 = 0 when x=-15

so the domain (values of x that we can use) are all Real numbers except for -15, or written simpler as:\[D=[ x\in R:x\ne-15]\]

Answer 10

Vertical asymptotes are defined for any non removable bad domain values.

since there is no factor on the top that can cancel any factor on the bottom .... its already in simplest form, then we can claim that x=-15 is the only VA we have

Answer 11

so it would be {x|x≠-15} ?

Answer 12

that would be one way of writing it yes, but its correctness depends on whos grading it :)

Answer 13

well this question is a multiple choice lol

Answer 14

then the closest one to that is fine

Answer 15

other option is {x|x≠0 and x≠-15}

Answer 16

that ones there to fool you, since your thought process is spose to be bad zeros they try to trick you into saying x != 0

Answer 17

ok =)

Answer 18

when x=0, the bottom is 15^3 is fine :) the top zeros out but thats perfectly acceptable for fraction to have a 0 on top

Answer 19

vertical asymps .. VA are bad zeros that simply cannot be factored out of the equation .... they cannot be removed by reducing the setup

Answer 20

oh ok. what about c and d?

Answer 21

d is just putting all the information from abc into a picture ....

c is expressed for large values of x

Answer 22

if we expand the top and bottom, we can compare leading terms to have a simple way to determine any horizontal asymptote

Answer 23

ok

Answer 24

x(x-20)^2
----------
(x+15)^3



x(x^2 + ...)
----------
x^3 + ...



x^3 + ...
--------
x^3 + ...


for large values of x, the higest degree terms always take control of the function, the leading terms

Answer 25

if the top is a bigger degree, then the value is not bounded and heads off into infinity


if the bottom a bigger degree, then the value IS bounded and heads off towards 0


if the degrees are equal, then the x parts are bounded by 1, and we are left with the coefficients:

Answer 26

in this case, x^3/x^3 limits out to 1 ... HA is then defined to be y=1

Answer 27

so how would you graph it? would it be -15 on the x and 1 on the y?

Answer 28

yep, and then some minor comparisons to find some points of interest, like y intercept and x intercept