Graph f(x)=x^2-3x-10/x-2.
Find the domain, range, VA, HA/SA

Question

Graph f(x)=x^2-3x-10/x-2.
Find the domain, range, VA, HA/SA

misty1212 · Answer

HI!!

anonymous · Answer

Hey:)

misty1212 · Answer

is it 
$$\frac{x^2-3x-10}{x-2}$$

anonymous · Answer

Yes

misty1212 · Answer

ok lets do them one at a time

misty1212 · Answer

domain is easy enough, set the denominator equal to zero and solve
then say "all real numbers except that one"

misty1212 · Answer

what do you get for that? (you can do it in your head)

anonymous · Answer

2

misty1212 · Answer

yeah so all numbers except 2

misty1212 · Answer

now before we continue, lets make sure this is not a trick question 
can you factor the numerator and cancel?

anonymous · Answer

Let me see...

misty1212 · Answer

hmmi think it does factor, but not cancel !

anonymous · Answer

(x-5)(x+2)/x-2

misty1212 · Answer

yeah but nothing cancels, so lets continue
but you should always check that first

misty1212 · Answer

once we have the domain, we also have the vertical asymptote too

misty1212 · Answer

clear or no?

anonymous · Answer

So it is also 2

anonymous · Answer

yes!

misty1212 · Answer

careful here

misty1212 · Answer

the vertical asymptote is a vertical line, not a number

misty1212 · Answer

so you want to return and answer of $x=2$ the vertical line, not just the number 2

misty1212 · Answer

clear?

anonymous · Answer

Oh! Yes! Clear

misty1212 · Answer

now for the horizontal  asympote 
the degree of the numerator is ?

anonymous · Answer

2

misty1212 · Answer

and the denominator?

anonymous · Answer

there is no horizontal then since it's greater than the denom.

misty1212 · Answer

bingo

misty1212 · Answer

but the degree of the numerator is one more than the degree of the denominator
that means there will be a slant asymptote 
do you know how to find it?

anonymous · Answer

I never learned about a slant asymptote so I dont know :(

misty1212 · Answer

ok it is easy to say what to do , but not easy for me to write 
in one word "divide"

misty1212 · Answer

when you divide a polynomial of degree 2 by a polynomial of degree 1 you get a polynomial of degree 1 ( a line) and a remainder
ignore the remainder, the line is the slant asymptote

misty1212 · Answer

you can use either long division or (much easier) synthetic division 
do you know how to do that?

anonymous · Answer

Yes :)

misty1212 · Answer

whew
like i said it is very hard for me to write division here
what do you get?

misty1212 · Answer

take your time

anonymous · Answer

I did it in a rush so I got x^2-1x-6.

misty1212 · Answer

hmm no  you should have a polynomial of degree 1, not a quadratic

misty1212 · Answer

1     -3     -10
2
______________
    1

misty1212 · Answer

that is the iniital set up 
then 
     1     -3     -10
2           2
______________
    1      -1

misty1212 · Answer

the next line is unimportant since we only need the line, not the remainder 
the line has the coefficients from the final row so $x-1$
slant asymptote is $y=x-1$

anonymous · Answer

Oh! Now I know what I did wrong! When did the fak divison, i fogot to add the "0^2" Lol but yes, I know understand

misty1212 · Answer

ok then we are done!

misty1212 · Answer

oh except for the range

anonymous · Answer

I'm so bad at finding that.

misty1212 · Answer

in this case it is real easy, since it has a vertical asympote and no horizontal asymptote
that means it goes from $-\infty$ to $\infty$

misty1212 · Answer

you can write your answer as an interval $(-\infty, \infty)$ or just say "all real numbers"

misty1212 · Answer

it is harder for some other ones, but not for this one

anonymous · Answer

Yay! Thanks so much:)))

misty1212 · Answer

$$\color\magenta\heartsuit$$