Question

graph the rational function f(x)=2x+3/x+4

Answer 1

\[f(x)=\frac{2x+3}{x+4}\]?

Answer 2

yes that is correct

Answer 3

I dont understand how to graph it...

Answer 4

you got the vertical and horizontal asymptotes?

Answer 5

vertical is -4 (i think), and horizontal I couldn't figure out

Answer 6

horizontal maybe infinity?

Answer 7

vertical in indeed \(x=-4\) since that is where the denominator is zero

Answer 8

no horizontal is not "infinity"
the numerator and denominator are both polynomials of degree 1
since the degrees are the same, the horizontal asymptote is at \(y\) = ratio of leading coefficients

Answer 9

the leading coefficient of the numerator is 2
the leading coefficient of the denominator is 1
the ratio is \(\frac{2}{1}=2\) and the horizontal asymptote is therefore
\[y=2\]

Answer 10

a simple way to think about this is if say \(x=1,000,000\) then you have
\[\frac{2,000,003}{1,000,004}\] which is basically 2

Answer 11

ok that makes sense...and then I have to find 2 points right?

Answer 12

well you need to plot something yes
some point where the x value is larger than -4 to the right of the vertical asymptote and some number to the left

Answer 13

lousy picture but you get the idea

Answer 14

ok so those are my two asymptotes...so i just plot a random point larger than negative 4?

Answer 15

well not random
lets pick a number larger than \(-4\) say \(-3\)

Answer 16

what is \(f(-3)\) ?

Answer 17

ok...-3

Answer 18

and what do you get ?

Answer 19

plug it into the equation right?

Answer 20

right

Answer 21

I got -3 again

Answer 22

yeah that looks right
so the point \((-3,-3)\) is on the graph

Answer 23

I plot (-3,-3)

Answer 24

yes. went ahead and plotted it

Answer 25

and there should be a second point
larger than 2?

Answer 26

know you know what it looks like to the right of the asymptote

Answer 27

*now you know