Question

My Notes SAlgTrig4 3.6.035. 27. -12 points Find all horizontal and vertical asymptotes (if any). (If an answer does not exist, enter DNE, Enter your answers as a comma-separated list of equations.) 4x2+1 s(x) 7x-4 vertical asymptote(s) horizontal asymptote

Answer 1

For a rational functions

$f(x)=\frac{g(x)}{h(x)}$

Vertical asymptote occur at point x=C where h(C)=0

For given funtion denominator (h(x)) is 2x²+7x-4. So find point where

2x²+7x+4=0

This is quadratic equation of form ax²+bx+c=0 . Solution for this is

$x=\frac{-b_+^-\sqrt{b^2-4ac}}{2a}$

Here a=2,b=7,c=4

$x=\frac{-7_+^-\sqrt{7^2-4*2*-4}}{2*2}=\frac{-7_+^-\sqrt{81}}{4}=\frac{-7_+^-9}{4}$

$x=\frac{-7+9}{4}=0.5$. Or. $x=\frac{-7-9}{4}=-4$

h(x) becomes zero when x=0.5 and x=-4

So vertical asymptote are at these points.

vertical asymptote at x=0.5 and x= -4

If g(x) and h(x) is same degree polynomial,then horizontal asymptote occur at point

$Y=\frac{m}{n}$

Where m is leading coefficient of g(x) and n is leading coefficient h(x).

Both numerater g(x) and denominator h(x) are quadratic equation.(degree 2)

Coefficient of leading term of numerator is 4 and leading term of denominator is 2.

So horizontal asymptote is at

$Y=\frac{4}{2}=2$

So horizontal asymptote at y=2