Question 1136718
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Again you should know parentheses are required....<br>
The function is<br>
{{{(-4x^2+1)/(x^2+x-6) = (-(2x+1)(2x-1))/((x+3)(x-2))}}}<br>
(1) y-intercept: when x=0: -1/6<br>
(2) x-intercept(s): where y=0; i.e., where the numerator is 0: at x=-1/2 and x=1/2<br>
(3) vertical asymptote(s): where the denominator is 0 -- at x=-3 and x=2<br>
(4) horizontal asymptote: y=-4 (degree of numerator and denominator are the same, so ratio of leading coefficients)<br>
A graph of the rational function and the horizontal asymptote:<br>
{{{graph(400,400,-10,10,-10,10,(-4x^2+1)/(x^2+x-6),-4)}}}<br>
The graph points out another thing you should look for when graphing a rational function: Does the graph cross the horizontal asymptote?<br>
To find out, set the function equal to -4 and solve:<br>
{{{(-4x^2+1)/(x^2+x-6)=-4}}}
{{{-4x^2+1 = -4x^2-4x+24}}}
{{{4x = 23}}}
{{{x = 23/4}}}<br>
The graph shows the graph of the rational function crossing the asymptote y=-4 at x= 23/4 = 5.75.