Analyze the graph of the function `x^2-9/(x^2+3x-4)`
(1) The y-intercept is (0,9/4) found by letting x=0.
(2) The x-intercepts are found by letting y=0. Then `x^2-9/(x^2+3x-4)=0=>x^2=9/(x^2+3x-4)` .
Then cross multiply to get `x^4+3x^3-4x^2-9=0` . There are two real roots to this equation, thus two x-intercepts. You can use a table of values to approximate these as `x~~-4.1,x~~1.61` .
(3) Rewrite the function, factoring the denominator to get `x^2-9/((x+4)(x-1))`
There are two vertical asymptotes: x=-4 and x=1. We know they are vertical asymptotes as there are no common factors in the numerator and denominator, and these values result in division by 0.
(4) The domain is `(-oo,-4)uu(-4,1)uu(1,oo)` . The only values of x not in the domain -4 and 1 where you would be dividing by 0.
(5) Rewriting the function as a single term yields `(x^4+3x^3-4x^2-9)/(x^2+3x-4)`
The end behavior is determined by the terms of highest degree in the numerator and denominator. As x goes to positive or negative infinity, the function approaches `x^4/x^2=x^2` ** You could consider `y=x^2` to be an asymptotic function to the given function **
(6) Create a table of values -- the easiest way is to let technology (graphing calculator, Excel, etc...) do the work. Some points: (-7,48.625),(-6,35.36),(-5,23.5),(-3,11.25),(-2,5.5),(-1,2.5),(0,2.25),(2(2.5),(3,8.36),(4,15.625),(5,24.75),(6,35.82)
(7) Here is the graph for -7<x<7 and then for -20<x<20