State the asymptotes, holes, roots and intercepts for 2x^2+10x+12/x^2-9
Include Vertical asymptote, horizontal asymptote, roots, x-int and y-int with end behaviours and describe how the graph will look like
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We can factor to get
We can simplify with the idea that x!=-3 or x!=3
Now we have vertical asymptotes at x=3, and a hole at x=-3.
Since deg(2(x+2)) = deg(x-3) and the leading coefficients are 2 and 1 we get a horizontal asymptote at y=2.
The x-intercept is when 2(x+2) = 0 or x=-2.
The y-intercept is when y=(2(0+2))/(0-3)=-4/3
The end behavior is y=2.
The below graph is how we would plot this funcion:
Note the hole at x=-3, it is not an asymptote.
1) Since x=3, and x=-3 get the denominator to be zero than they are your certical asymptote.
2) Since the numerator and denominator are of the same degree we have a horizontal asymptote, y=2.
3) No oblique asymptote.
Hence x-intercept (-2,0)
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