How do I do the following problem: Analyze the graph of a rational function R(x)= 8x^2+26x+15/2x^2-x-15?

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embizze | High School Teacher | (Level 2) Educator Emeritus

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Analyze the graph of `R(x)=(8x^2+26x+15)/(2x^2-x-15)` .

(1) Factor numerator and denominator:


(2) The domain of `R(x)` is all reals except `x=-2/5,x=3` .(Cannot divide by zero)

`R(x)` is equivalent to the function `y=(4x+3)/(x-3)` except at `x=-2/5` .(Cancel the identical binomials)  So the range of `R(x)` is all reals except 4. (An easy way to see this is to rewrite `(4x+3)/(x-3)` as `15/(x-3)+4` using long division, and realize that `15/(x-3)` can never be zero)

(3) The x-intercept is `x=-3/4` . The y-intercept is -1.

(4) There is a removable discontinuity at `x=-2/5` .(A "hole" in the graph). There is an infinite discontinuity at `x=3` .

(5) The graph has a horizontal asymptote at `y=4` and a vertical asymptote at `x=3` .

(6) The graph :

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