# 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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### 1 Answer

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

(1) Factor numerator and denominator:

`R(x)=((2x+5)(4x+3))/((2x+5)(x-3))`

(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 :