# How do you graph y =(8x-4x^2)/((x+2)^2)?

Apr 19, 2016

see explanantion

#### Explanation:

$\textcolor{b l u e}{\text{Investigating extremities}}$

color(brown)(y=Lim_(x->+-oo) (8x-4x^2)/((x+2)^2)->color(blue)((-4(+-x)^2)/((+-x)^2) = -4)

Note that x^2" " underline("'wins'") over the other values in this equations

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$\textcolor{b l u e}{\text{Investigating } x = 0}$

color(brown)(y= (8x-4x^2)/((x+2)^2)->color(blue)(0/4=0)

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$\textcolor{b l u e}{\text{Investigating } y = 0}$

$\implies 8 x = 4 {x}^{2}$

$\textcolor{b l u e}{\implies x = 2}$

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color(blue)(x_("intercepts") -> (x,y)-> (0,0)" and "(2,0)
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$\textcolor{b l u e}{{y}_{\text{intercepts}} \to \left(x , y\right) \to \left(0 , 0\right)}$
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$\textcolor{b r o w n}{\text{The equation is undefined at "x=-2" (excluded value)}}$

Let $\delta x$ be very small
Let $x - 2$

Then we have

$y = \frac{8 \left(- 2 + \delta x\right) - 4 {\left(- 2 + \delta x\right)}^{2}}{{\left[\left(- 2 + \delta x\right) + 2\right]}^{2}}$

$y = \frac{- 16 + 8 \delta x - 4 \left(4 - 4 \delta x + {\left(\delta x\right)}^{2}\right)}{{\left(\delta x\right)}^{2}}$

$y = \frac{- 16 + 8 \delta x - 16 + 16 \delta x - 4 {\left(\delta x\right)}^{2}}{{\left(\delta x\right)}^{2}}$

$y = - \frac{32}{{\left(\delta x\right)}^{2}} + \frac{24}{\delta x} - 4$

as $\delta x$ becomes increasingly small then $| - \frac{32}{{\left(\delta x\right)}^{2}} | > \frac{24}{\delta} x$ .

$\textcolor{b r o w n}{y = {\lim}_{\delta x \to 0} - \frac{32}{{\left(\delta x\right)}^{2}} + \frac{24}{\delta x} - 4} \textcolor{b l u e}{\to - \infty}$

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