Question

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

Answer 1

see explanantion

Explanation:

`color(blue)("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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`color(blue)("Investigating "x=0)`

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

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`color(blue)("Investigating "y=0)`

`=>8x=4x^2`

`color(blue)(=> x=2)`

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`color(blue)(x_("intercepts") -> (x,y)-> (0,0)" and "(2,0)`
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`color(blue)(y_("intercepts")->(x,y)-> (0,0) )`
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`color(brown)("The equation is undefined at "x=-2" (excluded value)")`

Let `deltax` be very small
Let `x-2`

Then we have

`y=(8(-2+deltax) -4(-2+deltax)^2)/([(-2+deltax)+2]^2)`

`y=(-16+8deltax-4(4-4deltax+(deltax)^2))/((deltax)^2)`

`y=(-16+8deltax-16+16deltax-4(deltax)^2)/((deltax)^2)`

`y=-32/((deltax)^2) +24/(deltax)-4`

as `deltax` becomes increasingly small then `|-32/((deltax)^2)| >24/deltax` .

`color(brown)(y=lim_(deltax->0) -32/((deltax)^2) +24/(deltax)-4)color(blue)(-> -oo)`

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