`y = x^2/(x + 8)` Sketch the curve by locating max/mins, asymptotes, points of inflection, etc.

Textbook Question

Chapter 4, Review - Problem 25 - Calculus: Early Transcendentals (7th Edition, James Stewart).
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gsarora17 | (Level 2) Associate Educator

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a) Asymptotes

Vertical asymptotes are the zeros of the denominator


Vertical asymptote is x=-8

Degree of numerator=2

Degree of denominator=1

Degree of numerator=1+Degree of denominator, so the asymptote is a slant asymptote of the form y=mx+b

For a rational function the slant asymptote is the quotient of the polynomial division.


 the slant asymptote is y=x-8

b) Maxima/Minima




Let's find critical numbers by solving x for y'=0,

`x(x+16)=0rArrx=0 , x=-16`



Local maximum=-32 at x=-16

Local minimum=0 at x=0

c) Inflection points





there is no solution for x , so there are no inflection points.

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