sketch the curve of y=((x^2 −1)/(x^2 −7x−12))


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Question Number 123199 by aurpeyz last updated on 23/Nov/20

$s k e t c h t h e c u r v e o f y = \frac{x^{2} - 1}{x^{2} - 7 x - 12}$
	Answered by MJS_new last updated on 24/Nov/20
	$y=((x^2 −1)/(x^2 +7x−12)) y′=−((7x^2 +22x+7)/((x^2 +7x−12)^2 )) y′′=((2(7x^3 +33x^2 +21x+83))/((x^2 −7x−12)^3 )) y=0 ⇒ zeros at x=±1 x^2 +7x−12=0 ⇒ asymptotes at x=((7±(√(97)))/2) (≈−1.42 and 8.42) y=1+((7x+11)/(x^2 −7x−12)) ⇒ asymptote at y=1 (for x=±∞) y′=0 ⇒ extremes at x=−((11±6(√2))/7) (≈−2.78 and −.36) y′′ { ((>0 for x=−((11+6(√2))/7) ⇒ local min at ≈ (((−2.78)),((.44)) ))),((<0 for x=−((11−6(√2))/7) ⇒ local max at ≈ (((−.36)),((.09)) ))) :} y′′=0 ⇒ inflection point ≈ (((−4.62)),((.49)) ) y′′ { ((<0 for x<≈−4.62 ⇒ curvature −)),((>0 for ≈−4.62<x<≈−1.42 ⇒ curvature +)),((<0 for ≈−1.42<x<≈8.42 ⇒ curvature −)),((>0 for x>≈8.42 ⇒ curvature + )) :} now sketch it$
	$y = \frac{x^{2} - 1}{x^{2} + 7 x - 12}$ $y^{'} = - \frac{7 x^{2} + 22 x + 7}{{(x^{2} + 7 x - 12)}^{2}}$ $y^{″} = \frac{2 (7 x^{3} + 33 x^{2} + 21 x + 83)}{{(x^{2} - 7 x - 12)}^{3}}$ $y = 0 \Rightarrow zeros at x = \pm 1$ $x^{2} + 7 x - 12 = 0 \Rightarrow asymptotes at x = \frac{7 \pm \sqrt{97}}{2} (\approx - 1.42 and 8.42)$ $y = 1 + \frac{7 x + 11}{x^{2} - 7 x - 12} \Rightarrow asymptote at y = 1 (for x = \pm \infty)$ $y^{'} = 0 \Rightarrow extremes at x = - \frac{11 \pm 6 \sqrt{2}}{7} (\approx - 2.78 and - . 36)$ $y^{″} {\begin{cases} > 0 for x = - \frac{11 + 6 \sqrt{2}}{7} \Rightarrow local \min at \approx (\begin{array}{c} - 2.78 \\ . 44 \end{array}) \\ < 0 for x = - \frac{11 - 6 \sqrt{2}}{7} \Rightarrow local \max at \approx (\begin{array}{c} - . 36 \\ . 09 \end{array}) \end{cases}$ $y^{″} = 0 \Rightarrow inflection point \approx (\begin{matrix} - 4.62 \\ . 49 \end{matrix})$ $y^{″} {\begin{cases} < 0 for x <\approx - 4.62 \Rightarrow curvature - \\ > 0 for \approx - 4.62 < x <\approx - 1.42 \Rightarrow curvature + \\ < 0 for \approx - 1.42 < x <\approx 8.42 \Rightarrow curvature - \\ > 0 for x >\approx 8.42 \Rightarrow curvature + \end{cases}$ $now sketch it$
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