# How do you sketch the curve y=x^2/(x^2+9) by finding local maximum, minimum, inflection points, asymptotes, and intercepts?

Mar 16, 2017

The minimum is $= \left(0 , 0\right)$
The points of inflexions are $= \left(- \sqrt{3} , \frac{1}{4}\right)$ and $= \left(\sqrt{3} , \frac{1}{4}\right)$
The intercept is $= \left(0 , 0\right)$
The horizontal asymptote is $y = 1$

#### Explanation:

Let $f \left(x\right) = {x}^{2} / \left({x}^{2} + 9\right)$

$f \left(- x\right) = {x}^{2} / \left({x}^{2} + 9\right)$

$f \left(x\right) = f \left(- x\right)$

The curve is symmetric about the y-axis

The derivative of a quotient is

$\left(\frac{u}{v}\right) ' = \frac{u ' v - u v '}{{v}^{2}}$

We start by calculating the first derivative

$y = {x}^{2} / \left({x}^{2} + 9\right)$

$u = {x}^{2}$, $\implies$, $u ' = 2 x$

$v = {x}^{2} + 9$, $\implies$, $v ' = 2 x$

$\frac{\mathrm{dy}}{\mathrm{dx}} = \frac{2 x \left({x}^{2} + 9\right) - 2 x \left({x}^{2}\right)}{{x}^{2} + 9} ^ 2$

$= \frac{2 {x}^{3} + 18 x - 2 {x}^{3}}{{x}^{2} + 9} ^ 2$

$= \frac{18 x}{{x}^{2} + 9}$

The critical values are when $\frac{\mathrm{dy}}{\mathrm{dx}} = 0$

$\frac{18 x}{{x}^{2} + 9} ^ 2 = 0$

When $x = 0$

We can build a chart

$\textcolor{w h i t e}{a a a a}$$x$$\textcolor{w h i t e}{a a a a}$$- \infty$$\textcolor{w h i t e}{a a a a a a}$$0$$\textcolor{w h i t e}{a a a a a a}$$+ \infty$

$\textcolor{w h i t e}{a a a a}$$\frac{\mathrm{dy}}{\mathrm{dx}}$$\textcolor{w h i t e}{a a a a a a}$$-$$\textcolor{w h i t e}{a a a}$$0$$\textcolor{w h i t e}{a a a a}$$+$

$\textcolor{w h i t e}{a a a a}$$y$$\textcolor{w h i t e}{a a a a a a a a}$↘$\textcolor{w h i t e}{a a a}$$0$$\textcolor{w h i t e}{a a a a}$↗

Now, we calculate the second derivative

$u = 18 x$, $\implies$, $u ' = 18$

$v = {\left({x}^{2} + 9\right)}^{2}$, $\implies$, $v ' = 2 \left({x}^{2} + 9\right) \cdot 2 x$

$\frac{{d}^{2} y}{\mathrm{dx}} ^ 2 = \frac{18 {\left({x}^{2} + 9\right)}^{2} - 18 x \cdot \left(4 x \left({x}^{2} + 9\right)\right)}{{x}^{2} + 9} ^ 4$

$= 18 \frac{\left({\left({x}^{2} + 9\right)}^{2} - 4 {x}^{2} \left({x}^{2} + 9\right)\right)}{{x}^{2} + 9} ^ 4$

$= \frac{18 \left({x}^{2} + 9\right) \left(\left({x}^{2} + 9\right) - 4 {x}^{2}\right)}{{x}^{2} + 9} ^ 4$

$= \frac{18 \left(9 - 3 {x}^{2}\right)}{{x}^{2} + 9} ^ 3$

$= \frac{54 \left(3 - {x}^{2}\right)}{{x}^{2} + 9} ^ 3$

$\frac{{d}^{2} y}{\mathrm{dx}} ^ 2 = 0$ when $x = - \sqrt{3}$ and $x = \sqrt{3}$

The points of inflexions are $\left(- \sqrt{3} , \frac{1}{4}\right)$ and $\left(\sqrt{3} , \frac{1}{4}\right)$

We can build the chart

$\textcolor{w h i t e}{a a}$$I n t e r v a l$$\textcolor{w h i t e}{a a a a}$]-oo,-sqrt3[$\textcolor{w h i t e}{a a a a}$]-sqrt3,sqrt3[$\textcolor{w h i t e}{a a a a}$]sqrt3,+oo[

$\textcolor{w h i t e}{a a}$$\frac{{d}^{2} y}{\mathrm{dx}} ^ 2$$\textcolor{w h i t e}{a a a a a a a a a a a a a}$$-$$\textcolor{w h i t e}{a a a a a a a a a a a a}$$+$$\textcolor{w h i t e}{a a a a a a a a a}$$-$

$\textcolor{w h i t e}{a a}$$y$$\textcolor{w h i t e}{a a a a a a a a a a a a a a a a}$$\cap$$\textcolor{w h i t e}{a a a a a a a a a a a a}$$\cup$$\textcolor{w h i t e}{a a a a a a a a a}$$\cap$

${\lim}_{x \to \pm \infty} y = {\lim}_{x \to \pm \infty} {x}^{2} / {x}^{2} = 1$

The horizontal asymptote is $y = 1$

graph{(y-(x^2)/(x^2+9))(y-1)=0 [-7.02, 7.024, -3.51, 3.51]}