How do you graph y=6/(x^2+3)?

Dec 21, 2017

See below.

Explanation:

First find significant points, these will help in sketching the graph.

y axis intercepts occur when $x = 0$:

$\frac{6}{{\left(0\right)}^{2} + 3} = 2 \textcolor{w h i t e}{88}$ coordinate $\left(0 , 2\right)$

x axis intercepts occur when $y = 0$:

$\frac{6}{{x}^{2} + 3} = 0$

This can only be zero when denominator is zero, which is undefined, so no x axis intercepts.

as $x \to \infty$ $\textcolor{w h i t e}{888} \frac{6}{{x}^{2} + 3} \to 0$

as $x \to - \infty$ $\textcolor{w h i t e}{888} \frac{6}{{x}^{2} + 3} \to 0$

So the x axis is a horizontal asymptote.

$y = 0$

Vertical asymptotes occur where the function is undefined:

$\frac{6}{{x}^{2} + 3}$ is not undefined for any real $x$, so no vertical asymptotes.

$\frac{6}{{x}^{2} + 3}$ attains a maximum value when the denominator is a minimum value, this can be seen to be 3 when $x = 0$

( this is the y axis intercept that was previously found )

Graph:

graph{y=6/(x^2+3) [-10, 10, -5, 5]}