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

Question

Sophie Adams · Accepted Answer

See below.

First find significant points, these will help in sketching the graph. y axis intercepts occur when #x=0#: #6/((0)^2+3)=2color(white)(88)# coordinate #( 0 , 2 )# x axis intercepts occur when #y=0#: #6/(x^2+3)=0# This can only be zero when denominator is zero, which is undefined, so no x axis intercepts. as #x->oo# #color(white)(888)6/(x^2+3)->0# as #x->-oo# #color(white)(888)6/(x^2+3)->0# So the x axis is a horizontal asymptote. #y=0# Vertical asymptotes occur where the function is undefined: #6/(x^2+3)# is not undefined for any real #x#, so no vertical asymptotes. #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]}

Chloe Alexander · Answer

undefined To graph $ y = \frac{6}{x^2 + 3} $, follow these steps:

1. Identify key points: Find the vertical asymptotes, horizontal asymptotes, x-intercepts, and y-intercepts, if any.
2. Plot the key points on the coordinate plane.
3. Determine the behavior of the function as x approaches positive and negative infinity.
4. Sketch the curve connecting the key points, ensuring it follows the behavior of the function.

Vertical asymptotes occur where the denominator equals zero. In this case, $ x^2 + 3 = 0 $ has no real solutions, so there are no vertical asymptotes.

Horizontal asymptotes occur as $ x $ approaches positive or negative infinity. Since the highest power of $ x $ in the denominator is $ x^2 $, and there is no corresponding term in the numerator, the horizontal asymptote is at $ y = 0 $.

To find the x-intercepts, set $ y = 0 $ and solve for $ x $:

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

This equation has no real solutions, so there are no x-intercepts.

To find the y-intercept, set $ x = 0 $:

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

So, the y-intercept is at $ (0, 2) $.

As $ x $ approaches positive or negative infinity, the function approaches the horizontal asymptote at $ y = 0 $.

Now, plot the y-intercept at $ (0, 2) $ and sketch the curve. Since the function approaches $ y = 0 $ as $ x $ approaches positive or negative infinity, the graph will decrease towards the x-axis in both directions.