Graphing

I want to discuss the methods used to hand draw the graph of this equation:



Lets start by looking for symmetry.



This equation is even because of its y-axis symmetry.

Next lets look at the domain of this function:



Now we look for x-int/y-int:

x-int:



y-int:

Does not exit.


Their isn't a x-int either because they are not within the domain. This is because they are imaginary. thanks a lot square root! :-)

Next we will look for asymptotes:

vertical:



Vertical asymptotes at -2 & 2.

Horizontal:

DNE

Oblique/curvilinear:

DNE

Next lets look for critical points:



Therefore critical points are located at

(zero would be a critical point if it was inside of the domain)

I will leave it to you to plug in test values and execute the first derivative test. Your result should be that f'(x) decreases from (-infinity,-square root of 7), (stationary at -square root of 7), increases (-square root of 7, -2), decreasing (2, square root of 7), and increasing (square root of 7, +infinity). With these results under consideration we conclude that plus/minus square root 7 is minimum.

Next we would examine concavity but your calculations should align with mine. I will give you the second derivative and allow you to find the intervals yourself. You should conclude that f''(x) is always increasing and has no inflection points. Here is the second derivative:



This is one hell of a goddamn derivative...

(good luck, hopefully of you can complete this you will be good for the test :-)