Short of using my calculator how do I go about finding out all the transition points, asymptotes and intercepts of an equation?

y= __1_x ^{2}+3

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- Nov 15th 2012, 05:23 PMChip7723Sketching a Graph
Short of using my calculator how do I go about finding out all the transition points, asymptotes and intercepts of an equation?

y= ____1___x ^{2}+3 - Nov 15th 2012, 06:03 PMMarkFLRe: Sketching a Graph
a) I am assuming transition points refer to local extrema (or other potential places the slope changes sign) and inflections points.

i) local extrema:

Equate the first derivative to zero, solve for .

ii) places where function is not differentiable:

Look at where the derivative is undefined or does not exist.

iii) inflection points:

Equate the second derivative to zero to find critical numbers, and check to see if the sign of the second derivative changes across these critical numbers.

b) asymptotes

i) vertical:

look at where the function is undefined, and ensure the singularity in not removable.

ii) horizontal:

Evaluate .

iii) oblique

If the degree of the numerator is one more than the degree of the denominator, perform polynomial division and the quotient gives the asymptote.

c) intercepts

i)

Set and solve for .

ii)

Evaluate .