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 $\displaystyle x$.

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 $\displaystyle \lim_{x\to\pm\infty}f(x)$.

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) $\displaystyle x$

Set $\displaystyle f(x)=0$ and solve for $\displaystyle x$.

ii) $\displaystyle y$

Evaluate $\displaystyle f(0)$.