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
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 $\displaystyle f(0)$.