Summary of Graphing - MAT 271

You can only take even roots of non-negative numbers

Real life restrictions (no negative distances, time, etc.)

Domain info can give you the start or end of a graph

You can only take a logarithm of a positive number

Denominator can not equal 0

y-intercept => set x = 0

x-intercept(s) => set y = 0

non-continuous - piece-wise functions

Minima

f'(c) = 0, second derivative is positive OR f'(c) = 0 and f' changes from - to + at c . The minimum value is f(c)

Absolute extrema is the largest or smallest value taken on by the function (Consider endpoints of closed intervals as well as all local extrema)

Local extrema is max/min on some interval of the function

Maxima

f'(c) = 0, second derivatie is negative OR f'(c) = 0 and f' changes from + to - at c. The maximum value is f(c).

Trig functions - period/amplitude, etc.

If f' changes from + to - at c, then f has a local max at c.

If f' cjanges from - to + at c, then f has a local min at c.

If f' does not change sign at c, then f has no local max or min at c.

Assumes c is a critical number and f is continuous

If f''(c) exists and f'' changes sign at c, then we have an inflection point at c. Inflection point is (c, f(c)). If f''(c) is an inflection point, then f''(c) = 0.

To verify a function has been drawn

oblique

If num. degree is greater than the denom. - found by long division

If the num. degree is 1 more than denom. - it's called a slant asymptote - found by long division

vertical

Rational function - where denominator = 0

horizontal

Rational Function - compare degrees of num. and denom.

If num. degree is less than the denom. - ha is y = 0

Calculus - evaluate the limit as x approaches + or - infinity

If same - ha is y = ratio of leading coefficients

Occur where num. and denom. have common factors

Decreasing where first derivative is negative

Increasing where first derivative is positive

Concave up if second derivative is positive

Concave down if second derivative is negative

If f'(c) = 0 and f''(c)<0, then f has a local maximum at c

If f'(c)=0 and f''(c)>0, then f has a local minimum at c

f(-x) = f(x) => f is an EVEN function => symmetry about the y-axis

Periodic functions

f(-x) = -f(x) => the function is and ODD function => the graph passes through (0,0) and is symmetrical about the origin (turn graph upside down = looks the same