Summary of Graphing - MAT 271

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Summary of Graphing - MAT 271

Symmetry

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

Periodic functions

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

Second Derivative Test

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

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

Concavity

Concave down if second derivative is negative

Concave up if second derivative is positive

Transformations

Increasing/Decreasing

Increasing where first derivative is positive

Decreasing where first derivative is negative

Holes

Occur where num. and denom. have common factors

Asymptotes

horizontal

Rational Function - compare degrees of num. and denom.

If same - ha is y = ratio of leading coefficients

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

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

vertical

Rational function - where denominator = 0

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 Line Test

To verify a function has been drawn

Plot a Few Points

Points of Inflection

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.

First Derivative Test

Assumes c is a critical number and f is continuous

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

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

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

Parent Function Graphs

Trig functions - period/amplitude, etc.

Extrema

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).

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

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

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)

Jumps

non-continuous - piece-wise functions

Intercepts

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

y-intercept => set x = 0

Domain of the function

Denominator can not equal 0

You can only take a logarithm of a positive number

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

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

You can only take even roots of non-negative numbers