1. Pre Calculus Functions and Graphs

2. Functions A function is a relation where each element of the domain is paired with exactly one element of the range independent variable - x dependent variable - y domain - set of all values taken by independent variable range - set of all values taken by the dependent variable

3. Mapping 3 -6 9 12 5 0 -8 2

4. Representing Functions notation - f(x) numerical model - table/list of ordered pairs, matching input (x) with output (y) US Prison Polulation (thousands) YearTotalMaleFemale 198032931613 198550247923 199077473044 19951125105768 20001391129893 200515261418108

5. graphical model - points on a graph; input (x) on horizontal axis … output (y) on vertical algebraic model - an equation in two variables

6. Vertical Line Test

7. Finding the range implied domain - set of all real numbers for which expression is defined example: Find the range

9. Continuity http://www.calculus-help.com/tutorials function is continuous if you can trace it with your pencil and not lift the pencil off the paper

10. Discontinuities point discontinuity –graph has a “hole” –called removable –example

12. jump discontinuity - gap between functions is a piecewise function example

14. infinite discontinuity - there is a vertical asymptote somewhere on the graph example

16. Finding discontinuities factor; find where function undefined sub. each value back into original f(x) results …

17. Increasing - Decreasing Functions function increasing on interval if, for any two points decreasing on interval if constant on interval if

18. Example:

20. Boundedness of a Function

23. Extremes of a Function local maximum - of a function is a value f(c) that is greater than all y- values on some interval containing point c. If f(c) is greater than all range values, then f(c) is called the absolute maximum

24. local minimum - of a function is a value f(c) that is less than all y-values on some interval containing point c. If f(c) is less than all range values, then f(c) is called the absolute minimum

25. A B C D E F G H I J K local maxima Absolute maximum Absolute minimum local minima

26. Example: Identify whether the function has any local maxima or minima

27. Symmetry graph looks same to left and right of some dividing line can be shown graphically, numerically, and algebraically graph: x f(x) -3 9 1 0 0 1 1 3 9 numerically

28. algebraically even function –symmetric about the y-axix –example

29. odd function –symmetric about the origin –example

30. Additional examples: even / odd

31. Asymptotes horizontal - any horizontal line the graph gets closer and closer to but not touch vertical - any vertical line(s) the graph gets closer and closer to but not touch Find vertical asymptote by setting denominator equal to zero and solving

32. End Behavior A function will ultimately behave as follows: –polynomial … term with the highest degree –rational function … f(x)/g(x) take highest degree in num. and highest degree in denom. and reduce those terms –example