1. Graphs of Polynomial Functions A-REI.10 Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve (which could be a line). F-IF.4 For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. F-IF.7 Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases. Graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior. A-REI.10 Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve (which could be a line). F-IF.4 For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. F-IF.7 Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases. Graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior.

2. Graphing Polynomial Functions  To graph:  Type equation into y =  2 nd graph (to get table)  Plot points from table  Graph  Look at graph to determine shape  To graph:  Type equation into y =  2 nd graph (to get table)  Plot points from table  Graph  Look at graph to determine shape

3. Describe the…  Odd or Even Function  End behavior  Number of zeros (roots/x-intercepts)  Number of turning points  Y-intercept  Where zeros are located  Where relative extrema are located  Interval on which the function is increasing or decreasing  Domain and range  Odd or Even Function  End behavior  Number of zeros (roots/x-intercepts)  Number of turning points  Y-intercept  Where zeros are located  Where relative extrema are located  Interval on which the function is increasing or decreasing  Domain and range

4. Tell if the function is odd or even.  Even: graph is symmetric about the y-axis  Odd: graph is symmetric about the origin  Neither: graph has no symmetry  Even: graph is symmetric about the y-axis  Odd: graph is symmetric about the origin  Neither: graph has no symmetry

5. Odd/Even Functions

6. End behavior

8. Zeros/ Roots/ X-intercepts and Y-intercept  Number of real zeros: Count the number of times the graph crosses the x-axis  Find the zeros: Tell where the x-intercepts are located  On an exact point  In between two consecutive x-values  Y-intercept: where graph crosses the y-axis  Number of real zeros: Count the number of times the graph crosses the x-axis  Find the zeros: Tell where the x-intercepts are located  On an exact point  In between two consecutive x-values  Y-intercept: where graph crosses the y-axis

9. X and Y intercepts

10. Relative Extrema  Turning points: where the graph changes direction  Relative extrema: minimum or maximum of the function on a specific interval  “Hills and Valleys”  Relative Minima: “valleys” low points of the graph  Relative Maxima: “hills” high points of the graph  Turning points: where the graph changes direction  Relative extrema: minimum or maximum of the function on a specific interval  “Hills and Valleys”  Relative Minima: “valleys” low points of the graph  Relative Maxima: “hills” high points of the graph

11. Relative Extrema

12. Interval on which the function is increasing or decreasing  Increasing: graph goes up from left to right  Decreasing: graph goes down from left to right  Use the x-values of the endpoints of the interval and write in interval notation  Use parentheses if not equal ( )  Use straight brackets if equal [ ]  Increasing: graph goes up from left to right  Decreasing: graph goes down from left to right  Use the x-values of the endpoints of the interval and write in interval notation  Use parentheses if not equal ( )  Use straight brackets if equal [ ]

13. Intervals of Increasing and Decreasing

14. Extra Examples

16. Parts of a Graph from the Equation Odd/Even/Neither Odd: all exponents odd Even: all exponents even Neither: Mixture of odd and even exponents End Behavior Odd Degree, Positive LC: Down and Up Odd Degree, Negative LC: Up and Down Even Degree, Positive LC: Up and Up Even Degree, Negative LC: Down and Down X-intercepts Solutions of the equation (2 nd trace 2 in calculator) Y-intercept (0, constant term) Relative Extrema and Interval of Increasing/Decreasing Number of turning points = One less than the degree Type into y = in calculator 2 nd trace 3 for min 2 nd trace 4 for max

17. Examples