Function Characteristics – End Behavior

1. Function Characteristics – End Behavior AII.7 - The student will investigate and analyze functions algebraically and graphically. Key concepts includea) domain and range, including limited and discontinuous domains and ranges; b) zeros; c) x- and y-intercepts; d) intervals in which a function is increasing or decreasing; e) asymptotes;f) end behavior; g) inverse of a function; and h) composition of multiple functions. Graphing calculators will be used as a tool to assist in investigation of functions.

2. End Behavior • The arrows at the end of a graph tell us the image goes on forever. In what direction would you say these graphs continue indefinitely?

3. End Behavior • The end behavior tells us about the far ends of the graph, when the x or y values are infinitely large or small. • Most graphs have two ends so we talk about the left-hand end behavior and the right-hand end behavior. • There are typically two dimensions to the end behavior: left/right and up/down. Most graphs do not have strictly horizontal or vertical end behavior.

4. End Behavior • Terminology – infinite end behavior • If a graph continues to the LEFT indefinitely: • “xapproaches -∞" or symbolically • If a graph continues to the RIGHT indefinitely: • “xapproaches ∞" or symbolically • If a graph continues down indefinitely: • “yapproaches -∞" or symbolically • If a graph continues up indefinitely: • “yapproaches ∞" or symbolically

5. End Behavior Terminology

6. End Behavior – visual approach Let’s see what happens to the graph if we ‘zoom’ out a bit. (note the change in the scales on the graph) Notice that the ends continue to extend in the same directions as we zoom out. What directions do they go?

7. End Behavior – visual approach • Right End Behavior • The right-hand side of this graph goes up indefinitely. • Our two directions are right () and up (). • So the right-hand end behavior is “as approaches , approaches ” • Left End Behavior • The left-hand side of this graph goes down indefinitely. • Our two directions are left () and down (). • So the left-hand end behavior is “as approaches , approaches ”

8. End Behavior – numerical approach • We said the right end behavior of this graph was “as approaches , approaches ”. • Let’s examine this numerically by checking out some large values of x and seeing the y value that go with them.

9. End Behavior – numerical approach Right End Behavior: Notice what happens the y values as x gets exponentially larger. • As x gets exponentially larger, y also continues to get exponentially larger, thus confirming the right end behavior “as approaches , approaches ”.

10. End Behavior – numerical approach Left End Behavior: Notice what happens the y values as x gets exponentially smaller (approaches -∞). • As x gets exponentially smaller, y also continues to get exponentially smaller, thus confirming the left end behavior “as approaches , approaches ”.

11. End Behavior – visual approach What would you predict is the end behavior for the quartic graph we looked at earlier? • Left-hand end behavior: • As xapproaches negative infinity (goes to the left), y approaches infinity (goes up) • Right-hand end behavior: • As xapproaches infinity(goes to the right), y approaches infinity (goes up) • End behavior: • As • Since both ends continued up, we combined the end behaviors into one statement. Goes up to the left Goes up to the right

12. End Behavior – visual approach • End behavior: As • Let’s visually confirm this by ‘zooming out’ on the graph.

13. End Behavior – numerical approach Left End Behavior Right End Behavior As x gets exponentially larger or smaller, y continues to rise exponentially confirming the end behavior: as

14. End Behavior – non-infinite • Predict the end behavior of the following functions. What difference do you notice about their shape compared to the functions we have been exploring?

15. End Behavior – non-infinite • These functions level off as they go to the left and/or right. The y-values do not necessarily approach ±∞.

16. End Behavior – non-infinite Fill in the following tables. Use the data you find to determine the end behavior of this exponential function. Left End Behavior * These values are rounded because the decimal exceeds the capabilities of the calculator. Left End Behavior: As x approaches −∞, yapproaches -1

17. End Behavior – non-infinite Fill in the following tables. Use the data you find to determine the end behavior of this exponential function. Right End Behavior * This value was so large that it exceeded the capabilities of my calculator. Right End Behavior: As x approaches ∞, yapproaches ∞

18. End Behavior – non-infinite Recap: Left End Behavior: As x approaches −∞, yapproaches -1 Right End Behavior: As x approaches ∞, yapproaches∞ OR As , and as

19. End Behavior – non-infinite Fill in the following tables. Use the data you find to determine the end behavior of this rational function. Left End Behavior Left End Behavior: As x approaches −∞, yapproaches 2

20. End Behavior – non-infinite Fill in the following tables. Use the data you find to determine the end behavior of this rational function. Right End Behavior Right End Behavior: As x approaches ∞, yapproaches 2

21. End Behavior – non-infinite Recap: Left End Behavior: as x approaches −∞, yapproaches 2 Right End Behavior: as x approaches ∞, yapproaches2 OR as x∞, y

22. End Behavior - Patterns • Which of the following have the same end behaviors? As As C B A D As ,

23. End Behavior - Patterns • How are these functions similar? • They are all polynomial functions • Their equations are made up of the sum/difference of terms with integer exponents • Their end behaviors always approach∞ or -∞. • A and D have even degrees • A is a quadratic () and D is quartic () • Even degree polynomial functions have the same left and right end behaviors. • Meaning, either both ends go up (as ) or both ends go down () .

24. End Behavior - Patterns • B and C have odd degrees • B is a cubic () and D is quintic() • Odd degree polynomial functions have opposite left and right end behaviors. • Meaning if the function goes down to the left (as then it goes up to the right (as ) and vice versa.

25. End Behavior - Patterns • The leading coefficient will determine whether the functions point up or down. • A negative leading coefficient will cause a reflection over the x-axis. Recap of the end behavior of polynomial functions