AII.7 - The student will investigate and analyze functions algebraically and graphically. Key concepts include a) 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.

 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?

 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.

Direction of Infinite Continuance We say Symbolic Notation Left x approaches negative infinity Right x approaches infinity Up y approaches infinity Down y approaches negative infinity

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?

xy ,000 10, ,000 1,000,000 Right End Behavior: Notice what happens the y values as x gets exponentially larger. xy ,000 10, ,000 1,000,000 xy ,000 10, ,000 1,000,000 xy ,0001,999 10, ,000 1,000,000 xy ,0001,999 10,00019, ,000 1,000,000 xy ,0001,999 10,00019, ,000199,999 1,000,000 xy ,0001,999 10,00019, ,000199,999 1,000,0001,999,999

xy , , ,000 -1,000,000 Left End Behavior: Notice what happens the y values as x gets exponentially smaller (approaches - ∞). xy , , ,000 -1,000,000 xy , , ,000 -1,000,000 xy ,000-2, , ,000 -1,000,000 xy ,000-2, ,000-20, ,000 -1,000,000 xy ,000-2, ,000-20, , ,001 -1,000,000 xy ,000-2, ,000-20, , ,001 -1,000,000 -2,000,001

What would you predict is the end behavior for the quartic graph we looked at earlier? Goes up to the left Goes up to the right

xy , , ,000 -1,000,000 xy ,088, , , ,000 -1,000,000 Left End Behavior xy ,000 10, ,000 1,000,000 xy ,888, ,000 10, ,000 1,000,000 xy ,888, ,000 10, ,000 1,000,000 xy ,888, ,000 10, ,000 1,000,000 xy ,888, ,000 10, ,000 1,000,000 xy ,888, ,000 10, ,000 1,000,000 xy ,000 10, ,000 1,000,000 Right End Behavior xy ,088, , , ,000 -1,000,000 xy , , ,000 -1,000,000 xy ,088, , , ,000 -1,000,000 xy ,088, , , ,000 -1,000,000 xy ,088, , , ,000 -1,000,000

 Predict the end behavior of the following functions. What difference do you notice about their shape compared to the functions we have been exploring?

 These functions level off as they go to the left and/or right. The y-values do not necessarily approach ±∞.

xy ,000 Fill in the following tables. Use the data you find to determine the end behavior of this exponential function. xy * -1,000-1* * These values are rounded because the decimal exceeds the capabilities of the calculator. Left End Behavior Left End Behavior: As x approaches − ∞, y approaches - 1

xy ,000 Fill in the following tables. Use the data you find to determine the end behavior of this exponential function. xy ,217, ,000Error* * This value was so large that it exceeded the capabilities of my calculator. Right End Behavior Right End Behavior: As x approaches ∞, y approaches ∞

xy , ,000 Fill in the following tables. Use the data you find to determine the end behavior of this rational function. xy , , Left End Behavior Left End Behavior: As x approaches − ∞, y approaches 2

xy ,000 10,000 Fill in the following tables. Use the data you find to determine the end behavior of this rational function. xy , , Right End Behavior Right End Behavior: As x approaches ∞, y approaches 2

 Which of the following have the same end behaviors? A B C D

 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 Degree Leading Coefficient Even Positive Negative Odd Positive Negative