ICM ~ Unit 4 ~ Day 3. Horizontal Asymptotes, End Behavior

ICM ~ Unit 4 ~ Day 3 Horizontal Asymptotes, End Behavior

Warm Up ~ Day 3 1. Find the domain, then convert to fractional/rational eponent. f ( ) 7. Simplify completely: 3( + 5). 3. Find the domain, & y intercepts, and label any discontinuities: h 16 3

Warm Up ~ Day 3 ANSWERS 1. Find the domain, then convert to fractional/rational eponent. Domain : f ( ) 7. Simplify completely: 3( + 5). 1 (7 ) 1 6075 (,7] 3. Find the domain, & y intercepts, and label any discontinuities: h 16 3 Domain : [ 4,3) (3,4] int : ( 4, 0)&(4,0) 4 y int : (0, ) 3 Nonremovable Discontinuity (Vertical Asymptote at = 3)

Homework Answers

What is End Behavior? A Graphical Approach

Let s do a little research on end behavior. This will help us write range and evaluate limits at and -. We ll try these after some definitions

Polynomials End Behavior of Graphs Rational Functions End Behavior: what the graph does as (What is the graph doing at huge values and itty bitty values?) or * End Behavior is similar to finding Horizontal Asymptotes (if there is a HA)!

Definition of a Limit If f() becomes arbitrarily close to a unique number L as approaches c from either side, the limit of f() as approaches c is L. L is a y-value! c is an -value! lim f ( ) L c

Before an investigation on End Behavior, let s evaluate limits at and - for these graphs.

Before an investigation on End Behavior, let s evaluate limits at and - for these graphs. lim f( ) lim f( ) lim f( ) lim f( ) lim f( ) lim f( ) lim f( ) lim f( )

Definition of Degree Degree of a polynomial in one variable: the value of the greatest eponent E: f ( ) 4 9 8 Degree: (Even Degree) E: 3 g( ) 5 6 4 Degree: 3 (Odd Degree) Degree can help us with determining end behavior of polynomials

End Behavior - Polynomials Graph the following in your calculator and take note of the end behavior. Find the limits at and - of each. Epress them using proper Limit notation. Then, determine a way to predict end behavior without graphing and using limits. y 4 9 8 3 y 5 6 4 y 5 3 4 y 9 y 3 7 y 5 4 9 7 3 Hint: look at Degree and Leading Coefficient

End Behavior - Polynomials Graph the following in your calculator and take note of the end behavior. Determine a way to predict end behavior without graphing and using limits. (Hint: Degree) y 4 9 8 3 y 5 6 4 y 5 3 4 y 9 y 3 7 y 5 4 9 7 3 lim f ( ) lim f ( ) lim f ( ) lim f ( ) lim f ( ) lim f ( ) lim f ( ) lim f ( ) lim f ( ) lim f ( ) lim f ( ) lim f ( )

End Behavior Summary Odd, + starts down, ends up (Like + Slope) Odd, starts up, ends down (Like Slope) Odd Ends go Opposite (of each other)

End Behavior Summary Even, + both ends point up (Like + Parabola) Even, both ends point down (Like Parabola) Even Ends go Eactly the same

End Behavior Summary Odd, + starts down, ends up Odd, starts up, ends down Odd Ends go Opposite (of each other) Even, + both ends point up Even, both ends point down Even Ends go Eactly the same

End Behavior: Rational Functions Rational functions: Ratio of two polynomials As you may have guessed, degrees of these two polynomials play a key role in determining the end behavior. Consider the following scenarios: 1) The degree of the numerator is bigger than the degree of the denominator. ) The degree of the numerator is the same as the degree of the denominator. 3) The degree of the numerator is smaller than the degree of the denominator. Determine a way to predict end behavior without graphing.

Definition of Degree Degree of a polynomial in one variable: the value of the greatest eponent E: f ( ) 4 9 8 Degree: E: 3 g( ) 5 6 4 Degree: 3 Degree can help us with determining the horizontal asymptote of rational functions...

Horizontal Asymptotes For horizontal asymptotes, think BOSTON for polynomials! Looking at the degree of top & bottom Bottom > Top y=o Same = ratio Top > Bottom O No HA. N f( ) g ( ) 3 3 5 4 3 5 h ( ) 7 3 H. A. : y 0 H. A. : No H. A. y 5

You Try! What is the EQUATION of the horizontal asymptote for the following functions? Then write the end behavior using limits. f( ) 3 9 74 11 H. A. : y 3 4 3 lim f( ) 4 3 lim f( ) 4 Bottom > Top y=o Same = ratio Top > Bottom O N No HA. g ( ) 4 3 5 9 lim g ( ) lim g ( ) H. A. : none h ( ) 7 15 lim h ( ) 0 lim h ( ) 0 H. A. : y 0

Finding the Range of a Function Use numeric, algebraic and graphical approaches simultaneously. Keep in mind we are finding ALL y-coordinates of points on the graph. Write the range of the following functions in interval notation. 8 f ( ) 7 g ( ) You 1 Try! m ( ) 9

Finding the Range of a Function Use numeric, algebraic and graphical approaches simultaneously. Keep in mind we are finding ALL y-coordinates of points on the graph. Write the range of the following functions in interval notation. 8 f ( ) 7 g ( ) 1 Range Range : [0, ) : (, 0) (0, ) m ( ) Range 9 : (, )

Domain: Summary Consider the vertical asymptotes and the -value of the hole Make sure values under the radical are positive Range: Consider the horizontal asymptotes and the y-value of the hole -intercept: Set y = 0 and solve for. y-intercept: Set = 0 and solve for y.

Practice! 4.) g ( ) 1 11.) f( ) ( 3)( 1) Find the -Domain - & y intercepts -End Behavior using limits -Range 3 1.) g ( ) 1 3.) f( ) 1 Tetbook: p.98

Tetbook: p.98 4.) g ( ) lim g ( ) 1 lim g ( ) 1 Domain : (,) (, ) Range : (,1) (1, ) int : (0, 0) y int : ( 0,0) Find the -Domain - & y intercepts -End Behavior using limits -Range 1 11.) f( ) ( 3)( 1) lim f( ) 0 lim f( ) 0 Domain : (, 3) ( 3, 1) (1, ) 1 1 Range : (, 0) (0, ) (, ) 4 4 int : none 1 y int : (0, ) 3 1 Hole :(1, ) 4

Tetbook: p.98 3 1.) g ( ) 1 lim g ( ) 1 lim g ( ) 1 Domain : (,0) (0, ) Range : (,1) (1, ) int : ( 3, 0) y int : none Find the -Domain - & y intercepts -End Behavior using limits -Range 3.) f( ) lim f( ) 1 lim f( ) 1 1 Domain : (,0) (0, ) Range : (, 1) [0 ) int: (1, 0) y int : none

Asymptote Lab Packet p. 4-5 HW Packet p. 6