Date: 11/5/12- Section: 1.2 Obj.: SWBAT identify horizontal and vertical asymptotes.

Transcription

1 Date: 11/5/12- Section: 1.2 Obj.: SWBAT identify horizontal and vertical asymptotes. Bell Ringer: Graded Quiz Evaluating Fucntions Homework Requests: Symmetry Worksheet #2 Lecture: Mini lecture on Asymptotes vertical and horizontal(end behavior) Classwork Work odd problems pg 99; odds Homework: Continue Symmetry WS #2 If you don t know it, Then you can t show it! Announcements: Symmetry WS #2 due Friday for reduced credit. No Calculator Sharing on tests from here on out. You may borrow a Calculator from me on Calculator loan program.

2 What is an asymptote? Asymptote - a line that a graph approaches, but does not intersect. Example, for y=1/x, the line approaches the x-axis (y=0), but never touches it. No matter how far we go into infinity, the line will not actually reach y=0, but will always get closer and closer. There are horizontal asymptotes. And vertical asymptotes Occur because the denominator of a fraction has gone to zero. Ex. Y =4/(x-2)

3 Horizontal and Vertical Asymptotes The line y b is a horizontal asymptote of the graph of a function y f ( x) if f ( x) approaches a limit of b as x approaches + or -. In limit notation: lim f ( x) b or lim f ( x) b. x x The line x a is a vertical asymptote of the graph of a function y f ( x) if f ( x) approaches a limit of + or - as x approaches a from either direction. In limit notation: lim f ( x) or lim f ( x). x a x a Slide 1-3

4 Date: 11/6/12- Section: 1.2 Obj.: SWBAT identify horizontal and vertical asymptotes. Bell Ringer: Work 2 problems from Symmetry Worksheet #2 Homework Requests: Symmetry Worksheet #2 Lecture: Mini lecture on Asymptotes vertical and horizontal(end behavior) Classwork Work odd problems pg 99; odds Homework: pg 99 #56-66 evens Announcements: If you don t know it, Then you can t show it! No Calculator Sharing on tests from here on out. You may borrow a Calculator from me on Calculator loan program.

5 Asymptote Tests: Vertical Asymptotes Look for places where the denominator = 0. Since, the denominator 0. The vertical asymptote is where the function goes to +/- infinity. y= x+2. Find where denom. = 0 Solve 2x+1 = 0 denom. 0 x -1/2 vert. asymptote at x = - ½ 2x+1 F(x) = (x+2)/(2x+1) Vertical asymptote occurs at x = -(1/2) lim 3x/(x+2) = - lim 3x/(x+2) = X-> -(1/2) - X-> -(1/2) + Examples: Find the vertical asymptotes of these functions. Sketch the graph. F(x) = 1/x f(x) = x / (x 2-1) Vertical asymptote occurs at x = 0 Vertical asymptotes occurs at x = +1, -1 lim 1/x = - lim 1/x = + lim x / (x 2-1) = + lim x / (x 2-1) = - X-> 0 - X-> 0 + X-> -1 - X-> -1 + lim x / (x 2-1) = - lim x / (x 2-1) = + X-> +1 - X-> +1 + F(x) = 3x/(x+2) Vertical asymptote occurs at x = -2 h(x) = 2/ (x 2 +1) lim 3x/(x+2) = lim 3x/(x+2) = - X-> -2 - X-> -2 +

6 Asymptote Tests: Horizontal Asymptotes (Case +, Case - ) and End Behavior End Behavior behavior as x goes to +/- Examine the Function: 1. Algebraic solution: y= x+2 2x+1 As x gets large this function begins to behave like y= x or y= 1 2x 2 Case + positive x gets large this fraction approaches ½ Case - negative x gets large this fraction approaches ½ 2. Numerical solution y= x+2. 2x+1 Case + For x substitute in a very large positive value like Then look at the value of y. Case - Substitute in a very large negative value like Then look at the value of y. Both cases: If y hangs around a value y=b, then this is the horizontal asymptote. Be very careful with parentheses. 3. Table: Look at table for large + and - values of x. If y hangs around a value y=b, then this is the horizontal asymptote.

7 Date: 11/7/12- Section: 1.2 Obj.: SWBAT identify horizontal and vertical asymptotes. Bell Ringer: pg 99 #55, 57 Homework Requests: pg 99 #56-66 evens Lecture: Mini lecture on Asymptotes vertical and horizontal(end behavior) Classwork Work odd problems pg 99; odds Homework: pg 99 #56-66 evens Finish problems finding Horiz. and Vert. asymptotes.; Sketch Graphs Read Section 1.3 If you don t know it, Then you can t show it! Announcements: Tutoring Today No Calculator Sharing on tests. You may borrow a Calculator from me on Calculator loan program.

8 Asymptote Tests: Horizontal Asymptotes (Case +, Case - ) and End Behavior End Behavior behavior as x goes to +/- Examine the Function: 1. Algebraic solution: y= x+2 2x+1 As x gets large this function begins to behave like y= x or y= 1 2x 2 Case + positive x gets large this fraction approaches ½ Case - negative x gets large this fraction approaches ½ 2. Numerical solution y= x+2. 2x+1 Case + For x substitute in a very large positive value like Then look at the value of y. Case - Substitute in a very large negative value like Then look at the value of y. Both cases: If y hangs around a value y=b, then this is the horizontal asymptote. Be very careful with parentheses. 3. Table: Look at table for large + and - values of x. If y hangs around a value y=b, then this is the horizontal asymptote.

9 Asymptote Tests: Horizontal Asymptotes (Case +, Case - ) and End Behavior End Behavior behavior as x goes to +/- Examine the Function: 1. Algebraic solution: y= x+2 2x+1 As x gets large this function begins to behave like y= x Case + positive x gets large this fraction approaches ½ Case - negative x gets large this fraction approaches ½ lim x+2 = 1 2x+1 2 lim x+2 2x+1 = 1 2 X-> - X-> + Examples: F(x) = 1/x Case + Case - F(x) = 1/+ F(x) = 1/+ Approaches 0 Horizontal asymptote approaches 0 lim 1/x = 0 lim 1/x= 0 X-> - X-> + 2x or y= 1 2

10 Asymptote Tests: Horizontal Asymptotes (Case +, Case - ) and End Behavior End Behavior behavior as x goes to +/- Examine the Function: Examples: 2. F(x) = 3x/(x+2) As x gets large 3x/x Case - Case + F(x) = 3(- )/(- ) = 3 F(x) = 3( )/( ) = 3 Horizontal asymptote approaches 3 lim 3x/(x+2) = 3 lim 3x/(x+2) = 3 X-> - X-> + 3. j(x) = 3x/(2-x) As x gets large 3x/(-x) Case - Case + j(x) = 3(- )/(- - ) = -3 j(x) = 3( )/(- ) = -3 Horizontal asymptote approaches 3 lim 3x/(2-x) = -3 lim 3x/(2-x) = -3 X-> - X-> +

11 Asymptote Tests: Horizontal Asymptotes (Case +, Case - ) and End Behavior End Behavior behavior as x goes to +/- Examine the Function: Examples: 4. F(x) = 2 x /(x-1) As x gets large 2 x /x Case - Case + F(x) = 2 /(- ) = -2 F(x) = 2 /( ) = 2 Horizontal asymptote approaches 3 lim 2 x /(x-1) = -2 lim 2 x /(x-1) ) = 2 X-> - X-> + 5. h(x) = 2/ (x 2 +1) As x gets large 2/(x 2 ) Case - Case + j(x) = 2/(- ) 2 = 0 j(x) = 2/( ) 2 = 0 Horizontal asymptote approaches 0 lim 2/ (x 2 +1) = 0 lim 2/ (x 2 +1) = 0 X-> - X-> +

12 Date: 11/8/12- Section: 1.2 Obj.: SWBAT identify and evaluate the 12 Basic Functions Bell Ringer: None Homework Requests: pg 99 #56-66 evens Classwork Work: Asymptote Worksheet Homework: Complete Asymptote Worksheet Read Section 1.3 Announcements: Tutoring Today If you don t know it, Then you can t show it! No Calculator Sharing on tests. You may borrow a Calculator from me on Calculator loan program.