ICM ~Unit 4 ~ Day 2. Section 1.2 Domain, Continuity, Discontinuities

Transcription

1 ICM ~Unit 4 ~ Day Section 1. Domain, Continuity, Discontinuities

2 Warm Up Day Find the domain, -intercepts and y-intercepts Factor completely Factor completely. 8 7

3 Practice Find the domain and & y intercepts of f 1 5 h g 4 1 1

4 Warm Up Day Find the domain, -intercepts and y-intercepts. 5 5 Domain : [, ) (, ) int : (,0) y int :(0, 5). 1 9 Domain : (, 3) ( 3,3) (3, ) int : none 1 y int :(0, ) 9

5 Warm Up Day 3. Factor completely Factor completely. 8 7 (34)( ) ( 3)(4 6 9)

6 Practice Find the domain and & y intercepts of f Domain : int : (0.5, 0) y int : None 1 5 h g (,0) (0,3) (3, ) Domain : [-, ] Domain : int : (, 1) ( 1,4] (4, 0) y int : (0, ) - int : (, 0), (-, 0) y - int : (0, - 3 )

7 Homework Questions?

8 Defining Continuity A function is continuous at a point if the graph does not come apart at that point. Try graphing these: E: y = - 3 E: f() = e + 7 E: g() = 6 E: h() =

9 A function is continuous at all -values if... There are NO breaks in the curve. Notice: You can trace the entire graph without lifting your pencil!

10 Different Types of discontinuities: 1. Removable also referred to as a hole. Nonremovable a. Jump (occur in piecewise functions or greatest integer) b. Vertical Asymptote (infinite discontinuity)

11 E: f Removable discontinuity hole ( ) There is a y-value for every ecept at = -. hole ( ) The function is undefined and discontinuous at = - This occurs when there is a zero in the denominator that can be canceled out when algebraic steps are taken.

12 Jump Discontinunity the curve jumps from one y-value to the net (NONREMOVABLE) Notice: this graph still has a y-value for every. Notice: this one doesn t

13 Nonremovable Discontinunity (Vertical Asymptote) Infinite Discontinuity E: f( ) 3 There is a zero in the denominator of a function that cannot be canceled out through algebra.

14 Finding Discontinuities Summary To find vertical asymptotes and holes, factor the problem and simplify. If the factor in the bottom cancels out, it gives us a hole. Removable f( ) ( )( 1) 1 How do you find the y-value of the hole? (, ) Simplify and substitute the 3 1 -value into the equation! f ( ) 1 If the factor does not cancel out, it gives us a vertical asymptote. Nonremovable So the hole occurs at = - The vertical asymptote for f() is at = 1

15 More Eamples: Find the discontinuities and determine whether they are removable or nonremovable. Hint: Factor first! 1) f( ) 4 6 ) f( )

16 More Eamples ANSWERS: Find the discontinuities and determine whether they are removable or nonremovable. Hint: Factor first! 1) f( ) Hole : (, ) 5 V. A. 3 ) f( ) Hole : (0, ) and (3, 1) 4 V. A. 4

17 You Try! Find the discontinuities and determine whether they are removable or nonremovable. Hint: Factor first! 5 3) )

18 You Try! Find the discontinuities and determine whether they are removable or nonremovable. Hint: Factor first! 5 3) ANSWERS 5 ( 5)(3) 1 (3 ) 1 Hole : ( 5, ) 17 V. A. 3 4) ( 3)( 3) 3( 3) Hole V. A. : ( 3, 18) None 3( 3)

19 Student Practice: A) Classify the function as continuous or discontinuous. B) If discontinuous, specify the type of discontinuity and where it eists. C) State the domain and & y intercepts. 1) f ( ) ( 3)( ) ) f( ) 4

20 Student Practice: ANSWERS A) Classify the function as continuous or discontinuous. B) If discontinuous, specify the type of discontinuity and where it eists. C) State the domain and & y intercepts. 1) f ( ) ( 3)( ) ) f( ) 4 Domain : Continuous Discontinuous Domain : (-, ) - int : (-3, 0), (, 0) y - int : (0, - 6) Hole at (, 4) int : (, 0) y int : (0, ) removable( hole) (,) (, )

21 Try this one! h ( ) 3 Domain: & y intercepts: Continuous? Type of discontinuity? Where is the discontinuity?

22 Try this one! ANSWERS h ( ) 3 Domain: & y intercepts: Continuous? Type of discontinuity? Domain : (,0) (0, ) No Where is the discontinuity? int : None yint : None Removable (Hole) Hole at (0, 1)

23 Let s look at this one together Domain: & y intercepts: f ( ) Continuous? Type of discontinuity?

24 Let s look at this one ANSWERS Domain: & y intercepts: f ( ) (, ] [0] [, ) int : (,0), (0,0), (,0) 4 4 y int : (0,0) Continuous? No Type of discontinuity? None

25 Eamine the table of values below. All of the following statements are true EXCEPT A. = - is a vertical asymptote in y 1 B. = - is an infinite discontinuity in y 1 C. = - is a removable discontinuity in y D. = - is a vertical asymptote in y Tomorrow, you ll do an Asymptote Lab to learn more about this. Looking at y, the y s are in order, but y = -4 was skipped, so there is a Removable Discontinuity (Hole) at (-, -4)

26 Definition of Degree Degree of a polynomial in one variable: the value of the greatest eponent E: f ( ) Degree: E: 3 g( ) Degree: 3 Degree can help us determine the horizontal asymptote of rational functions...

27 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 ( ) h ( ) 7 3 H. A. : y 0 H. A. : No H. A. y 5

28 You Try! What is the EQUATION of the horizontal asymptote for the following functions? f( ) Bottom > Top y=o Same = ratio Top > Bottom O No HA. N g ( ) h ( ) 7 15

29 You Try! What is the EQUATION of the horizontal asymptote for the following functions? f( ) g ( ) H. A. : H. A. : y 3 4 none Bottom > Top y=o Same = ratio Top > Bottom O No HA. N h ( ) 7 15 H. A. : y 0

30 You Try: True or False 1) The graph of function f is defined as the set of all points (, f()) where is in the domain of f. Justify your answer. ) If a function is not continuous, then the domain cannot be all real numbers.

31 True or False ANSWERS 1) The graph of function f is defined as the set of all points (, f()) where is in the domain of f. Justify your answer. True! This is the definition of a function. ) If a function is not continuous, then the domain cannot be all real numbers. False! It could be a piecewise function.

32 Is this continuous? State whether the scenario is continuous or discontinuous. A) Outdoor temperature as a function of time. Continuous B) Number of soft drinks sold at a ballpark as a function of outdoor temp. Discontinuous C) Your hair length as a function of days in a year Continuous

33 Packet p. -3 Homework Day