Greatest Integer Function (GIF)

Transcription

1 Greatest Integer Function (GIF) Greatest Integer Notation:,, (sometimes ). To evaluate, drop the brackets and replace the real number with the greatest integer less than or equal to the number. Eamples : = 9 = 6 = 4 = 8 = 0 = 1 1

2 Basic GIF Dom: Rule: Ran: Graph: 5 Inc: 4 Dec: Never 3 2 Pos: 1 Neg: Ma: Min: None None intercept: 0 intercept: 2

3 Recall that in a basic function... and Parameters a and b are both equal to 1 Parameters h and k are both equal to 0. 3

4 Transformed GIF Rule: Parameter a

5 Summar: Parameter a determines the distance between each step (called the counterstep) The counterstep = If a +, the function is increasing. If a, the function is reflected about the ais and is decreasing. Steps: 5

6 Parameter b length length 6

7 Summar: Parameter b determines the length of each step. The length = i.e., the reciprocal of b. If b +, the function is increasing. Steps: If b, the function is decreasing and is reflected about the ais. Steps: 7

8 If a and b are both negative, the function is increasing again, but each step starts with an empt dot and ends with a full dot

9 Remember: If a and b have the same sign, or if, then the function is increasing : If a and b have opposite signs, or if then the function is decreasing :, 9

10 Notice that the origin,, has been a solid (coloured) dot. Parameter h translates the function h units left or right; Parameter k translates the function k units up or down. This means that will alwas be a solid dot. 10

11 Eample: }

12 Using all of the information we know about a, b, h & k, we will graph Greatest Integer Functions and find the equation of a GIF from a given graph. 12

13 Eamples: 1) Graph

14 2) Graph

15 3) Graph

16

17 Given the function and provide a stud. Dom: Ran: Inc: Dec: Pos: Neg: -int: Zeros: *, draw its graph

18 * For a transformed GIF, the range can be written as Ran set-builder notation i.e. The range is the set of all values, such that = am + k, where m is an integer. So... Ran 18

19 Rachel, Bernard, Christian & Julie cannot agree on the equation of the function f that is represented b the graph. The obtained these equations: Rachel: Bernard: Christian: Julie: Which of the four friends is/are right?

20 Finding the Rule If ou can graph a GIF when given an equation, then ou can find the rule when given a graph : reverse the process. Eample: Determine the rule

21 Eample: Determine the equation

22 Eample: Determine the equation

23 Equations Given the function, determine... a) b) c) 23

24 Determine the zeros of the function. let = 0 Isolate the greatest integer part. 24

25 When solving for, the solution must be an interval. 25

26 Solve the equation. Algebraicall 26

27 Graphicall

28 Solve the equation. If the isolated greatest integer equals a fraction or decimal, then there can be no solution. 28

29 Given, find. 29

30 Solving Word Problems 1. Claudia works in a boutique. Her weekl earnings are made up of a base salar of $175 plus a 12% commission for ever $100 in sales. 30

31 a) Construct a graph showing Claudia's earnings for the first $500 of sales. 31

32 b) Determine the equation of the greatest integer function associated with this situation. 32

33 c) If Claudia sells $950 worth o f merchandise this week, then what will her salar be? 33

34 d) How much would Claudia have to sell in order to make $667? Claudia would have to sell $. 34

35 2. Martina rents a tennis court at her local park. The rate is $5 for the first hour and $4 for ever additional hour or portion thereof. a) How much would Martina pa if she plaed for 12 hours & 42 minutes this week? She would pa $

36 b) How long did she pla if she paid $37.00? She plaed for hours. 36

37 c) Martina onl brought $15.00 toda; how long can she pla? She can pla up to 3 hours, or h

38 d) Knowing that the park is open from 8 AM to 10 PM, determine the domain and range for this situation. Dom = Ran = 38

39 1. Greatest Integer Function.notebook 39

40 length = $25.00 Shoes: Shoes & socks: $150 is the ke value $5.49 put the total up to or over $150 40

41 The price of the shoes was $ or. 41