Chapter 4 - Functions and Relations

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1 Warm Up In your notebook, write down a relationship between two different quantities that could be represented by this graph: Quantity 2 Quantity 1 Sep 21 7:55 AM Chapter 4 - Functions and Relations Vocabulary - 1. Relation 2. Independent Variable 3. Dependent Variable 4. Input 5. Output 6. Function 7. Linear Function Sep 21 8:02 AM 1

2 Relating Quantities Cost Grass Height Pounds of Apples Time Amount of Paint in Can Area Painted Sep 21 8:19 AM The graph represents the number of hours in a journey to PA and how many miles have been driven total. Miles Driven Hours 1. When was the car traveling the fastest? 2. What might have happened between 3 and 4? And between 5 and 7? 3. When was the car driving the slowest? 4. When did the longest uninterrupted time of driving occur? 5. Explain what might have been happening at the beginning an end of the trip. Sep 21 8:23 AM 2

3 Flight of a Model Rocket D F A C E H G B Slow Descent Landing Quick Ascent Elevation Take Off Stops Rising Time Parachute Opens Jan 29 8:03 AM Independent Variable Dependent Variable Sep 21 8:30 AM 3

4 Determine the Independent and Dependent Variables Hours Driving Miles Driven Price of Dinner Number of Entrees Test Score Hours Studied Games Played Points Scored Sep 21 8:31 AM Function - A function is a relationship that pairs each input with exactly one output value Hours Driving Miles Driven Price of Dinner Number of Entrees Test Score Hours Studied Games Played Points Scored Sep 21 8:33 AM 4

5 A linear function is a function whose rate of change remains constant. Jan 29 7:59 AM What is the relationship between the number of rectangles and the perimeter of the figure they form? Rect, X Perim, y Ordered Pair, (x,y) Sep 21 8:04 AM 5

6 Examples of Linear Functions Mountain Climbing Hours Elevation Climbing Shopping Soup Cans Bill 0 $ $ $ $55.94 Gas Left in Tank Miles Gallons Left Sep 21 8:35 AM The table shows the relationship between the number of photos, x, you take and the amount of memory, y, that is left on your phone. Photos, x Memory, y Describe the relationship using words and an equation. Sep 21 8:14 AM 6

An automaker makes a car that can travel 40 miles on its charged battery, before it begins to use gas. Then the car can travel 50 miles per gallon of gas used. 1. What is the independent variable?

7 An automaker makes a car that can travel 40 miles on its charged battery, before it begins to use gas. Then the car can travel 50 miles per gallon of gas used. 1. What is the independent variable? Dependent? 2. Use a table to show the relationship. 3. Write a function rule to show the relationship. 4. Using the calculator, graph the function. 5. If the car has an 18 gallon tank, how far can the car go? Sep 21 10:41 AM Bell Ringer Sketch a graph of... The temperature during the day increases rapidly after 9AM, levels off between 1PM and 5PM and decreases slowly after 5. The temperature then levels off at 11PM Independent Variable Dependent Variable Sketch a graph which relates the two quantities. Sep 21 4:52 PM 7

8 Men's printed T-shirts cost $12.99 each for the first 5 shirts purchased at target. After that, each additional shirt costs $4.99. Which graph correctly describes this relationship. A C Total Cost Total Cost # of Shirts # of Shirts B Total Cost D # of Shirts # of Shirts Total Cost Sep 22 7:53 AM Jennifer runs deliveries for her flower shop. She makes stops heading away from the shop, then two on her way back. Which graph represents the scenario correctly? A Time C Distance from Store Distance from Store Time B Distance from Store D Time Time Distance from Store Feb 1 11:17 AM 8

9 What is a function? What is a linear function? Sep 21 5:20 PM Ocatagon 2 Octagons 3 Octagons Octagons and Perimeter 1. What is the independent variable? Dependent? 2. Use a table to show the relationship. 3. Write a function rule to show the relationship. 4. Using the calculator, graph the function. 5. If there are 6 octagons, what is the perimeter? Sep 21 10:54 AM 9

10 Building a Staircase A stair case will be built using long concrete blocks. Determine the number of blocks needed. Step 1 Step 2 Step 3 # of Steps Blocks Needed Ordered Pair Sep 22 8:00 AM The previous example is an example of what are called Triangle Numbers Why are they called triangle numbers? Step 1 Step 2 Step 3 Step 4 Square Numbers work similarly Step 1 Step 2 Step 3 Step 4 The rule for Square Numbers is easy, as it's in the name... Number of Squares = Step # Sep 22 8:23 AM 10

11 Bonus: Figure out the function rule which produces the chart and graph from the previous page. 100 for a quiz grade # of Steps Blocks Needed Ordered Pair Recursive Formula Explicit Formula Sep 22 8:06 AM A non-linear function is a function whose graph is not a line, or part of a line. Sep 22 8:09 AM 11

12 Determine if the following charts are representative of linear or non-linear functions by graphing. x y x y x y Sep 22 8:10 AM Determining Linear vs Non-Linear with an Equation Using a graphing calculator graph: 1. y = 2x y = x y = x y = -3x + 6 What is the common thread? Sep 22 10:27 AM 12

13 What is the definition of a function? Feb 4 8:44 AM Functions : Formal Terminology Function Notation f(x) Sep 22 3:50 PM 13

14 f(x) = 6x - 3 y = 6x - 3 Sep 23 8:47 AM Evaluate a function f(x) = 6x - 3 f(3) = f(-2) = Sep 23 8:47 AM 14

15 Evaluate a function r(x) = x 2 x + 6 r(5) = r( 2) = Feb 1 10:38 AM Please take a calculator Bell Ringer g(x) = x g(-3) = Feb 4 9:25 AM 15

16 Agenda: 1: Learn stuff 2. Video 3. Some more stuff 4. Practice problems 5. USA Test Prep, (if time). Feb 4 10:58 AM Domain and Range in Tables Input Output Domain Range Gas in Gallons 3 6 Cost Domain Range Sep 23 8:45 AM 16

17 The Domain and Range of a Function Determine the range of the function over a domain of {-2, 0, 4, 8}. f(x) = -5x - 8 f(-2) = f(0) = f(4) = f(8) = Range { } Sep 23 9:02 AM Functions A function is a pairing of inputs and outputs, such that each input is paired with exactly one output. Domain - Contains all numbers that are inputs. Range - Contains all numbers that are outputs. Sep 23 8:45 AM 17

18 To Function or Not to Function In Out In Out In Out Yes or No Yes or No Yes or No Sep 23 8:59 AM Domain and Range as Points (2, 3); (4, 6); (6, 8); (8, 3) x y Function Mapping Sep 23 8:59 AM 18

19 Function Mapping Yes or No Yes or No Sep 23 8:58 AM Independent Input x Domain Dependent Output Range f(x) y Sep 23 8:52 AM 19

20 Video of Dude Feb 4 10:52 AM Functions or not? x y x y x y Yes or No Yes or No Yes or No Sep 23 10:00 AM 20

21 BELL RINGER x y Is the chart a function or not? Yes No 2. If yes, is the function linear? Yes No Sep 28 1:34 PM Functions A function is a pairing of inputs and outputs, such that each input is paired with exactly one output. Domain - Contains all numbers that are inputs. Range - Contains all numbers that are outputs. Sep 23 8:45 AM 21

22 Is this a graph of a function? Yes or No? Sep 23 10:01 AM VLT y x Sep 23 10:06 AM 22

23 Determine the domain and range of the following coordinates and determine if they are functions or not. 1. {(8, 1); (6, 3); (-2, 0); (8, -5)} 2. {(-1, 4); (-2, 5); (6, 5); (-3, 10)} X Y Sep 24 7:57 AM Determine which diagram models the relations best, and whether or not they are functions. {(3, 0); (1, 8); (4, 0); (6, 8)} A X Y 0 8 B X Y 0 8 Is not a function Is a function C X 0 8 Y D Is a function X 0 8 Y Is not a function Sep 24 8:14 AM 23

24 Using a Graph to Evaluate Functions f Find the value of f(1) = 2. f(5) = Sep 25 7:59 AM f 1. f(-1) 2. f(-2) 3. f(-4) 4. f(0) Sep 25 8:03 AM 24

25 Slightly different... g g(x) = 1 g(x) = -3 Sep 25 8:06 AM Feb 4 12:59 PM 25

26 Feb 4 12:59 PM f(x) = 4x f(-2) = 2. f(3) = 3. f(x) = f(x) = 23 Sep 25 8:09 AM 26

27 Grab a Calculator BELL RINGER Find the missing information 1. f(3) = 2. f(0) = 3. f(x) = 3 Sep 25 8:26 AM Grab a Calculator Bell Ringer f(x) = 3x 2 5 f( 2) = Evaluate Sep 24 12:54 PM 27

28 Plan for Today 1. Bell Ringer 2. Continuos Vs Discrete 3. Brief Review 4. USA Test Prep "Quiz" 5. Review Sheet Sep 25 12:30 PM Continuous and Discrete Graphs A continuous graph is a graph with no breaks. A discrete graph is composed of distinct, isolated points. Continuous graph Discrete graph Sep 29 7:56 AM 28

29 The cost a coach will pay at Dick's Sporting Goods is a function of how many basketballs he purchases. Each ball costs $30. Write the equation that models this scenario. Is the graph continuous or discrete? Cost Basketballs Sep 29 8:08 AM The distance Summer runs is a function of the amount of time she runs for. She runs 10 miles per hour. Write the equation that models this scenario. Is it continuous or discrete? Distance Hours Sep 29 8:13 AM 29

30 Sample Questions 1. X Y Over the domain {2, 6}, what is the corresponding range? Domain Range 2. When the domain is -2, what is the range of the function graphed? When the range is 3, what is the domain of the function graphed? 3. Are the points plotted representative of a function? (-5, 2); (-2, 4); (-1, -1); (3, 3); (3, 6); (5, 2) x y Which is the graph of 2x 3 + 5? 5. What is the table of 2x 3 + 5? Sep 28 10:35 AM Linear Function? Yes or No (-2, -3) (0, -1) (1, 0) (3, 2) (5, 4) x y Linear Function? Yes or No (0, 0) (1, 1) (4, 2) (4, -2) (9, -3) (9, 3) Sep 21 9:31 AM 30

31 functions and relations (1).notebook August 23, 2016 Feb 5 12:49 PM Feb 5 12:49 PM 31

32 functions and relations (1).notebook August 23, 2016 Feb 5 12:51 PM Feb 5 12:51 PM 32

33 Complete the Review for a Grade Take a Laptop and Complete USA Test Prep Assignment: Functions Practice Feb 5 9:21 AM Plan for Today 1. Bell Ringer 2. One last note on functions. 3. Complete practice test. 4. Review practice test. Actions for Today 1. Work quietly with one other person. 2. Keep desks where they are. 3. All problems must be completed and work turned in. Sep 29 8:25 AM 33

34 functions and relations (1).notebook August 23, 2016 Sep 29 8:34 AM Things to Materials List Consider Theme Color Spirit Texture 3 D Sep 29 11:18 AM 34

35 For Today DON'T MOVE DESKS! 1. Brief Review 2. Don't talk during the test. 3. Eyes on your own paper. 4. When you are done - Read Homework for another class Remain silent Sep 29 12:24 PM Function vs Not Function Matching Points/Charts to Equations Matching Points/Charts to Graphs Sep 30 9:36 AM 35

36 A designer wants to make a circular fountain inside a square lawn as follows... r What is a rule for the area of the grass as a function of r? Sep 22 1:28 PM Grab a Calculator Bell Ringer Find the range of the function whose domain is {-3, -1, 6} f(x) = -2x range: { } Sep 23 8:52 AM 36

37 Benchmark Stats USA Test Prep Stats Mean Median Mode Mean Median Mode Feb 8 11:28 AM Also... Writing on the walls. Garbage on the floor. Constant interruptions Feb 8 11:49 AM 37

38 Feb 8 10:40 AM Feb 8 11:27 AM 38

39 Feb 8 11:46 AM Feb 8 11:31 AM 39

40 How am I going to help you win? 1. No cell phones 2. Quiet during lessons 3. Make-up on Thursdays While we're at it No eating 5. No swearing Feb 8 11:32 AM How are you going to help you win? 1. Use class time for work 2. Take notes 3. Study those notes 4. Come for tutoring 5. Abide by the rules Feb 8 11:37 AM 40

41 Please take a calculator Stow away those phones Bell Ringer The ordered pairs (2, 12), (3, 27), and (4, 48) represent a function. What is the rule which represents the function? A. y = 6x B. y = 2 x C. y = 3x 2 D. y = 2x 2 Feb 8 1:10 PM Feb 9 11:38 AM 41

42 functions and relations (1).notebook August 23, 2016 Feb 9 11:41 AM Feb 9 11:42 AM 42

43 functions and relations (1).notebook August 23, 2016 Feb 9 11:42 AM Feb 9 11:42 AM 43

44 functions and relations (1).notebook August 23, 2016 Feb 9 11:42 AM Feb 9 11:42 AM 44

45 Linear or Non Linear? Hour Temp 8pm Feb 9 10:47 AM The number of people at the party can be represented by the function p(x) = 5x, where x is the number of cars outside the house. How many people are at the party when there are 8 cars out front? What does "per" or "for each" typically signify? Feb 9 10:58 AM 45

46 Function or not? (3, 6); (8, 10); (10, 10); (12, 6) x y Feb 9 11:01 AM For the test 1. Please remain quiet for the duration of the test. 2. No cell phones. (Referral) 3. Answer all questions, even the ones where you need to write in the answer. 4. When you're done, use a computer if you want. 5. Still no cell phones when completed with the test. 6. Put your head down, or work on something else. 7. Remain seated until the bell. Feb 9 11:05 AM 46

Using Graphs to Relate Two Quantities

- Think About a Plan Using Graphs to Relate Two Quantities Skiing Sketch a graph of each situation. Are the graphs the same? Explain. a. your speed as you travel from the bottom of a ski slope to the top