Math 1201 Unit 5: Relations & Functions. Ch. 5 Notes

Transcription

1 Math 1201 Unit 5: Relations & Functions Read Building On, Big Ideas, and New Vocabulary, p. 254 text. 5.1 Representing Relations (0.5 class) Read Lesson Focus p. 256 text. Outcomes Ch. 5 Notes 1. Define and give an example of a set. p Define and give an example of an element of a set. p Define and give an example of a relation. p Represent a relation in various ways. pp n Def : A set is a collection of distinct objects. For example, the types of animals found in Newfoundland could be a set. n Def : An element of a set is one of the objects in the set. For example, in the set of the types of animals found in Newfoundland, beaver would be an element. Some other elements of this set would be moose, rabbits, and bears. Note: The elements of a set are often listed inside a set of curly brackets called braces. For example, the set of the types animals found in Newfoundland could be written as beaver, moose, rabbit, bear, duck, geese,... n Def : A relation shows how the elements in 2 or more sets are connected to each other. For example, if pairs moose, walk, duck, fly we have 2 sets moose, duck and walk, fly then one relation could be written as a set of ordered which shows how some animals in Newfoundland can move. Note that the previous relation was written as a set of ordered pairs. However, a relation can be written in other ways. 1

2 Ways to Represent a Relation Ordered Pairs Arrow (Mapping) Diagram Table or Table of Values (TOV) Graph (E.g.: Line Graph, Bar Graph, Histogram) Equation OR Words For the sets 2010, 2011, 2012, 2013 and 22, 24,18, 20, the relation 2010,22, 2011,24, 2012,18, 2013,20 could represent the number of students in Math 1201 at Heritage Collegiate for the given years. Do # s 3 a, 4, 6 a, b, 10, pp text in your homework booklet. 2

3 5.2 Properties of Functions (2 classes) Read Lesson Focus p. 256 text. Outcomes 1. Define and give an example of a function. p Determine if a relation is a function. pp Represent a function in a variety of ways. pp Determine the domain of a function. p Explain how the domain of a function is related to the independent variable. p Determine the range of a function. p Explain how the range of a function is related to the dependent variable. p. 267 Recall that a relation shows how the elements in two sets are related or connected. n Def : The elements in the first set is called the domain. n Def : The elements in the second set is called the range. E.g.: In the relation described by the arrow (mapping) diagram below, the domain is 4, 2, 0, 3 and the range is 3, 3, 5,6,7. E.g.: In the relation described by the arrow (mapping) diagram below, the domain is 8, 9,10,13 and the range is 3, 1, 5. 3

4 E.g.: In the relation described by the bar graph below, the domain is comedy, action, romance, drama, sci-fi and the range is 1, 4, 5, 6. Note that the order of the elements does not matter and that if an element is repeated, it is only written once in the domain and/or range. In this example, you may have wondered why the domain was the movie type and why the range was the number of people choosing that movie type as their favorite instead of the reverse. The answer lies what the independent and dependent variables are in this relation. The domain is the elements that make up the independent variable and the range is the elements that make up the dependent variable. In this relation, the independent variable is movie type so the domain is comedy, action, romance, drama, sci-fi and the dependent variable is the number selecting that movie type as their favorite so the range is 1, 4, 5, 6. We can now expand our definitions of domain and range. n Def : The elements/values associated with the independent variable is called the domain. n Def : The elements/values associated with the dependent variable is called the range. Function A Special Type of Relation n Def : A function is a special type of relation. It is a relation in which each element in the domain is associated with exactly one element in the range. E.g.: The relation described by the arrow (mapping) diagram below is a function because each element in set A is associated with exactly one element in set B. a is associated with x b is associated with y c is associated with y d is associated with z 4

5 E.g.: The relation described by the arrow (mapping) diagram below is NOT a function because the element 3 is associated with 5 and with 7. E.g.: Is the relation below a function? Why or why not? E.g.: Explain why the left relation is a function while the right relation is not as function. 5

6 Complete the following table. The first row is done for you. Relation Function? (Y or N) Independent Variable Domain Dependent Variable Range N x 0,1, 2, 3 y 2,1, 2, 4 x 2,1, 0,1, 2 y 0,1, 4 N/A N/A Student Name 9,10 Number of Students Do # s 4, 5 c, d, 9, pp text in your homework booklet. 6

7 A Function as an Input-Output Machine A function can be thought of as a machine that turns an element in the domain (input) into exactly one element in the range (output) (see below). A specific example of changing in an input value into an output value can be seen below. The input value (3) goes into the function machine which multiplies it by 3, subtracts 4, and outputs the value 5. Note that the 3 that is input must come from the domain of the relation and the 5 that is output becomes part of the range of the relation. E.g.: Complete the table of values (TOV) below for the function machine directly above. Input Value (# that goes into the function machine) Output Value (# that comes out of the machine) OR (# that replaces the x in 3x 4) E.g.: Complete the table of values (TOV) below for the function machine above. Input Value (# that goes into the function machine) OR (# that replaces the x in 3x 4) Output Value (# that comes out of the machine)

8 Function Notation Functions typically take the form of a rule of some sort (see below). A value from the domain (the input value) is changed according to a rule into an output value that becomes part of the range. For example, if the input value was 10 and the function rule was multiply by 2 then the output would be 20. E.g.: Complete the table below. Input Function Rule Output 10 Cut in half -5 Triple 9 Square 49 Take the square root We give the rule a name, often f because it is the first letter in the word function, but it could be any name. The function rule in the function machine below is called Fred. Now the function rule is often written as an algebraic expression of some sort, like 3x 4, so when we give this rule a name we might write something like 8

9 f ( x) 3x 4 Name of function Function Rule Variable in the function rule This is called function notation and it is pronounced f of x equals Function notation comes in handy when we want to quickly paraphrase an English phrase mathematically. Compare the statements in the table below. English Phrase If the function rule is multiply by 3 then subtract 4 find the output value when the input value is 5. If the function rule is double the number find the output value when the input value is -8. If the function rule is square the number find the output value when the input value is 11. If the function rule is find the square root of the number find the output value when the input value is 49. Mathematical Equivalent x f 5. If f x 3 4, find If g x 2x, find 8 g. 2 If hx x, find 11 If r x h. x, find r 49. E.g.: Complete the table for the given functions. The first two are done for you. Function Function Name Function Rule 9 Variable in the Rule Input Value f x 3x 4 f 3x 4 x 6 f 2 d t 0.5t 6 d 2 0.5t 6 t -4 d t 25t h t t t r x x 1 4 Output Value 6 OR 14 4 OR 2 We have used notation similar to function notation quite often in this course. For example, when we wrote an area or volume formula, we often wrote it using function notation without indicating the variable(s) used in the function rule. Some examples are given below. 2D or 3D Figure Common Form Function Notation Form Circumference of a Circle C 2 r Area of a Square 2 A s Area of a Circle 2 A r Area of a Triangle Cr 2 A s r s 2 A r r 2 bh 2 A, A b h bh 2

10 Area of a Rectangle A lw Surface Area of a Cube 2 SA 6s Surface Area of a Cylinder 2 SA 2r 2rh Surface Area of a Sphere 2 SA 4 r Volume of a Cube 3 V s Volume of a Sphere 4 3 V r 3 Al, w SA s lw 6s 2 SA r, h 2r 2rh 2 2 SAr 4 r V s s 3 3 V r r 4 3 Do # s 6, 20, p text in your homework booklet. E.g.: If f x 3x 4, find 5 Finding the Output Value given the Input Value f. The input value is 5, so 5 is substituted for x. f So, f If the input value is 5 and the function rule is 3x 4 then the output value is E.g.: If SAs 6s, find 2 SA. The input value is 2, so 2 is substituted for s. 3 SA So, 5 11 SA2 48 If the input value is 2 and the function rule is Do # s 14, 19 a, p. 272 text in your homework booklet. 3 6s then the output value is 48. Finding the Input Value given the Output Value E.g.: If f x 3x 4, find the value of x for which f x. Since f x 26, we substitute 26 for f x in f x 3x 4 to get 26 3x 4, and solve for x.

11 26 3x x x 30 3x x So if the output value is 26 and the function rule is f x 3x 4 E.g.: If gn 6n 8, find the value of n for which 1 Since gn 1, we substitute -1 for 1 6n n n 9 6n n then the input value was 10. gn. gn in gn 6n 8 So if the output value is -1 and the function rule is gn 6n 8 Do # s 15, 19 b, p. 272 text in your homework booklet. to get 1 6n 8, and solve for n. then the input value was

12 5.3 Interpreting and Sketching Graphs (1 class) Read Lesson Focus p. 276 text. Outcomes 1. Interpret a given graph. p Describe a situation for a given graph. p Sketch a graph for a given situation. p. 280 Interpreting a Given Graph E.g.: Use the graph below to answer the following questions. a) Over how many days was the water temperature taken? b) On what two consecutive days was the water temperature the same? c) What was the highest temperature recorded? d) What was the lowest temperature recorded? e) On how many days was the temperature lower than the day before? f) On how many days was the temperature higher than the day before? g) The greatest temperature increase occurred between what two consecutive days? Before you do the next example, you may want to look at the graph on the bottom of page 277 of the text and note which line segments indicate a steep or moderate increase, which segments indicate a steep or moderate decrease, and which segment indicates no change. 12

13 E.g.: Use the graph below to answer the following questions. a) When did the journey start? b) How far was Picnic Park from home? c) How long did it take to get to Picnic Park? d) How long was the stop at Picnic Park? e) How far was the Campground from Picnic Park? f) How long did it take to drive from home to the Campground? g) How long was the stop at the Campground? h) How long was the entire trip? i) Was the speed greater from home to Picnic Park or from Picnic Park to the Campground? j) When was the speed greatest? Do # s 3, 4, 8, pp text in your homework booklet. Describing a Given Graph E.g.: Describe what is happening for each line segment in the graph below. 13

14 i. From 10am to 11am, the number of people in the store increased by 3 from 2 to 5 people. ii. From 11am to 12pm, the number of people in the store increased by 5. iii. From 12pm to 1pm, the number of people in the store increased by 12. iv. From 1pm to 2pm, the number of people in the store decreased by 7. v. From 2pm to 3pm, the number of people in the store decreased by 10. vi. From 3pm to 4pm, the number of people in the store decreased by 1. vii. From 4pm to 5pm, the number of people in the store stayed the same. viii. From 5pm to 6pm, the number of people in the store decreased by 1 to 3 people. E.g.: Describe what the graph below indicates with respect to women s smoking rates. The rate of smoking for women doubled between 1960 and 1965 from about 90 smokers per 1000 to about 180 smokers per It increased only slightly between 1965 and The smoking rate for women increased sharply again between 1970 and 1975 to 320 smokers per 1000 and then remained stable until It then decreased at a nearly constant rate to 280 smokers per 1000 over the next 10 years, declined more sharply to 220 smokers per 1000 between 1990 and 1995, and declined further to 200 smokers per 1000 by the year E.g.: Describe what is happening in the distance-time graph below. 14

15 Dan walks with a speed of 2km/h for the first hour and then stops to rest for 30 minutes. He then walks with a speed of 2km/h for another hour and stops for an hour. Dan then walks all the way home in two hours with a constant speed of 2km/h. Do # s 6, 7, 9, pp text in your homework booklet. Sketching a Graph for a Given Situation E.g.: Sketch a distance-time graph that represents the following situation. A cyclist leaves home at 9am and travels at a constant speed of 20km/h for one hour. The cyclist then travels at a slower constant speed of 6.6km/h for the next 45 minutes. She then stops at 10:45am to rest for 1 hour. The cyclist then starts to travel home at a constant speed of 30km/h for 30 minutes, stops to rest for 15 minutes, and then continues with a constant speed of 40km/h, arriving home at 12:45pm. Do # s 10, 12, 13, pp text in your homework booklet. 15

16 5.4 Graphing Data (1 class) Read Lesson Focus p. 284 text. Outcomes 1. Graph a given set of data. pp Determine if the data is continuous or discrete. pp Determine if the data on the graph should be joined. pp Determine the domain of the relation from the graph of the data and write it using set notation and interval notation. pp Determine the range of the relation from the graph of the data and write it using set notation and interval notation. pp E.g.: For the data given in the table below: Altitude (A) (1000 ft.) Air Temperature (T) F i. Determine if the data is discrete or continuous. ii. Decide if the points can be joined. iii. Graph the data. iv. Give the domain of the relation. v. Give the range of the relation. i. The data is continuous because the air temperature can be measured at any height. ii. Since the data is continuous, the points can be joined. iii. iv. The domain is all the altitude values from 1000ft. to 30000ft. This can be written as A 1000 A , using interval notation. using set notation or as v. The range is all the temperature values from 47 F to 55 F. This can be written as T -47 T 55 47, 55 using interval notation. using set notation or as 16

17 E.g.: For the data given in the table below: Cost per Donut (C) (cents) # Donuts Sold (n) i. Determine if the data is discrete or continuous. ii. Decide if the points can be joined. iii. Graph the data. iv. Give the domain of the relation. v. Give the range of the relation. i. The data is discrete because part of a donut would not be sold. ii. Since the data is discrete, the points cannot be joined. iii. iv. The domain is all the values for the cost of a donut 30,35,40,45,50,55. v. The range is all the values for the number of donuts sold 120,275,450,630,825,1020. Do # 2, p. 286 text in your homework booklet. 17

18 5.5 Graphs of Relations and Functions (2 classes) Read Lesson Focus p. 287 text. Outcomes 1. Indentify the independent and dependent variables in a given context. p Explain how the domain of a relation is related to the independent variable. p Explain how the range of a relation is related to the dependent variable. p Given the graph of a relation, determine if a relation is a function. p Determine the domain of the relation from the graph of the data and write it using set notation and interval notation. p Determine the range of the relation from the graph of the data and write it using set notation and interval notation. p. 289 Independent and Dependent Variables Recall that in an experiment, the independent variable is the variable that is varied or manipulated by the researcher, and the dependent variable is the response that is measured. For example, in a study to determine how the amount of time studying affects test marks, the amount of study time would be the independent variable and the test marks would be the dependent variable. E.g.: Complete the table below. The first one is done for you. Situation Independent Variable Dependent Variable The amount of Vitamin C one consumes can Amount of Vitamin C Life Expectancy influence life expectancy. A farmer wants to determine the influence of different quantities of fertilizer on plant growth. The time spent studying will influence test scores. The weight of a package will determine the amount of postage paid. Shots on net will influence the number of goals scored. Determining if a Relation is a Function Recall that if a relation is written as a set of ordered pairs, as a TOV, or as a mapping diagram, you can tell that the relation is a NOT function if the first coordinate appears more than once. Note that the relation on the right is NOT a function because -2 occurs as the first coordinate more than once in the TOV. The relation on the left is a function because -3, 0, 3, 8, and -10 only occur once. 18

19 Determining from a Graph if a Relation is a Function If you are given a graph instead of a set of ordered pairs, a TOV, or an arrow (mapping) diagram, you can still easily determine if the relation is a function. Let s look at the graph below and list the coordinates of some of the points on the graph. Average Temp F # Beach Visitors If you look at the TOV closely, you should see that 86 (and 94) appears more than once as a first coordinate. This means that the relationship between the average daily temperature and the number of beach visitors is NOT a function. If you look at those points on the graph, you should see that the two points with a first coordinate of 86 can be joined by a vertical line. This is also true for the two points with a first coordinate of 94. This property leads us to a simple way of determining if a relation is a function given the graph of the relation. Vertical Line Test (VLT) for a Function A graph represents a function if no points on the graph can be joined with a vertical line. So the graph to the left is a function because it passes the vertical line test whereas the graph in the centre is NOT a function because it does NOT pass the vertical line test. Likewise, the graph of a circle is NOT a function because it does NOT pass the vertical line test. 19

20 Complete the following table. Graph Function? (Y or N) 20

21 Determining the Domain and Range of a Relation from a Graph E.g.: For the graph given to the right: i. Determine if the graph represents a function. ii. Give the domain and range of the relation. i. This graph represents a function because the graph passes the VLT. Domain = 1,3, 4,6 ; Range = 2, 2,5 ii. E.g.: For the graph given to the right: i. Determine if the graph represents a function. ii. Give the domain and range of the relation. i. This graph does NOT represent a function because the graph does NOT pass the VLT. Domain = x 4 x 4 OR 4, 4 ii. Range = y 2.7 y 3 OR 2.7,3 Do # s 7, 6, 4, 8, 9, 12, pp text in your homework booklet. E.g.: For the graph given to the right: i. Identify the independent variable. ii. Identify the dependent variable. iii. Determine the domain and range of the relation. iv. Determine if the relation is a function. i. Time of Day ii. Average Water Level iii. Domain = Time 12 midnight Time 12 midnight OR 12am, 12am Range = Height 4.2 Height 5.1 OR -4.2, 5.1 iv. The relation is a function because the graph passes the VLT. 21

22 E.g.: For the graph given to the right: i. Identify the independent variable. ii. Identify the dependent variable. iii. Determine the domain and range of the relation. iv. Determine if the relation is a function. i. Weight ii. Cost Domain = Weight 0 Weight 6 OR (0,6]; Range = 39, 41, 43, 45, 47, 48 iii. iv. The relation is a function because the graph passes the VLT. (See below) Do # s 11, 13, 19, pp text in your homework booklet. Determining the Domain Values and the Range Values from the Graph of a Function E.g. Use the graph below to find: i. the range (output) value when the domain (input) value is -1. ii. the range (output) value when the domain (input) value is -2. iii. the domain (input) value when the range (output) value is -2. i. Since 1,1 is a point on the graph, if the domain value is -1 then the range value is 1. 22

23 ii. Since 2,2 is a point on the graph, if the domain value is -2 then the range value is 2. iii. Since 4, 2 is a point on the graph, if the range value is -2 then the domain value is -4. E.g. Use the graph below to find: i. the range (output) value when the domain (input) value is -3. ii. the range (output) value when the domain (input) value is 0. iii. the domain (input) value when the range (output) value is 4. i. Since 3,0.25 is a point on the graph, if the domain value is -3 then the range value is ,1 is a point on the graph, if the domain value is -0 ii. Since then the range value is 1. 3,4 is a point on the graph, if the range value is 4 iii. Since then the domain value is 3. E.g. Use the graph to the right to find: i. f 2 ii. iii. iv. f f f v. A range of values for x for which f x 10. vi. A range of values for x for which f x 0. vii. A range of values for x for which f x 0. i. Since 2,10 is a point on the graph, then ii. Since 0,10 is a point on the graph, then f f iii. Since 1,0 is a point on the graph, then 1 0 iv. Since 2, 10 is a point on the graph, then v. f x 10 for 2x 3 vi. f x 0 for 2 x 1 vii. f x 0 for 1x 3 f. f

24 E.g. Use the graph to the right to find: i. f 1 ii. iii. f f 1 3 iv. x such that f x 4. v. x such that f x 1. i. 1 Since 1, 2 is a point on the graph, then 1 f 1. 2 ii. Since 1,2 is a point on the graph, then f 1 2. iii. Since 3,8 is a point on the graph, then f 3 8. iv. Since 2,4 is a point on the graph, then f x 4 when x 2. v. Since 0,1 is a point on the graph, then f x 1 when x 0. Do # s 16, 17, p. 296 text in your homework booklet. 24

25 5.6 Properties of Linear Relations (2 classes) Read Lesson Focus p. 300 text. Outcomes 1. Determine if a TOV represents a linear relation. p Determine if a graph represents a linear relation. p Determine the rate of change of a linear relation from its graph. p Determine the domain and range of a linear relation and express them in a variety of ways. pp Determining if a Table of Values (TOV) Represents a Linear Relation Examine the two TOV s below. Both scatter plots above seem to indicate that each relation is linear, but one of them is not. In the left TOV s, the change in the independent (x) variable is constant (it is -6). Each time the value of x decreases by 6. The change in the dependent (y) variable is also constant (it is -4). Each time the value of y decreases by 4. This TOV represents a linear relation. In the right TOV, the change in the independent (x) variable is constant (it is 1). Each time the value of x increases by 1. However, the change in the dependent (y) variable is NOT constant. This TOV does NOT represent a linear relation. *** If a constant change in the independent variable produces a constant change in the dependent variable, then the relation is linear. *** Watch (0-7:25) 25

26 E.g.: Complete the table below. The first one is done for you. TOV Linear Relation? (Y or N) Linear Function? (Y or N) Yes. As the value of x increases by 1 the value of y increases by 0.5. Yes, no x-coordinate appears more than once. x y Do # s 3, 4, 11, pp text in your home work booklet. 26

27 Determining if an Equation Represents a Linear Relation You can also tell if a relation is linear by looking at the equation. E.g.: Draw the graph for each equation below and determine if the equation represents a linear function. i. f x 2x 18 or y 2x 18 f x x 7x 18 or y x 7x 18 ii. 2 2 f x x x 9x 9 or y x x 9x 9 iii iv. f x 3 or y 3 v. x 4 f x 2x 18 or y 2x 18 Equation Graph Linear Function? (Y or N) 2 2 f x x 7x 18 or y x 7x 18 27

28 f x x x 9x 9 or y x x 9x 9 3 or y 3 f x x 4 The graphs of f x 2x 18 or y 2x 18, f x 3 or y 3, and x 4 are linear relations because their graphs are lines. The first equation represents an oblique line, the second a horizontal line, and the third a vertical line. In general, equations of the form: f x ax b or y ax b, a, b are constant (oblique line) f x k or y k, k is constant (horizontal line) x k, k is constant represent linear relations (vertical line) 28

29 E.g.: Complete the table below. Equation Linear Relation? (Y or N) y y f x 3x 7 or y 3x f x 2x 6x 5 or y 2x 6x 5 n n or y 11 f x 3x 2x 6x 4 or y 3x 2x 6x 4 n n f x y y x 6 y n Linear Function? (Y or N) 2y7 11 y y 2x 8 6 y n Watch (7:26-end) Watch (0-3:00) Do # s 6 b, 13, 16, 18, pp text in your home work booklet. Determining the Rate of Change of a Linear Relation from Its Graph To determine the rate of change from the linear graph we need to find the coordinates of 2 points on the line and determine the rise and the run between the points. If the graph rises to the right, the rate of change is positive and if the graph falls to the right, the rate of change is negative. **Rises to right = positive rate of change **Falls to right = negative rate of change E.g.: For the graph to the right: Rise = 7 2 = 5 Run = 4 1 = 3 5 Rate of change 3 29

30 E.g.: For the graph to the right: Rise = -2 4 = -6 Run = 8 (-4) = Rate of change 12 2 In general, the rate of change is the: change in the y-coordinate ***** ***** change in the x-coordinate E.g.: For the graph to the right find the: a) dependent variable. b) independent variable. c) rate of change. a) The dependent variable is engine repair cost. b) The independent variable is the number of oil changes per year. c) The coordinates of 2 points on the line would be 4,400 and 5,320 The rise is = $80 The run is 4 5 = -1 oil changes. The rate of change is $80 $80/ oil change 1 oil changes This tells us that, on average, every oil change decreases the cost of engine repair by $80. 30

31 E.g.: For the graph to the right find the: a) dependent variable. b) independent variable. c) rate of change. a) The dependent variable is price. b) The independent variable is the distance. c) The coordinates of 2 points on the line would be 10,1.5 and 20,3.1 The rise is = $1.6 The run is = 10 miles. The rate of change is $1.6 $0.16/mile 10miles This tells us that, on average, each mile driven increases the cost by $0.16 or 16 cents. E.g.: For the graph to the right find the: a) dependent variable. b) independent variable. c) rate of change. a) The dependent variable is jump height. b) The independent variable is the bike weight. c) The coordinates of 2 points on the line would be 19.5,10.3 and 22.5,9.8 The rise is = 0.5ft The run is = -3. The rate of change is $0.5ft 0.17ft/lbs 3lbs This tells us that, on average, each pound of weight of the bike decreases the jump height by 0.17ft. Watch (3:00-end) Do # s 7, 12, 14, 15, 17 pp text in your homework booklet. 31

32 5.7 Interpreting Graphs of Linear Relations (2 classes) Read Lesson Focus p. 311 text. Outcomes 1. Define the horizontal intercept of a graph. p Define the vertical intercept of a graph. p Given the graph of a linear function, determine its intercepts, domain, and range pp Given the equation of a linear function, sketch its graph. P Given the rate of change of a linear function and an intercept of its graph, identify the graph of the linear function. P Solve problems involving linear functions. pp n Def : The first coordinate of the point where the line intersects the horizontal axis (x-axis) is called the horizontal intercept (xintercept). n Def : The second coordinate of the point where the line intersects the vertical axis (y-axis) is called the vertical intercept (y-intercept). E.g.: For the graph to the right, the horizontal (x) intercept is 5.1. The vertical (y) intercept is E.g.: What is the horizontal intercept of the graph to the right? The horizontal intercept is What is the vertical intercept of the graph to the right? The vertical intercept is 32

33 E.g.: Use the graph to the right to answer the following questions. a) What is the independent variable? b) What is the dependent variable? c) What is the domain? d) What is the range? e) What is the cost of the cell phone after 9 months? f) What is the vertical intercept of the graph to the right? What does it represent? g) What is the rate of change of this linear relation? a) The independent variable is b) The dependent variable is Domain x 0 x 11 c) d) Range y 70 y 500 e) The cost of the cell phone after 9 months is f) The vertical intercept is. It represents the cost to purchase the cell phone. 11,500. The rate of change is g) 2 points on the line are 6,300 and $ $200 $40/ mnth 116mnths 5mnths E.g.: Use the graph to the right to answer the following questions. a) What is the independent variable? b) What is the dependent variable? c) What is the domain? d) What is the range? e) What is the length of the candle after 30 minutes? f) What is the vertical intercept of the graph to the right? What does it represent? g) What is the horizontal intercept of the graph to the right? What does it represent? h) What is the rate of change of this linear relation? a) The independent variable is b) The dependent variable is c) Domain d) Range e) The length of the candle after 30 minutes is f) The vertical intercept is. It represents the length of the candle before it is lit. g) The horizontal intercept is. It represents the time when. 33

34 h) 2 points on the line are 50,5 and 5 4in 1in 0.1in / min 50 60min -10min 60,4. The rate of change is. This is the amount by which the length of the candle decreases each minute. Do # s 5, 4, p. 319 text in your homework booklet. Sketching the Graph of a Linear Function Written in Function Notation E.g.: Sketch the graph of f x 4x 5 Method 1: Intercept-Intercept Method Find the x-intercept 0 4x x x 5 4x x let f x 0 Find the y-intercept (let x = 0) f x f x f x The x-intercept has coordinates 1.25,0 The y-intercept has coordinates 0,5. Where does the 5 show up in the equation f x 4x 5? We now plot both intercepts and join them to complete the graph. The fact that the line extends completely across the grid and that there are no solid dots at the endpoints indicate the line continues indefinitely. You can also place arrows at each end to show that the line continues indefinitely. 34

35 What is the rate of change for this graph? Where does this number appear in the equation? Two points on the graph are 1,1 and 0,5, so the rate of change is This number is the 01-1 number in front of the variable (x) in the equation. f ( x) 4x 5 Rate of Change y-intercept Method 2: y-intercept & Rate of Change The 5 in f x 4x 5 indicates that the y-intercept is 5. The " 4" the rate of change is in f x 4x 5 indicates that 4 4 or. This indicates that when the change in x is 1, the change in y is So we can plot the y-intercept (5) and go 1 unit right and 4 units down (-4) and plot a second point. We can then join these points to get the graph. 35

36 Summary Do # s 6, 15, pp text in your homework booklet. Matching a Graph to a Given Rate of Change and Vertical Intercept E.g.: Which graph below has a vertical intercept of -5 and a rate of change of 5 2.5? 2 4 The left graph has a vertical intercept of 3 and a rate of change of 0.8, so it is NOT the correct 5 5 graph. The right graph has a vertical intercept of -5 and a rate of change of 2.5, so it is the correct 2 graph. Do # 8, p. 320 text in your homework booklet. Problem Solving Involving Linear Functions 36

37 E.g.: The length of a baby for its first 16 months is modeled in the graph below. a) What is the vertical intercept? What does it represent? b) Determine the rate of change for the length of the baby? What does it represent? c) What is the domain? d) What is the range? e) How old is the baby when it is 22 inches long? f) What is the length of the baby when it is 14 months old? a) The vertical intercept is 20 inches. It represents the length of the bay when it was born. 8,22. The rate of change is b) Two points on the graph are 4,21 and 22 21in 1in 0.25in/mnth 84mnths 4mnths baby grows each month. Domain = x 0 x 16 c) d) Range = x 21x 24 e) The baby is 8 months old when it is 22in long. f) The baby is 23.5in when it is 14 months old.. This rate of change represents the amount by which the Do # s 9, 10, 11, 13, 14, 16, pp text in your homework booklet. Do # s 2, 3 b, d, 4, 5, 6 a, 7, 8, 9, 10, 11, 12, 13, 14, 17, 18, pp text in your homework booklet. 37