Math 10. Chapter 6: Linear Relations and Functions. Develop algebraic and graphical reasoning through the study of relations.

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1 Name: Date: Math 10 Chapter 6: Linear Relations and Functions General Outcome: Develop algebraic and graphical reasoning through the study of relations. Specific Outcomes: RF1. Interpret and explain the relationships among data, graphs and situations. [C, CN, R, T, V] [ICT: C6 4.3, C7 4.2] RF2. Demonstrate an understanding of relations and functions. [C, R, V] RF3. Demonstrate an understanding of slope with respect to: rise and run line segments and lines rate of change parallel lines perpendicular lines. [PS, R, V] RF4. Describe and represent linear relations, using: words ordered pairs tables of values graphs equations. [C, CN, R, V] RF5. Determine the characteristics of the graphs of linear relations, including the: intercepts slope domain range. [CN, PS, R, V] RF8. Represent a linear function, using function notation. [CN, ME, V] Learning Goals: Create graphs that represent different situations and interpret graphs Apply the various characteristics of linear relations to graphing Determine an acceptable range of values for a situation Work with function notation in a variety of ways Work with slopes and solve problems involving rates of change

2 Mark Assignment 6.1: Page #1, 2, 4-6, 10, 11, : Page #1-3, 5, 7, 9, : Page #1-4, : Page #1-4, 6-9, 16, : Page #1-5, 7, 8, 17 Review: Page #1-18 Quiz Chapter Test

3 Name: Date: 6.1 Graphs and Relations Match each graph with a situation from the list below: a) the temperature of a cup of hot chocolate over time b) a car accelerating to a constant speed c) the distance a person walks during a hike d) the height of a soccer ball kicked across a field Draw a sketch of a graph with each rate of change: a) Faster increase: b) Faster decrease: c) No rate of change: d) Rate of change that is NOT constant:

4 Example 1: Wakeboarding has grown to be a popular water sport. The graph shows the distance that a wakeboarder is from her starting point on Last Mountain Lake in Saskatchewan. Describe what the boarder is doing. Example 2: Which graph best represents bacteria growth if the bacteria s food supply is limited? Explain. Example 3: Josaphee leaves her home and walks to the store. After buying a drink, she slowly jogs to her friends house. Josaphee visits with her friend for a while and then runs directly home. Using the distances shown, draw a distance-time graph that shows Josaphee s distance from her house. Explain each section of the graph.

5 Name: Date: 6.2 Linear Relations Vocabulary: Relation Linear Relation Non-linear Relation Discrete Data Continuous Data Independent Variable Dependent Variable A relation can be presented in a variety of ways:

6 Example 1: The Canadian National Frog Jumping Championship is part of Les Folies Grenouilles. This annual festival is in St-Pierre-Jolys, MB. The first champion, a frog named Georges, jumped a distance of just over 2 m in a single leap. Assume that Georges could maintain a distance of 2 m on every jump and that the total distance travelled from the start is measured after every jump. Consider the relationship between the number of jumps Georges takes and the total distance the frog travels. a) Identify the relationship as linear or a non-linear. Expalin how you know. b) Create a variable to represent each quantity in the relation. Which is the dependent variable? Which is the independent variable? c) Create a table of values for this relation. What are appropriate values for the independent variable? d) Create a graph for the relation. Are the data discrete or continuous?

7 Name: Date: Example 2: Consider each relation. Determine whether the relation is linear. Explain why or why not. a) the relation described by {, (-9, -10), (-7,-5), (-5,0), (-3,5), (-1,10),..} b) The graph shows the relationship between the amount, A, of a radioactive isotope present and the age of a rock sample over time, t, in years. c) the relation described by the equation m 17 = 0.8n Example 3: Match each linear relation with possible representations in the selections that are given. Justify your choices. a) The pressure, P, that a scuba diver experiences under water increases at a constant rate relative to the diver s depth, d, below the surface. b) y = 1/2x + 4 E One number is half another number increased by four. F (0, 101), (25,176), (50,251), (75,326), (100,401), (125,476)

8 Section Domain & Range Mathematics 10C Definitions: Domain - Range - List - Set Notation - Interval Notation Example 1 - Determine the Domain & Range from a Graph Give the domain and range using words, a number line, set notation, and interval notation. Words: Words: Number Line: Number Line:

9 Name: Set Notation: Set Notation: Date: Interval Notation: Interval Notation: Example 2 - Domain & Range for a Situation A motorized Ferris wheel has a radius of 22 cm. The support structure keeps the bottom of the wheel 3 cm above the base. It takes 10 s to complete one revolution. The graph shows the height of one of the chairs during two rotations of the wheel, starting at the lowest point. a) What are the values of A, B, C, D, and E? What do they represent? Point Value It represents... A B C D E

10 b) What are the domain and range of the graph? Words: Number Line: Set Notation: Interval Notation: Example 3 - Domain & Range for Discrete Data Data for a relation are recorded in the table of values. Give the domain and range using set notation and lists. a b Set Notation: Lists:

11 Name: Date: Example 4 - Using Technology with a Restricted Domain The same species of corn grows at an average rate of 5 cm per day from the start of week 7 until the end of week 9. The plant s growth in this period is modelled using the formula h = 5a + 214, where h is the height of the plant, in centimetres, and a is the age of the plant, in days. Use a graphing calculator to show a graph of the plant s height for these three weeks. Key Ideas - The is the set of numbers for which the independent variable is defined. - The is the set of numbers for which the dependent variables is defined. - The domain and range can be written as words, a number line, interval notation,, and a list.

12 Section Functions Mathematics 10C After looking at the following groups, describe the differences between relations that are functions and those that are not. What do you think makes a relation a function? What makes a relation not a function?

13 Name: Date: Definitions: Function - Function Notation - Input - Output - Vertical Line Test - Function notation highlights the input/output aspect of a function. In the example f(x) = 4x + 1, the function is, with a variable name of. The rule is and it assigns a unique value for each value of. If the input is 2, the output is represented by and equals. Example 1 - Determine if a Relation is a Function For each set of relations, determine whether or not it is a function. Explain your choice.

14 Example 2 - Working with Function Notation The function F(C) = 1.8C + 32 is used to convert temperature in recorded in Celsius to a temperature recorded in Fahrenheit. a) Determine F(25). Explain your answer. a b) Determine C so that F(C) = 100. Explain your answer. Example 3 - Graphing Linear Functions Use the relation y = 3x - 1. a) Write the relation in function notation using f for the name of a function. b) Make a table of values. Graph the function. c) Determine the value of x if f(x) = 53.

15 Name: Date: Key Ideas All are, but not all are. A relation is classified as a function if each value in the corresponds to exactly one value in the. Each function has its own formula or rule, that is given in special notation called. Section Slope Mathematics 10C Definition: Slope - Slope = = A line segment that rises from left to right has a slope. A line segment that falls from left to right has a slope. Horizontal Lines Vertical Lines

16 Example 1 - Classifying the Slope of a Line Classify the slope of each line as positive, negative, or neither. Example 2 - Determine the Value of a Slope When discussing a roof truss, carpenters refer to the span instead the width, and the pitch rather than slope. Determine the pitch of the supported by the truss shown. of the roof

17 Name: Date: Example 3 - Determine the Slope of a Line Segment What is the slope, m, of each line segment with the given end points? a) S(-3, 6) and T(5, 2) b) H(4, 3) and K(4, 8) c) M(-9, -7) and N(-1, -7) Example 4 - Use Slope to Draw a Line The point (-6, 1) is on a line with a slope of 1/3. List three other points on the line.

18 Example 5 - Use Slope as a Rate of Change Determine the average rate of change for the Brentwood Regatta using the graph shown. Summary Slope can be positive,,, or. Slope is the ratio of : and usually given in fraction form. You can use two points on the line to find the slope using the formula: m = Slope can also give the average rate of change.