Chapter 4: Linear Relations

Transcription

1 Chapter 4: Linear Relations How many people can sit around 1 table? If you put two tables together, how many will the new arrangement seat? What if there are 10 tables? What if there are 378 tables in a row... how many people would that seat? 4.1 Writing Equations to Describe Patterns Ex. A landscape designer uses wooden boards for the square plots in a garden: 1 plot 2 plots 3 plots 4 plots Is the number of boards related to the number of plots? How?

2 4.1 Writing Equations to Describe Patterns Relationships between two variables can be represented in different ways: i) using pictures 1 plot 2 plots 3 plots 4 plots ii) using a table of values # of plots # of boards OR # of plots # of boards Writing Equations to Describe Patterns iii) Using an equation (usually table of values done first) # of plots # of boards b = 3p + 1 where b = number of boards p = number of plots iv) in words The number of boards is related to the number of plots. The number of boards will be equal to one more than three times the number of plots.

3 4.1 Writing Equations to Describe Patterns v) As a graph # of plots # of boards Number of Boards Number of Plots 4.1 Writing Equations to Describe Patterns So... if you put 378 tables together, how many people will it fit? - Make a picture... - Make a table... - Find the formula...

4 4.1 Writing Equations to Describe Patterns Example from page 16: A plane starts at 10,000 m above sea level and descends according to the following pattern. Time (mins) Height (m) 0 10, , , , ,800 Can you make find a formula that describes this pattern? 4.1 Writing Equations to Describe Patterns Strategies to establish a formula that relates two variables: Tip 1: Build a table of values Tip 2: Determine what the "repeating" sequence is. You are going to have to multiply this value. Tip 3: Determine what the constant or the part that doesn't repeat. You are going to have to add or subtract this value Use this information to determine your formula

5 4.1 Writing Equations to Describe Patterns Example from page 17: Practice P #4ac, ac, 6, 7, 8, 9, 10, Challenge #20

6 4.2 Graphing Linear Equations Before plotting the relationship between two variables on a graph you need to know which is the dependant (responding) variable and which is the independent (manipulated) variable. The dependant variable depends on the value of the other variable in the relationship. Ex. The height of the plane depends on how long it has been descending for, therefore the height is the dependant variable. Ex. The number or boards depends on the number of garden plots therefore the number of boards is the dependant variable. Time (mins) Height (m) 0 10, , , , ,800 Ex. The number of people a table can seat depends on the number of tables that have been put together therefore the number of people is the dependant variable. 4.2 Graphing Linear Equations The independent variable always goes on the x (horizontal) axis and the dependent variable goes on the y (vertical) axis. Number of Boards Height of the plane (m) 10,000 8,000 6,000 4,000 What is a linear relation? How can you tell whether a graph is linear? What about a table of values? 2, Number of Plots Number of minutes of descent Why would I join the points on one graph, but not the other?

7 4.2 Graphing Linear Equations Homework: p.170 #2, #4,, 6ac, 8, 9ac, 10ace, Challenge: 18 Print or #10c and #14 (all parts, not just the graph) 4.2 Graphing Linear Equations Suppose you had a section of ribbon 20 cm long. How many different ways could you cut it into two pieces? How are the lengths of the two pieces related? Show this relations: - in words - in a table - in a graph - as an equation

8 4.2 Graphing Linear Equations Ex. For the equation 3x - 2y = 6 a) make a table of values b) graph the equation 4.3 Graphing Equations with only one Variable Now suppose there was only one variable in an equation: - What would the table of values look like? - What would the graph look like? y = 3 3y = 4 2x - 1 = 0 x = -2 y = -18 4y + 1 = -

9 4.3 Graphing Equations with only one Variable What does the graph of x = 3 look like? Why might some students mistakenly think that the graph of x = 3 makes a horizontal line? How might students reason to avoid making this mistake? 4.3 Graphing Equations with only one Variable Homework: p.178 #4 # (no need to "describe") #7 #8 #13 (use graph paper) #14 (you may use computer or graph paper) #1ad (you may use computer or graph paper) Challenge: 21e

10 4.4 Matching Equations and Graphs Investigate P. 183 & Discuss as a class Sari Monica Bruce 4.4 Matching Equations and Graphs How do you match an equation with a graph? What strategies can you use? y = -x + 2 or y = 3x -2 A. Create a Table of Values for one of the equations and see which graph "works"

11 4.4 Matching Equations and Graphs How do you match an equation with a graph? What strategies can you use? y = -x + 2 or y = 3x -2 B. Choose a point (x,y) that is only on one of the graphs and plug it into one of the equations. What happens if the point you choose is shared by both graphs? 4.4 Matching Equations and Graphs How do you match an equation with a graph? What strategies can you use? y = -x + 2 or y = 3x -2 C. (Advanced)... make x or y = 0 and look at the constant to see where it crosses the axis...

12 4.4 Matching Equations and Graphs Homework: p.188 #-8, Using Graphs to Estimate Values In 1998, the price of gasoline was about $0. / L In 2008, the price of gasoline was about $1.3 / L Is the price of gasoline a linear relationship? Assuming the average price of gasoline has a roughly linear relationship with time, what would you expect the price of gasoline to be in 2018? How did you determine that?

13 4. Using Graphs to Estimate Values What is the approximate temperature of the water after it has been heated for 1. minutes? Explain how you know. 4. Using Graphs to Estimate Values What year was the US population approximately 26 million? Explain how you know.

14 4. Using Graphs to Estimate Values What is the approximate distance travelled after 100 minutes? Explain how you know. 4. Using Graphs to Estimate Values Interpolation is the estimation of values between two data points. What is the value of x when y = 8? 10 y What is the value of y when x = -3? x - -10

15 4. Using Graphs to Estimate Values In what year would you expect the US population to hit 400 million? Explain how you know. 4. Using Graphs to Estimate Values After approximately how many minutes would the distance travelled equal 400m? Explain how you know. ~ 100m after 80 min ~ 3000m after 160 min Therefore... ~ 400m after 240 min

16 4. Using Graphs to Estimate Values Extrapolation is when we expect or assume a graph to continue to be linear and use it to predict a value that is not on the graph graph. 10 y What is the value of x when y = -4? What is the value of y when x = -10? x Using Graphs to Estimate Values Homework: p.196 #, 6, 8-11, 13, 1-10