Slide 1 / 122 Slide 2 / 122 8th Grade Modeling Relationships 2015-11-30 www.njctl.org Slide 3 / 122 Slide 4 / 122 Table of Contents Click on the topic to go to that section Interpreting with Functions Analyzing a Graph Interpreting with Functions Comparing Different Representations of Functions Glossary Return to Table of Contents Slide 5 / 122 Slide 6 / 122 Two Values to Determine Slope Review Look at the given table. We can use any two values to determine Any function can be written as a table, graph, verbal slope. We can find where x = 0 to determine the y-intercept. model, or equation. We can find the rate of change and the y-intercept from any of these representations. x -2 -1 0 1 2 Remember, to find slope we can use the formula: y -5 -2 1 4 7 y 2 - y 1 Slope = y 2 - y 1 7 - 4 3 x 2 - x 1 Slope = = = = 3 x 2 - x 1 2 - 1 1 To find the y-intercept (initial value) we look to where x = 0 y-intercept = 1 since when x = 0, y = 1
Slide 7 / 122 Slide 8 / 122 Two Ways to Find y-Intercept Continue the Table Sometimes a table will not show the x-coordinate of zero. In y = 3x - 4 that case you need to figure it out. There are a few ways to do x 1 2 3 4 5 6 7 0 it. y -4 -1 2 5 8 11 14 17 y = 3x - 4 x 1 2 3 4 5 6 7 -3 +3 +3 +3 y -1 2 5 8 11 14 17 We can simply continue the table so that we can find y when x = 0. We see that the y's are moving at intervals of +3. In order to work There are two ways to find the y-intercept here. backwards to where x = 0 we need to subtract 3 from y. -3 - (-1) = -4 so when x = 0, y = -4. Note: This technique works best when the table is close to x = 0 Slide 9 / 122 Slide 10 / 122 Substitute for X 1 What is the slope of the following table? x y y = 3x - 4 -2 -1 x 1 2 3 4 5 6 7 -1 1 y -1 2 5 8 11 14 17 0 3 Answer 1 5 Another technique would be to substitute for x and solve 2 7 for y using the equation. 3 9 y = 3x - 4 y = 3(0) - 4 y = 0 - 4 y = -4 Note: This technique only works if you have the equation. Slide 11 / 122 Slide 12 / 122 2 What is the y-intercept of the following table? 3 What is the slope of this table? x y x y -2 -1 10 4 -1 1 11 5 0 3 Answer Answer 12 6 1 5 13 7 2 7 14 8 3 9
Slide 13 / 122 Slide 14 / 122 4 What is the y-intercept of this table? 5 What is the y-intercept of this table? y = x - 6 x y 3 7 x y 4 9 10 4 Answer Answer 5 11 11 5 6 13 12 6 7 15 13 7 14 8 Slide 15 / 122 Slide 16 / 122 How Much Per Week? What is the Slope? If we look at the slope or rate of change we can figure out how Carla puts away a certain amount of money per week. She much she puts in each week. So what is the slope of this table? started out with a certain amount. How could we figure out what she started out with and what she puts in per week? week 0 1 2 3 4 5 6 Answer week of 0 1 2 3 4 5 6 saving amount in 74 82 90 98 106 114 122 account amount in 74 82 90 98 106 114 122 account Slide 17 / 122 Slide 18 / 122 What is the Initial Value? Try one. If we look at the initial value or y-intercept we can figure out how Rob is training for a marathon. He is increasing his run each much Carla started with because the y-intercept is where Carla week. Based on the table: started without any weeks of savings. What is the initial value? How much more will he run each week? How many miles was he running before training? week of 0 1 2 3 4 5 6 Answer saving How many weeks until he runs a full marathon 26.2 miles? amount in Weeks since 74 82 90 98 106 114 122 2 3 4 5 6 7 account training began Miles 7 10 13 16 19 22
Slide 19 / 122 Slide 20 / 122 Look at the Given Scenario. 6 Evan is buying pizza from a store that sells each pie for a given price and each topping for set price as well. Use the table to identify the cost of each topping. Luke wants to buy cookies online for $12 per box with a flat fee for shipping and handling of $4.00. # of 1 2 3 4 5 toppings # of Notice in this scenario that the Answer boxes price initial value would be $4.00. Cost 15.25 16.50 17.75 19.00 20.25 However, in a real-life scenario, 1 16 one does not pay for shipping and 2 20 handling if nothing is being 3 24 bought. BE CAREFUL with your interpretation so that it makes 4 28 sense with the scenario. 5 32 Slide 21 / 122 Slide 22 / 122 8 Evan is buying pizza from a store that sells each pie for a 7 Evan is buying pizza from a store that sells each pie given price and each topping for set price as well. The for a given price and each topping for set price as toppings would be the: well. How much is a pie without any toppings. Rate of Change A # of B Initial Value Answer 1 2 3 4 5 toppings Cost 15.25 16.50 17.75 19.00 20.25 Slide 23 / 122 Slide 24 / 122 9 Evan is buying pizza from a store that sells each pie Slope and Y-Intercept for a given price and each topping for set price as well. The pizza would be the: Look at the given graph. A Rate of Change What is the slope? Initial Value B What is the y-intercept? Slope is often referred to +2 rise as . To find the rise +2 run count the number up or down to the next point. To find the run count the number left or right to the next point. The rise is +2 and the run is 2.
Slide 25 / 122 Slide 26 / 122 Y-Intercept 10 What is the slope of this graph? Now let's look at the y-intercept. All you need to do for this is find the point that crosses the y-axis and that is the y-intercept. In this case, the y-intercept is -4. +2 +2 Slide 27 / 122 Slide 28 / 122 Find the Answer 11 What is the y-intercept of this graph? 2000 Jorge is emptying his pool. 1750 According to this graph, 1500 how many gallons were in the pool to start? 1250 Gallons 1000 How quickly is the pool 750 draining? 500 250 0 3 6 9 12 15 18 21 24 Time in hours Slide 29 / 122 Slide 30 / 122 Find the Answer Find the Answer Jorge is emptying his pool. 2000 Jorge is emptying his pool. 2000 1750 1750 According to this graph, how How quickly is the pool many gallons were in the pool 1500 1500 draining? to start? 1250 1250 -1000 Gallons Gallons To find this we can find the rate 1000 1000 The initial value is where we of change or the slope. + start at the beginning of the 750 750 9 scenario. Find two well plotted points 500 500 and find the rise over run. Notice that when x = 0 there 250 250 are 2000 gallons in the pool. -1000 = -111 approx. Therefore Jorge started with 0 3 6 9 12 15 18 21 24 9 0 3 6 9 12 15 18 21 24 Time in hours Time in hours 2000 gallons. This can be interpreted as emptying 1000 gallons every 9 hours or approximately 111 gallons per hour.
Slide 31 / 122 Slide 32 / 122 Find the Answer 12 Sandra is going to a buffet. The meal is a fixed price but she has to pay for each soda she drinks. What is the Jorge is emptying his pool. initial value? Be prepared to explain how it relates to the 2000 scenario. 1750 How quickly is the pool draining? 1500 35 To find this we can find the rate of 1250 -1000 Gallons change or the slope. 30 Answer 1000 25 + Another way to find this would be 750 9 20 Cost to make ordered pairs and use the 15 slope formula. 500 10 250 (0, 2000) (18, 0) 5 0 0 3 6 9 12 15 18 21 24 1 2 3 4 5 6 Time in hours 0 - 2000 -2000 -1000 = = = Number of drinks 18-0 18 9 Notice the slope is negative because the pool is being -111 approximately emptied. Slide 33 / 122 Slide 34 / 122 13 Sandra is going to a buffet. The meal is a fixed price 14 Nora is selling hoodies for a fundraiser at but she has to pay for each soda she drinks. What is $15 per item. She pays a total of $225 for them. the slope? Use the points (0, 15) and (6, 25). Be How many does she have to sell to break even? prepared to explain how it relates to the scenario. 400 Answer 350 35 Answer 300 30 25 250 20 Total 200 Cost 15 150 10 100 5 50 0 1 2 3 4 5 6 Number of drinks 0 3 6 9 12 15 18 21 24 Hoodies Sold Slide 35 / 122 Slide 36 / 122 16 How many feet does the tree grow every year? 15 What does the circled coordinate mean? A This tree grows 4 feet 32 A 1 foot 32 every year. 28 28 24 2 feet This tree was planted B 24 B Height 20 Answer when it was 4 feet. Height 20 Answer 16 C 4 feet 16 There are 4 trees planted. C 12 D 12 feet 12 8 The tree was planted 8 4 D 4 when it was 4 years old. 0 1 2 3 4 5 6 7 8 0 1 2 3 4 5 6 7 8 Years Since Planting Years Since Planting
Recommend
More recommend
