Extra Practice Chapter 5. Topics Include: Direct & Partial Variation Slope Slope as a Rate of Change First Differences

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1 Extra Practice Chapter 5 Topics Include: Direct & Partial Variation Slope Slope as a Rate of Change First Differences

2 5.1 - Practice: Direct Variation 1. Find the constant of variation for each direct variation. a) The cost for a long-distance telephone call varies directly with time. A 1-min phone call cost $0.96. The total mass of magazines varies directly with the number of magazines. The mass of 8 magazines is.6 kg. c) The distance travelled varies directly with time. In h, Alex drove 195 km.. The cost, C, in dollars, of wood required to frame a sandbox varies directly with the perimeter, P, in metres, of the sandbox. a) A sandbox has perimeter 9 m. The wood cost $0.70. Find the constant of variation for this relationship. What does this represent? Write an equation relating C and P. c) Use the equation to find the cost of wood for a sandbox with perimeter 15 m.. The cost, C, in dollars, to park in a downtown parking lot varies directly with the time, t, in hours. The table shows the cost for different times. t (h) C ($) a) Graph the data in the table. Write the constant of variation for this relationship. What does it represent? c) Write an equation relating C and t. 4. The distance, d, in kilometres, Kim travels varies directly with the time, t, in hours, she drives. Kim is travelling at 80 km/h. a) Assign letters for variables. Make a table of values to show the distance Kim travelled after 0 h, 1 h, h, and h. Graph the relationship. c) What is the constant of variation for this relationship? d) Write an equation in the form y = kx. 5. a) Describe a situation this graph could represent. Write an equation for this relationship.

3 5. - Practice: Partial Variation 1. Identify each relation as a direct variation or a partial variation. a). Identify each relation as a direct variation or a partial variation. a) y = x + y = x c) C = 0.65n d) h = 5t +. The relationship in the table is a partial variation a) Use the table to identify the initial value of y and the constant of variation. Write an equation in the form y = mx + b. c) Graph the relation. Describe the graph. 4. Latoya is a sales representative. She earns a weekly salary of $40 plus 15% commission on her sales. a) Copy and complete the table of values. c) Sales ($) Earnings ($) Identify the initial value and the constant of variation. c) Write an equation relating Latoya s earnings, E, and her sales, S. d) Graph the relation.

4 5. - Practice: Slope 1. Find the slope of each object. a) c) d). Find the slope of each line segment. a). For safety, the slope of a staircase must be greater than 0.58 and less than A staircase has a vertical rise of.4 m over a horizontal run of.5 m. a) Find the slope of the staircase. Is the staircase safe?

5 5.4 Practice: Slope as a Rate of Change 1. At rest, Vicky takes 6 breaths every 5 min. What is Vicky s rate of change of number of breaths? 5. a) Make a table relating the number of squares to the diagram number. Graph the data in the table.. When he is sleeping, Jeffrey s heart beats 768 times in 1 min. What is his rate of change of number of heartbeats?. A racecar driver completed a 500-km closed course in.8 h. What is the rate of change, or speed? 4. The graph shows the speed of the cars on a roller coaster once the brakes are applied. Find the slope of the graph. c) Interpret the slope as a rate of change. a) Find the slope of the graph. Interpret the slope as a rate of change.

6 5.5 - Practice: First Differences 1. Consider the relation y = x. a) Make a table of values for x-values from 0 to 5. Graph the relation. c) Classify the relation as linear or non-linear. d) Add a third column to the table in part a). Label the column First Differences. Find the differences between consecutive y-values and record them in this column.. Consider the relation y = 1 x. a) Make a table of values for x-values from 0 to 5. Graph the relation. c) Classify the relation as linear or non-linear. d) Find the differences between consecutive y-values. Add a column to your table in part a) to record the first differences.. Refer to your answers to questions 1 and. How can you use first differences to tell if a relation is linear or non-linear? 4. Copy and complete each table. State whether each relation is linear or non-linear. a) First Differences First Differences Copy and complete the table for each equation. Identify each relation as linear or non-linear. First Differences 1 4 a) y = x y = x c) y = x + 1 d) y = x Collect like terms. Then, simplify. a) 4b + b + 1 p 7 p + 4 c) 1 + y y d) 5 x 1 x e) 6a b + b + a f) 7r + + r r 1 g) 9s s + 5t 4s h) g h + 5h + g h 7. Simplify. a) 4 + v + 5v 10 7a b a b c) 8k k 5k k d) x 4x + 8x + 5x e) 1 4m 8 m + m f) 6y + 4y + 10 y 6 y g) 5 + h + h 4 + h h h) 4p + q p + p 7q 8. Simplify. a) a + 6b + b 4 + a 4x + xy + y + 5x xy y c) m 4 m m + m 4 d) x + xy + y x + xy y 9. The length of a rectangle is times the width of the rectangle. Let x represent the width of the rectangle. a) Write an expression to represent the length of the rectangle. Write a simplified expression for the perimeter of the rectangle. c) Suppose the width is 6 cm. Find the perimeter of the rectangle.

7 5.6 - Practice: Connecting Variation, Slope, and First Differences 1. Look at this graph of a relation. a) Calculate the slope. Determine the vertical intercept. c) Write an equation for the relation of the form y = mx + b.. Look at this graph of a relation. Identify the initial value of y. c) Identify the constant of variation. d) Make a table of values for x-values from 0 to 4. e) Graph the relation. 4. The table represents a relation a) Use a graph to represent the relation. Use words to represent the relation. c) Identify the vertical intercept and the slope. Write an equation to represent the relation. 5. A large cheese pizza costs $8.00, plus $0.50 for each topping. a) Make a table to represent the relation. Graph the relation. c) Write an equation to represent the relation. 6. Represent this relation using words, with numbers, and with an equation. a) Calculate the slope. Determine the vertical intercept. c) Write an equation for the relation.. A relation is represented by the equation y = x 5. a) Is this relation a direct variation or a partial variation?

8 Chapter 5 Review 5.1 Direct Variation, pages a) Graph the data in the table What is the constant of variation for this relationship? c) Write an equation relating y and x.. Evan earns $7/h babysitting. The amount he earns, in dollars, varies directly with the time, in hours, he babysits. a) Assign variables. Make a table of values showing Evan s earnings for 0 h, 1 h, h, h, and 4 h. Graph the relationship. c) Identify the constant of variation. What does this represent? d) Write an equation in the form y = kx. 5. Partial Variation, pages Classify each relation as a direct variation, a partial variation, or neither. Explain. a) d = 45t y = x + c) y = x + d) d = 45t The relationship between the variables in the table is a partial variation c) Graph the relation. Describe the graph. 5. The owner of a small business is having brochures printed. The design cost is $1500. Printing costs $0.08 per brochure. The relationship between cost and the number of brochures is a partial variation. a) Identify the fixed cost and the variable cost. Write an equation for this relationship. c) What is the total cost for 800 brochures? 5. Slope, pages Find the slope of each line segment. a) a) Identify the initial value of y and the constant of variation. Write an equation in the form y = mx + b.

9 7. Calculate the slope of each line segment. a) EF GH c) JK 8. One endpoint of line segment AB is A(, 4). The slope of this line segment is. Find possible coordinates for B. 5.4 Slope as a Rate of Change, pages It took 8 min to fill a 5-L bucket. a) What is the rate of change of the volume of water? Graph the volume of water in the bucket over time. 10. Tom and Ana ran a race. The graph shows the distance each person ran in 10 s. Who ran faster? How much faster? 5.5 First Differences, pages Use first differences. Is each relation linear or non-linear? a) a) Make a table comparing the side length of a square to its perimeter for side lengths 1,,, 4, and 5. Is the relationship between side length and perimeter linear or non-linear? 5.6 Connecting Variation, Slope, and First Differences, pages Represent the relation y = x + using numbers, a graph, and words.

10 Chapter 5 Practice Test Multiple Choice For each question, select the best answer. 1. Which relation is a partial variation? A y = 5x B y = x C y = 5x D y = x 5. Sophie s earnings vary directly with the number of hours she works. She earned $5 in 4 h. What is the constant of variation? A 0.16 B 6.5 C 100 D 1. What is the slope of this staircase? Short Response 6. a) Calculate the slope. Find the vertical intercept. c) Write an equation for the relation. A 6 C 4. Which equation represents this relation? B D A y = x B y = x 1 C y = x 1 D y = x The cost to cater a party is $00 plus $15 for each guest. Which equation represents this relation? A C = 15n + 00 B C = 15n 00 C C = 00n + 15 D C = 00n The distance travelled varies directly with time. Anthony ran 49.6 m in 8 s. a) Write an equation for this relationship. Graph the relation. 8. Is this relation linear or non-linear? How can you tell without graphing? The cost to install wood trim is $50, plus $6/m of trim installed. a) Write an equation for this relationship. 18 m of trim were installed. What was the total cost?

11 Extend Show all your work. 10. This graph shows the relationship between the cost of a taxi trip and the length of the trip. a) Calculate the rate of change of cost. How does the rate of change relate to the graph? Write an equation for the relationship. c) Suppose the flat fee changed to $.00. How would the equation change? How would the graph change?

12 Chapter 5 Test Multiple Choice For each question, select the best answer. Short Response 6. a) Calculate the slope. 1. Which relation is a direct variation? A y = 5x B y = x C y = 5x D y = 5x. The cost of tea varies directly with the mass. Liz bought 4.5 kg of tea for $10.5. What is the constant of variation? A 0.4 B C 5.85 D.0. What is the slope of this ramp? A B 9 C 18 D 9 4. Which equation represents this relation? A y = x + 4 B y = 4x C y = x + 4 D y = x 4 Find the vertical intercept. c) Write an equation for the relation. 7. The cost to ship goods varies directly with the mass. Paul paid $0.40 to ship a package with mass 4 kg. Write an equation for this relationship. 8. Is this relation linear or non-linear? How can you tell without graphing? Sheila works in a bookstore. She earns $40 per week, plus $0.15 for every bestseller she sells. a) Write an equation for this relationship. Last week, Sheila sold 19 bestsellers. How much did she earn? 5. The cost of a newspaper advertisement is $750 plus $80 for each day it runs. Which equation represents this relation? A C = 80n 750 B C = 80n C C = 750n + 80 D C = 750n 80

13 Extend Show all your work. 10. This graph shows the volume of water in a child s pool over time as the pool is draining. a) Calculate the rate of change of the volume of water. How does the rate of change relate to the graph? Write an equation for the relationship. c) Suppose the rate of change changes to 4 L/min. How long will it take the pool to empty?

14 BLM Practice: Direct Variation 1. a) c) 65. a).0; the cost per metre, in dollars, of wood C =.P c) $4.50. a) BLM 5..1 Practice: Partial Variation 1. a) partial variation partial variation c) direct variation. a) partial variation direct variation c) direct variation d) partial variation. a) ; 1 y = x + c).00; the cost per hour to park in this parking lot c) C =.00t 4. a) t (h) d (km) a) The graph intersects the y-axis at (0, ). As the x-values increase by 1, the y-values also increase by 1. Sales ($) Earnings ($) ; 0.15 c) E = 0.15S + 40 d) c) 80 d) d = 80t 5. a) Tomatoes cost $.50 per kg. C =.5m

15 BLM 5..1 Practice: Slope 1. a) a) c) d) 6 5. a) 0.69 Yes a) c) 4 d) 5 5. Answer may vary. Possible answer: B(6, 1) c) Each diagram has two more squares than the previous diagram. BLM Practice: Slope as a Rate of Change breaths/min. 64 beats/min. 179 km/h 4. a).6 Once the brakes are applied, the speed of the cars decreases at a rate of.6 m/s. 5. a) Diagram Number Number of Squares BLM Practice: First Differences 1. a, d) First Differences c) linear

16 . a, d) First Differences c) non-linear. If the first differences are equal, the relation is linear. If the first differences are not equal, the relation is non-linear. 4. a) First Differences non-linear First Differences linear 5. a) First Differences non-linear First Differences linear c) First Differences linear d) First Differences non-linear 6. a) b + 4 p c) 4y + 5 d) x + 4 e) 8a + b f) 9r + 1 g) s + 5t h) h + g 7. a) 6v 6 6a 5b c) 7k + 5 d) 10x + x e) m + 4 f) 5y + 4 g) 7h + 7 h) 6p 5q 8. a) a + 7b 6 9x + xy y c) m 4 m + 4 d) 5xy + y 9. a) x P = 6x c) 6 cm BLM Practice: Connecting Variation, Slope, and First Differences 1. a) 4 c) y = x a) c) y = x. a) partial variation 5 c) d)

17 e) c) C = 0.50n y varies partially with x, the initial value is and the constant of variation is y = x 4 BLM 5.CR.1 Chapter 5 Review 1. a) 4. a). a) 1 1 c) y = x Time, t Earnings, E The relationship between x and y is a partial variation. The initial value is 1 and the constant of variation is 4. c) 1; 4; y = 4x 1 5. a) Number of Toppings, n Cost ($), C c) 7; the amount Evan earns each hour he babysits d) y = 7x. a) direct variation neither

18 c) partial variation d) partial variation 4. a) 1; 5 y = 5x + 1 c) 1. Side Length Perimeter linear The graph intersects the y-axis at (0, 1). As the x-values increase by 1, the y-values increase by a) fixed cost: $1500 variable cost: $0.08 times the number of brochures C = 0.08n c) $ a) a) 5 c) Answers may vary. Possible answer: B(6, 6) 9. a) 6.5 L/min The length of a rectangle is more than its width. BLM 5.PT.1 Chapter 5 Practice Test 1. D. B. D 4. C 5. A 6. a) 6 c) y = x a) d = 6.t 10. Tom; 1 m/s 11. a) linear linear 1. a)

19 8. Linear; I found the first differences and noticed they were all equal. 9. a) C = 6l + 50 $ a) $0.95/km; the rate of change is the slope C = 0.95d +.50 c) The fixed portion of the equation would change from.50 to.00. The graph would shift up so the vertical intercept is. BLM 5.CT.1 Chapter 5 Test 1. A. D. B 4. A 5. B 6. a) c) y = x 7. C = 0.85m 8. Non-linear; I found the first differences and noticed they were not equal. 9. a) E = 0.15n + 40 $ a) L/min; the rate of change is the slope V = 00 t c) 50 min

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