3-1 Constant Rate of Change

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1 Determine whether the relationship between the two quantities shown in the table or graph is linear. If so, find the constant rate of change. If not, explain your reasoning. 1. Analyze the table. The rate of change from 5 hours to 8 hours is or 3 cents per hour. This is the same as the rate of change from 8 hours to 12 hours, or 3 cents per hour, and from 12 hours to 24 hours, or 3 cents per hour. So, the rate of change is a constant 3 per hour. 2. Analyze the table. The rate of change from 1 to 2 seconds, or 14.7 m/s, is not the same as the rate of change from 2 to 3 seconds, or 24.5 m/s. So, the rate of change is not constant. esolutions Manual - Powered by Cognero Page 1

2 3. Analyze the table. As the amount of oil increases by 2 cups, the amount of vinegar increases by cup. Since the rate of change is constant, it is linear. So, the constant rate of change is or cup vinegar per cup of oil. 4. Two points highlighted on the graph are (1, 2) and (3, 6). Find the rate of change between them. (1, 2) 1 minute, 2 inches (3, 6) 3 minutes, 6 inches esolutions Manual - Powered by Cognero Page 2

3 5. Two points highlighted on the graph are (2, 15) and (6, 45). Find the rate of change between them. (2, 15) 2 inches, 15 miles (6, 45) 6 inches, 45 miles 6. The two points on the graph are (5, 100) and (15, 250). Find the rate of change between them. (5, 100) 5 people, $100 (15, 250) 15 people, $250 esolutions Manual - Powered by Cognero Page 3

4 Determine whether a proportional linear relationship exists between the two quantities shown. Explain your reasoning. 7. To determine if the two quantities are proportional, express the relationship between cost and time from the values in the table as a ratio. The ratio is a constant 3 per hour, so the relationship is proportional. 8. To determine if the two quantities are proportional, express the relationship between vinegar and oil from the values in the table as a ratio. The ratio is a constant cup of vinegar per cup of oil, so the relationship is proportional. esolutions Manual - Powered by Cognero Page 4

5 9. To determine if the two distances are proportional, express the relationship between actual distance and map distance for the points on the graph as a ratio. The ratio is a constant miles per inch, so the relationship is proportional. 10. Use Math Tools Match each table with its rate of change. Find the rate of change for each table. or 0.8 ft/min; or 0.8 ft/min esolutions Manual - Powered by Cognero Page 5

6 or 10 ft/min; or 10 ft/min or 0.25 ft/min; or 0.25 ft/min or 2.4 ft/min; or 2.4 ft/min Draw a line connecting each table with the correct rate of change. esolutions Manual - Powered by Cognero Page 6

7 Determine whether the relationship between the two quantities shown in the table is linear. If so, find the constant rate of change. If not, explain your reasoning. 14. Find the rates of change. The sale price increases by $5 as the retail price increases by $10, so the rate of change is a constant value of. esolutions Manual - Powered by Cognero Page 7

8 15. Analyze the table. The rate of change from 1 to 2 hours, or 12 per hour, is not the same as the rate of change from 3 to 4 hours, or 24 per hour. So, the rate of change is not constant. 16. Determine whether a proportional relationship exists between the two quantities in Exercise 14. Explain your reasoning. To determine if the two quantities are proportional, express some of the relationships between the sale price and the retail price from the values in the table as a ratio. The ratio is a constant, so the relationship is proportional. esolutions Manual - Powered by Cognero Page 8

9 Reason Abstractly Find the constant rate of change for the graph and interpret its meaning. 17. The constant rate of change is. This means the distance decreased by 50 miles every hour. 18. The constant rate of change is. This means the earnings were $5 per hour. 19. The constant rate of change is. This means the sale price is of retail price. esolutions Manual - Powered by Cognero Page 9

10 20. The table shows the amount of money in Will s savings account. Graph the points on the coordinate plane and connect them with a straight line. What is the constant rate of change? Two points on the graph are (1, 40) and (3, 70). Find the rate of change between them. (1, 40) 1 week, $40 savings (3, 70) 3 weeks, $70 savings esolutions Manual - Powered by Cognero Page 10

11 21. The graph shows the distance Bianca traveled on her 2-hour bike ride. Determine if each statement is true or false. a. She traveled at a constant speed of 12 miles per hour for the entire ride. True False b. She traveled at a constant speed of 12 miles per hour for the first hour. True False c. She traveled at a constant speed of 4 miles per hour for the last hour. True False a. If the points are connected, the line would not be straight. This means the rate of change is not consistent, so a. is false. b. Two points from the first hour on the graph are (0, 0) and (1, 12). Find the rate of change between them. (0, 0) 0 hour, 0 miles (1, 12) 1 hour, 12 miles The constant speed during the first hour of Bianca s bike ride is 12 miles per hour. So, b. is true. c. Two points from the last hour on the graph are (1, 12) and (2, 16). Find the rate of change between them. (1, 12) 1 hour, 12 miles (2, 16) 2 hours, 16 miles The constant speed during the last hour of Bianca s bike ride is 4 miles per hour. So, c. is true. esolutions Manual - Powered by Cognero Page 11

12 Find the unit rate. Round to the nearest hundredth if necessary miles on 2.5 gallons Write the rate as a fraction with a denominator of ,500 kilobytes in 6 minutes Write the rate as a fraction with a denominator of red peppers for $5.50 Write the rate as a fraction with a denominator of 1. esolutions Manual - Powered by Cognero Page 12

3-2 Estimate Products. Estimate the product SOLUTION: Use rounding. So, is about SOLUTION: Use rounding.

3-2 Estimate Products. Estimate the product SOLUTION: Use rounding. So, is about SOLUTION: Use rounding. Estimate the product. 1. 9.7 3.3 So, 9.7 3.3 is about 30. 2. 3.4 5.6 So, 3.4 5.6 is about 18. 3. 17.5 8.4 So, 17.5 8.4 is about 160. 4. 44.8 5.1 225 So, 44.8 5.1 is about 225. 5. 28.21 8.02 So, 28.21 8.02