Rate of Change

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Answer Key & Explanations

Rate of Change · Grade 8 · Worksheet 2

1. A linear function has a slope of 6 and passes through the point (0, 11). What is the initial value of the function? Answer: 11 Solution: The initial value of a linear function is the y-intercept, which is the value of y when x = 0. The problem states the function passes through the point (0, 11). At x = 0, the y-coordinate is 11.
   Full step-by-step solution

   Step 1: The initial value of a linear function is the y-intercept, which is the value of y when x = 0. Step 2: The problem states the function passes through the point (0, 11). Step 3: At x = 0, the y-coordinate is 11. Step 4: Therefore, the initial value is 11. The answer is 11.

2. A scientist is studying bacterial growth in a lab. The initial population is 500 bacteria, and the population doubles every 3 hours. Write an exponential function in the form y = a * b^x that models the bacterial population, where y is the total population and x is the number of 3-hour periods that have passed. Answer: y = 500 * 2^x Solution: We start with an initial population of 500 bacteria. The population doubles every 3 hours.
   Full step-by-step solution

   Let's go step-by-step. --- **Step 1: Understand the problem** We start with an initial population of 500 bacteria. The population doubles every 3 hours. We are told to write an exponential function in the form: y = a * b^x where: y = total population x = number of 3-hour periods that have passed. --- **Step 2: Identify the initial value (a)** The initial population is when x = 0 (no time has passed). At x = 0, the population is 500. So: a = 500 --- **Step 3: Identify the growth factor (b)** The population doubles every 3 hours. Since x is measured in 3-hour periods, in 1 period (x = 1) the population should double. Doubling means multiplying by 2. So: b = 2 --- **Step 4: Write the equation** Substitute a = 500 and b = 2 into y = a * b^x: y = 500 * 2^x --- **Step 5: Check the reasoning** At x = 0: y = 500 * 2^0 = 500 * 1 = 500 (correct initial). At x = 1: y = 500 * 2^1 = 500 * 2 = 1000 (doubled after 3 hours). At x = 2: y = 500 * 4 = 2000 (doubled again after 6 hours). Matches the condition. --- **Final answer:** y = 500 * 2^x

3. Noah is filling a pool with water. The pool already has 40 gallons of water in it. A hose adds 18 gallons of water each minute. If the pool fills at a constant rate, what is the rate of change (gallons per minute) and the initial value (starting amount of water)? Answer: 18 Solution: The rate of change is the amount of water added each minute: 18 gallons per minute. The initial value is the amount of water already in the pool: 40 gallons.
   Full step-by-step solution

   The rate of change is the amount of water added each minute: 18 gallons per minute. The initial value is the amount of water already in the pool: 40 gallons. So the rate of change is 18 and the initial value is 40.

4. A right triangle is drawn on a coordinate plane with vertices at (0,0), (6,0), and (6,8). A second triangle is created by applying the transformation (x, y) → (1.5x, 1.5y) to all vertices of the first triangle. What is the area of the second triangle? Answer: 54 Solution: A = (0, 0) B = (6, 0) C = (6, 8) This is a right triangle with legs along the x-axis and vertical line x = 6. Leg AB is horizontal from (0,0) to (6,0), length = 6. Leg BC is vertical from (6,0) to (6,8), length = 8.
   Full step-by-step solution

   Let's go step by step. --- **Step 1: Identify the first triangle's vertices and area** Vertices of the first triangle: A = (0, 0) B = (6, 0) C = (6, 8) This is a right triangle with legs along the x-axis and vertical line x = 6. Leg AB is horizontal from (0,0) to (6,0), length = 6. Leg BC is vertical from (6,0) to (6,8), length = 8. Area of first triangle = (1/2) × base × height = (1/2) × 6 × 8 = (1/2) × 48 = 24 So the first triangle's area is 24. --- **Step 2: Apply the transformation to each vertex** Transformation rule: (x, y) → (1.5x, 1.5y) A' = (1.5×0, 1.5×0) = (0, 0) B' = (1.5×6, 1.5×0) = (9, 0) C' = (1.5×6, 1.5×8) = (9, 12) --- **Step 3: Find the side lengths of the second triangle** A'B' is horizontal from (0,0) to (9,0), length = 9. B'C' is vertical from (9,0) to (9,12), length = 12. So the second triangle is also a right triangle with legs 9 and 12. --- **Step 4: Calculate the area of the second triangle** Area = (1/2) × 9 × 12 = (1/2) × 108 = 54 --- **Step 5: Alternate reasoning using scale factor effect on area** The transformation (x, y) → (1.5x, 1.5y) is a dilation with scale factor 1.5. When all lengths are multiplied by 1.5, the area is multiplied by (1.5)^2 = 2.25. So area of second triangle = 24 × 2.25 = 54. --- **Final answer:** 54

5. A linear function has a slope of 8 and passes through the point (0, 15). What is the initial value of the function? Answer: 15 Solution: The initial value of a linear function is the y-intercept, which is the value of y when x = 0. The problem states the function passes through the point (0, 15). At x = 0, the y-coordinate is 15.
   Full step-by-step solution

   Step 1: The initial value of a linear function is the y-intercept, which is the value of y when x = 0. Step 2: The problem states the function passes through the point (0, 15). Step 3: At x = 0, the y-coordinate is 15. Step 4: Therefore, the initial value is 15. The answer is 15.

6. Liam is tracking the growth of his sunflower plant. In the first week, it grew 8 centimeters. After that, it grew at a constant rate of 5 centimeters per week. Write an equation in slope-intercept form (y = mx + b) that represents the total height of the sunflower, y, after x weeks. Answer: y = 5x + 8 Solution: In a linear equation of the form y = mx + b, the 'b' term represents the initial value or starting point before any changes occur.
   Full step-by-step solution

   In a linear equation of the form y = mx + b, the 'b' term represents the initial value or starting point before any changes occur. The 'm' term represents the constant rate of change, which is how much the dependent variable increases for each unit increase in the independent variable. This applies to many real-world situations like savings accounts, plant growth, or speed.

7. Isabella is reading a book for a school project. The book has 308 pages, and Isabella reads 29 pages each day. If Isabella reads at a constant rate, what is the rate of change (pages per day) and the initial value (pages read before starting, which is 0)? Answer: 29 Solution: The rate of change is the number of pages read each day: 29 pages per day. The initial value is the number of pages read at the start: 0 pages.
   Full step-by-step solution

   The rate of change is the number of pages read each day: 29 pages per day. The initial value is the number of pages read at the start: 0 pages. So the rate of change is 29 and the initial value is 0.

8. Moana is tracking the growth of a plant. The plant was 8 inches tall when Moana started measuring it. Each week, the plant grows by 3 inches. If the plant grows at a constant rate, what is the rate of change (growth per week) and the initial value (starting height)? Answer: 3 Solution: The rate of change is the amount the plant grows each week: 3 inches per week. The initial value is the height at the start: 8 inches.
   Full step-by-step solution

   The rate of change is the amount the plant grows each week: 3 inches per week. The initial value is the height at the start: 8 inches. So the rate of change is 3 inches per week and the initial value is 8 inches.