Functional Relationships

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Functional Relationships · Grade 8 · Worksheet 2

1. Noah is tracking the water level in a large tank as it is being drained for cleaning. The tank starts with 1800 gallons of water. He opens the drain valve, and water flows out at a constant rate of 12 gallons per minute. Describe qualitatively how the amount of water in the tank changes over time from the moment the valve is opened until the tank is empty. Be sure to state whether the relationship is increasing, decreasing, or constant, and explain why. Answer: The amount of water in the tank is decreasing at a constant rate over time until it reaches zero. Solution: At the start, the tank has 1800 gallons of water. As time passes, water flows out at 12 gallons per minute, so the total amount of water in the tank decreases.
   Full step-by-step solution

   Step 1: At the start, the tank has 1800 gallons of water. As time passes, water flows out at 12 gallons per minute, so the total amount of water in the tank decreases. Step 2: The rate of decrease is constant (12 gallons every minute), so the relationship is decreasing at a constant rate. Step 3: This continues until the tank is empty (0 gallons). Therefore, the relationship is decreasing and constant (linear decrease).

2. Emma is tracking the height of a plant over 10 days. On day 0, the plant is 5 cm tall. It grows at a constant rate of 2 cm per day for the first 5 days. Then, from day 5 to day 10, it grows at a constant rate of 1 cm per day. Describe qualitatively how the height of the plant changes over the 10-day period, including where it is increasing, decreasing, or constant. Answer: The height is increasing over the entire 10-day period, but at a faster rate from day 0 to day 5 (2 cm per day) and at a slower rate from day 5 to day 10 (1 cm per day). It is never constant or decreasing. Solution: From day 0 to day 5, the plant grows 2 cm per day. This is a positive rate, so the height is increasing. The graph would be a straight line with a slope of 2.
   Full step-by-step solution

   Step 1: From day 0 to day 5, the plant grows 2 cm per day. This is a positive rate, so the height is increasing. The graph would be a straight line with a slope of 2. Step 2: From day 5 to day 10, the plant grows 1 cm per day. This is also a positive rate, so the height is still increasing, but at a slower rate. The graph would be a straight line with a slope of 1. Step 3: There is no time when the height stays the same (constant) or goes down (decreasing). Step 4: Therefore, the height is increasing over the entire 10-day period, with a faster increase in the first 5 days and a slower increase in the last 5 days.

3. Emma is tracking the height of a plant over 8 weeks. The height in centimeters is given by the function h(w) = 3w + 5, where w is the number of weeks since planting. Describe how the height changes as w increases from 0 to 8. Answer: The height increases at a constant rate of 3 cm per week. Solution: Identify the function: h(w) = 3w + 5. This is a linear function in slope-intercept form (y = mx + b), where m = 3 and b = 5.
   Full step-by-step solution

   Step 1: Identify the function: h(w) = 3w + 5. This is a linear function in slope-intercept form (y = mx + b), where m = 3 and b = 5. Step 2: The slope m = 3 means that for every increase of 1 in w (each week), the height h increases by 3 centimeters. Step 3: The y-intercept b = 5 means the initial height at week 0 is 5 cm. Step 4: As w increases from 0 to 8, the height increases steadily. For example, at w = 0, h = 5 cm; at w = 4, h = 3(4) + 5 = 17 cm; at w = 8, h = 3(8) + 5 = 29 cm. Step 5: Since the slope is positive and constant, the function is increasing at a constant rate. The height increases by 3 cm each week. The answer is: The height increases at a constant rate of 3 cm per week.

4. A right triangle is drawn on a coordinate plane with vertices at (0,0), (5,0), and (5,12). A second triangle is created by applying the transformation (x, y) → (x + 3, y - 4) to all vertices of the first triangle. What is the perimeter of the new triangle? Answer: 30 Solution: Original vertices: (0,0), (5,0), (5,12) Transformation: (x, y) → (x + 3, y - 4) (0,0) → (0+3, 0-4) = (3,-4) (5,0) → (5+3, 0-4) = (8,-4) (5,12) → (5+3, 12-4) = (8,8) Between (3,-4) and (8,-4): horizontal distance = 8 - 3 = 5 units Between (8,-4) and (8,8): vertical distance = 8 - (-4) = 12 units…
   Full step-by-step solution

   Step 1: Apply the transformation to each vertex of the original triangle. Original vertices: (0,0), (5,0), (5,12) Transformation: (x, y) → (x + 3, y - 4) Step 2: Calculate new vertices: (0,0) → (0+3, 0-4) = (3,-4) (5,0) → (5+3, 0-4) = (8,-4) (5,12) → (5+3, 12-4) = (8,8) Step 3: Calculate side lengths of new triangle: Between (3,-4) and (8,-4): horizontal distance = 8 - 3 = 5 units Between (8,-4) and (8,8): vertical distance = 8 - (-4) = 12 units Between (3,-4) and (8,8): use distance formula = sqrt((8-3)^2 + (8-(-4))^2) = sqrt(5^2 + 12^2) = sqrt(25 + 144) = sqrt(169) = 13 units Step 4: Calculate perimeter: Perimeter = 5 + 12 + 13 = 30 units The answer is 30.

5. Describe how the function y = 3x² - 5 changes as x increases from -3 to 0, and then from 0 to 3. Is the function increasing, decreasing, or constant in each interval? Answer: From x = -3 to 0, the function is decreasing. From x = 0 to 3, the function is increasing. Solution: Recognize that y = 3x² - 5 is a quadratic function with a positive coefficient (3) for x². This means its graph is a parabola opening upward, with a minimum point at the vertex.
   Full step-by-step solution

   Step 1: Recognize that y = 3x² - 5 is a quadratic function with a positive coefficient (3) for x². This means its graph is a parabola opening upward, with a minimum point at the vertex. Step 2: The vertex of y = 3x² - 5 occurs at x = 0 (since there is no x term). At x = 0, y = 3(0)² - 5 = -5. Step 3: Consider the interval from x = -3 to 0. As x increases from -3 to 0, the value of x² decreases (since (-3)² = 9, (-2)² = 4, (-1)² = 1, 0² = 0). So 3x² decreases, and therefore y = 3x² - 5 also decreases. The function is decreasing on this interval. Step 4: Consider the interval from x = 0 to 3. As x increases from 0 to 3, the value of x² increases (0² = 0, 1² = 1, 2² = 4, 3² = 9). So 3x² increases, and therefore y = 3x² - 5 also increases. The function is increasing on this interval. Step 5: Conclusion: From x = -3 to 0, the function is decreasing. From x = 0 to 3, the function is increasing.

6. Aroha is tracking the temperature of a chemical solution during an experiment. She starts heating the solution at 9:00 AM. From 9:00 AM to 9:30 AM, the temperature rises steadily from 20°C to 35°C. Then, from 9:30 AM to 10:00 AM, she adds a cooling agent, and the temperature drops steadily back to 20°C. Finally, from 10:00 AM to 10:30 AM, she stops adding anything and the temperature stays constant at 20°C. Describe how the temperature of the solution changes over time from 9:00 AM to 10:30 AM. In each time interval, is the temperature increasing, decreasing, or constant? Answer: From 9:00 AM to 9:30 AM, the temperature is increasing. From 9:30 AM to 10:00 AM, the temperature is decreasing. From 10:00 AM to 10:30 AM, the temperature is constant. Solution: Identify the first time interval: 9:00 AM to 9:30 AM. The temperature rises from 20°C to 35°C. Since it is going up, the temperature is increasing during this interval.
   Full step-by-step solution

   Step 1: Identify the first time interval: 9:00 AM to 9:30 AM. The temperature rises from 20°C to 35°C. Since it is going up, the temperature is increasing during this interval. Step 2: Identify the second time interval: 9:30 AM to 10:00 AM. The temperature drops from 35°C to 20°C. Since it is going down, the temperature is decreasing during this interval. Step 3: Identify the third time interval: 10:00 AM to 10:30 AM. The temperature stays at 20°C. Since it does not change, the temperature is constant during this interval. The answer: From 9:00 AM to 9:30 AM, the temperature is increasing. From 9:30 AM to 10:00 AM, the temperature is decreasing. From 10:00 AM to 10:30 AM, the temperature is constant.

7. (2.5 × 10⁴) ÷ (5 × 10⁻²) = ? Answer: 500000 Solution: Divide the coefficients: 2.5 ÷ 5 = 0.5 Subtract the exponents: 4 - (-2) = 4 + 2 = 6 Combine the results: 0.5 × 10⁶ Convert to standard form: 0.5 × 1,000,000 = 500,000 The answer is 500000.
   Full step-by-step solution

   Step 1: Divide the coefficients: 2.5 ÷ 5 = 0.5 Step 2: Subtract the exponents: 4 - (-2) = 4 + 2 = 6 Step 3: Combine the results: 0.5 × 10⁶ Step 4: Convert to standard form: 0.5 × 1,000,000 = 500,000 The answer is 500000.