Function Fitting

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Answer Key & Explanations

Function Fitting · Grade 11 · Worksheet 2

1. Mason collects data on the population of bacteria in a Petri dish over time. He records the number of bacteria (in thousands) at specific hours. The data points are: (0, 12), (1, 24), (2, 48), (3, 96), (4, 192). Plot these points on a coordinate grid (time on the x-axis, population on the y-axis) and determine which type of function (linear, quadratic, or exponential) best fits the data. Then, sketch a smooth curve through the points and estimate the population at 5 hours. Answer: 384 thousand bacteria Solution: Write the data points: (0,12), (1,24), (2,48), (3,96), (4,192). Check for linear pattern: Differences between consecutive y-values: 24-12=12, 48-24=24, 96-48=48, 192-96=96.
   Full step-by-step solution

   Step 1: Write the data points: (0,12), (1,24), (2,48), (3,96), (4,192). Step 2: Check for linear pattern: Differences between consecutive y-values: 24-12=12, 48-24=24, 96-48=48, 192-96=96. The differences are not constant (12, 24, 48, 96), so it is not linear. Step 3: Check for quadratic pattern: Second differences: 24-12=12, 48-24=24, 96-48=48. The second differences are not constant (12, 24, 48), so it is not quadratic. Step 4: Check for exponential pattern: Ratios of consecutive y-values: 24/12=2, 48/24=2, 96/48=2, 192/96=2. The ratio is constant (2), so the data is exponential. Step 5: The exponential model has the form y = a * b^x. From (0,12): a = 12. From ratio: b = 2. So the function is y = 12 * 2^x. Step 6: Estimate population at x = 5: y = 12 * 2^5 = 12 * 32 = 384. The answer is 384 thousand bacteria.

2. Kaia is studying the cooling of a freshly baked loaf of bread. She records the temperature of the bread (in degrees Celsius) at various times (in minutes) after removing it from the oven. The data is shown in the table below: Time (min): 0, 10, 20, 30, 40, 50 Temperature (°C): 180, 90, 45, 22.5, 11.25, 5.625 Plot these points on a coordinate plane (time on the horizontal axis, temperature on the vertical axis). Determine which type of function (linear, quadratic, or exponential) best fits this data. Then, estimate the temperature of the bread after 60 minutes based on the pattern you observe. Answer: Approximately 2.8°C (exact: 2.8125°C) Solution: Examine the data. At time 0 min, temperature is 180°C. At 10 min, it is 90°C (half of 180).
   Full step-by-step solution

   Step 1: Examine the data. At time 0 min, temperature is 180°C. At 10 min, it is 90°C (half of 180). At 20 min, it is 45°C (half of 90). At 30 min, it is 22.5°C (half of 45). At 40 min, it is 11.25°C (half of 22.5). At 50 min, it is 5.625°C (half of 11.25). Step 2: The temperature is halved every 10 minutes. This means the ratio of successive temperatures is constant (0.5). A constant ratio indicates an exponential decay model. Step 3: The general form of an exponential decay function is T(t) = a * b^t, where a is the initial value and b is the decay factor per unit time. Here, the initial temperature a = 180°C, and the decay factor per 10 minutes is 0.5. To express per minute, note that after 10 minutes, T(10) = 180 * b^10 = 90, so b^10 = 0.5, thus b = (0.5)^(1/10). Alternatively, we can write the function as T(t) = 180 * (0.5)^(t/10). Step 4: Estimate temperature after 60 minutes. t = 60. T(60) = 180 * (0.5)^(60/10) = 180 * (0.5)^6 = 180 * (1/64) = 180 / 64 = 2.8125°C. Step 5: Rounding to one decimal place, the temperature is approximately 2.8°C. Conclusion: The data fits an exponential decay model, and the estimated temperature after 60 minutes is 2.8125°C (or about 2.8°C).

3. Hana records the following data: (0, 2), (1, 6), (2, 18), (3, 54), (4, 162). Determine which function type (linear, quadratic, or exponential) best fits the data, and write the equation of the model in the form y = a * b^x or y = ax^2 + bx + c or y = mx + b. Answer: y = 2 * 3^x Solution: Examine the y-values: 2, 6, 18, 54, 162. Calculate first differences: 6-2=4, 18-6=12, 54-18=36, 162-54=108. Not constant, so not linear.
   Full step-by-step solution

   Step 1: Examine the y-values: 2, 6, 18, 54, 162. Step 2: Calculate first differences: 6-2=4, 18-6=12, 54-18=36, 162-54=108. Not constant, so not linear. Step 3: Calculate second differences: 12-4=8, 36-12=24, 108-36=72. Not constant, so not quadratic. Step 4: Calculate ratios of consecutive y-values: 6/2=3, 18/6=3, 54/18=3, 162/54=3. The ratio is constant (3), so the data is exponential. Step 5: The exponential model is y = a * b^x. Here b = 3 (the constant ratio). When x=0, y=2, so a = 2. Step 6: The equation is y = 2 * 3^x. Final answer: y = 2 * 3^x.

4. Aroha records the following data: (1, 7), (2, 13), (3, 25), (4, 43), (5, 67). Determine which function type (linear, quadratic, or exponential) best fits this data and provide the equation of the model. Answer: Quadratic; y = 3x² + 3x + 1 Solution: x=1 to 2: 13 - 7 = 6 x=2 to 3: 25 - 13 = 12 x=3 to 4: 43 - 25 = 18 x=4 to 5: 67 - 43 = 24 First differences: 6, 12, 18, 24 (not constant, so not linear).
   Full step-by-step solution

   Step 1: Calculate first differences (Δy): x=1 to 2: 13 - 7 = 6 x=2 to 3: 25 - 13 = 12 x=3 to 4: 43 - 25 = 18 x=4 to 5: 67 - 43 = 24 First differences: 6, 12, 18, 24 (not constant, so not linear). Step 2: Calculate second differences: 12 - 6 = 6 18 - 12 = 6 24 - 18 = 6 Second differences are constant (6), so the data is quadratic. Step 3: Assume quadratic model y = ax² + bx + c. Use three points to solve. Using (1, 7): a(1)² + b(1) + c = 7 → a + b + c = 7 Using (2, 13): a(4) + b(2) + c = 13 → 4a + 2b + c = 13 Using (3, 25): a(9) + b(3) + c = 25 → 9a + 3b + c = 25 Step 4: Subtract first equation from second: (4a + 2b + c) - (a + b + c) = 13 - 7 → 3a + b = 6 Step 5: Subtract second equation from third: (9a + 3b + c) - (4a + 2b + c) = 25 - 13 → 5a + b = 12 Step 6: Subtract the result of Step 4 from Step 5: (5a + b) - (3a + b) = 12 - 6 → 2a = 6 → a = 3 Step 7: Substitute a = 3 into 3a + b = 6: 3(3) + b = 6 → 9 + b = 6 → b = -3 Step 8: Substitute a = 3, b = -3 into a + b + c = 7: 3 + (-3) + c = 7 → 0 + c = 7 → c = 7 Step 9: The quadratic model is y = 3x² - 3x + 7. Verify with point (4, 43): 3(16) - 3(4) + 7 = 48 - 12 + 7 = 43. Correct. Step 10: Verify with point (5, 67): 3(25) - 3(5) + 7 = 75 - 15 + 7 = 67. Correct. The answer is Quadratic; y = 3x² - 3x + 7.

5. The following data points are recorded: (1, 3), (3, 27), (5, 75), (7, 147). Determine which type of function (linear, quadratic, or exponential) best fits the data, and estimate the value of the function at x = 9. Answer: Quadratic; f(9) = 243 Solution: Compute first differences: 27 - 3 = 24, 75 - 27 = 48, 147 - 75 = 72. These are not constant (24, 48, 72), so not linear. Compute second differences: 48 - 24 = 24, 72 - 48 = 24.
   Full step-by-step solution

   Step 1: Compute first differences: 27 - 3 = 24, 75 - 27 = 48, 147 - 75 = 72. These are not constant (24, 48, 72), so not linear. Step 2: Compute second differences: 48 - 24 = 24, 72 - 48 = 24. The second differences are constant (24), so the data is quadratic. Step 3: Assume the quadratic function is f(x) = ax^2 + bx + c. Use the point (1, 3): a(1)^2 + b(1) + c = 3, so a + b + c = 3. Use (3, 27): 9a + 3b + c = 27. Use (5, 75): 25a + 5b + c = 75. Step 4: Subtract the first equation from the second: (9a + 3b + c) - (a + b + c) = 27 - 3, so 8a + 2b = 24, divide by 2: 4a + b = 12. Subtract the second from the third: (25a + 5b + c) - (9a + 3b + c) = 75 - 27, so 16a + 2b = 48, divide by 2: 8a + b = 24. Step 5: Subtract 4a + b = 12 from 8a + b = 24: (8a + b) - (4a + b) = 24 - 12, so 4a = 12, a = 3. Then 4(3) + b = 12, so 12 + b = 12, b = 0. From a + b + c = 3: 3 + 0 + c = 3, so c = 0. Thus f(x) = 3x^2. Step 6: Estimate at x = 9: f(9) = 3(9)^2 = 3(81) = 243. The answer is quadratic; f(9) = 243.

6. Olivia is monitoring the height of a burning candle over time. She records the following data: at time 0 minutes, the candle height is 15 cm; at 5 minutes, it is 12 cm; at 10 minutes, it is 9 cm; at 15 minutes, it is 6 cm; and at 20 minutes, it is 3 cm. Determine which type of function (linear, quadratic, or exponential) best models the relationship between the time and the height of the candle. Provide a brief justification based on the pattern of change in the data, and sketch the approximate curve. Answer: Linear function, because the height decreases by a constant amount (3 cm) every 5 minutes. Solution: Examine the data. Time (minutes): 0, 5, 10, 15, 20. Height (cm): 15, 12, 9, 6, 3.
   Full step-by-step solution

   Step 1: Examine the data. Time (minutes): 0, 5, 10, 15, 20. Height (cm): 15, 12, 9, 6, 3. Step 2: Calculate the change in height over each 5-minute interval. - From 0 to 5 minutes: 12 - 15 = -3 cm. - From 5 to 10 minutes: 9 - 12 = -3 cm. - From 10 to 15 minutes: 6 - 9 = -3 cm. - From 15 to 20 minutes: 3 - 6 = -3 cm. Step 3: The first differences (change in height) are constant at -3 cm for each equal time interval of 5 minutes. This constant rate of change indicates a linear relationship. Step 4: Check for quadratic or exponential patterns. For a quadratic function, the second differences (change in the first differences) would be constant. Here, the first differences are all -3, so the second differences are 0 (constant), which is a special case of a linear function, not quadratic. For an exponential function, the ratio of consecutive heights would be constant: 12/15 = 0.8, 9/12 = 0.75, 6/9 = 2/3, 3/6 = 0.5. These ratios are not equal, so it is not exponential. Step 5: Conclusion: The data is best modeled by a linear function. The height decreases by 3 cm every 5 minutes, so the relationship is linear. A sketch would show points (0,15), (5,12), (10,9), (15,6), (20,3) falling on a straight line with a negative slope. The answer is Linear function, because the height decreases by a constant amount (3 cm) every 5 minutes.