Function Fitting

lessonbunny.com

Answer Key & Explanations

Function Fitting · Grade 11 · Worksheet 1

1. Emma is monitoring the height of a sunflower plant over several days. She records the following data: Day 0: 15 cm, Day 5: 30 cm, Day 10: 60 cm, Day 15: 120 cm, Day 20: 240 cm. Determine which type of function (linear, quadratic, or exponential) best models the height of the sunflower over time. Explain your reasoning and write the function that fits the data. Answer: Exponential; H(t) = 15 * 2^(t/5) Solution: Check for linear model by calculating first differences: 30-15 = 15, 60-30 = 30, 120-60 = 60, 240-120 = 120. The differences are not constant, so not linear.
   Full step-by-step solution

   Step 1: Check for linear model by calculating first differences: 30-15 = 15, 60-30 = 30, 120-60 = 60, 240-120 = 120. The differences are not constant, so not linear. Step 2: Check for quadratic model by calculating second differences: 30-15 = 15, 60-30 = 30, 120-60 = 60. Second differences: 30-15 = 15, 60-30 = 30, not constant, so not quadratic. Step 3: Check for exponential model by calculating ratios: 30/15 = 2, 60/30 = 2, 120/60 = 2, 240/120 = 2. The ratio is constant at 2 every 5 days. Step 4: Write the exponential function. Starting height is 15 cm at day 0. Height doubles every 5 days. The function is H(t) = 15 * 2^(t/5), where t is time in days. The answer is exponential, H(t) = 15 * 2^(t/5).

2. A scientist records the following data for a population of bacteria over time (hours): (0, 50), (1, 100), (2, 200), (3, 400), (4, 800). Determine which function type (linear, quadratic, or exponential) best fits the data, and estimate the population at t = 5 hours. Answer: Exponential; 1600 Solution: Examine the y-values: 50, 100, 200, 400, 800. Calculate first differences: 100-50=50, 200-100=100, 400-200=200, 800-400=400. Not constant, so not linear.
   Full step-by-step solution

   Step 1: Examine the y-values: 50, 100, 200, 400, 800. Step 2: Calculate first differences: 100-50=50, 200-100=100, 400-200=200, 800-400=400. Not constant, so not linear. Step 3: Calculate second differences: 100-50=50, 200-100=100, 400-200=200. Not constant, so not quadratic. Step 4: Calculate ratios: 100/50=2, 200/100=2, 400/200=2, 800/400=2. The ratio is constant (2), so the data is exponential. Step 5: The exponential model is y = 50 * (2)^t. At t=5, y = 50 * 2^5 = 50 * 32 = 1600. The answer is exponential; 1600.

3. Noah is studying the spread of a new plant species in a forest. He records the area covered by the plants (in square meters) at different times (in years since the first observation). The data is: t (years): 1, 2, 3, 4, 5 A(t) (m²): 6, 16, 36, 66, 106 Determine which type of function (linear, quadratic, or exponential) best models the data, and estimate the area covered after 6 years. Answer: Quadratic; 156 m² Solution: Calculate first differences (change in area each year): 16 - 6 = 10 36 - 16 = 20 66 - 36 = 30 106 - 66 = 40 First differences: 10, 20, 30, 40 (not constant, so not linear).
   Full step-by-step solution

   Step 1: Calculate first differences (change in area each year): 16 - 6 = 10 36 - 16 = 20 66 - 36 = 30 106 - 66 = 40 First differences: 10, 20, 30, 40 (not constant, so not linear). Step 2: Calculate second differences (differences of first differences): 20 - 10 = 10 30 - 20 = 10 40 - 30 = 10 Second differences are constant at 10, so the data fits a quadratic model. Step 3: The quadratic function has the form A(t) = at² + bt + c. Using points (1, 6), (2, 16), (3, 36): For t=1: a + b + c = 6 For t=2: 4a + 2b + c = 16 For t=3: 9a + 3b + c = 36 Subtract first equation from second: (4a + 2b + c) - (a + b + c) = 16 - 6 → 3a + b = 10 Subtract second equation from third: (9a + 3b + c) - (4a + 2b + c) = 36 - 16 → 5a + b = 20 Subtract these two results: (5a + b) - (3a + b) = 20 - 10 → 2a = 10 → a = 5 Substitute a = 5 into 3a + b = 10: 3(5) + b = 10 → 15 + b = 10 → b = -5 Substitute a = 5, b = -5 into a + b + c = 6: 5 + (-5) + c = 6 → c = 6 So the quadratic model is A(t) = 5t² - 5t + 6. Step 4: Estimate area after 6 years: A(6) = 5(36) - 5(6) + 6 = 180 - 30 + 6 = 156. The best model is quadratic, and the estimated area after 6 years is 156 m².

4. The population of a certain bacterial culture grows according to the model P(t) = 1200e^(0.05t), where t is time in hours. After how many hours will the population reach 3000 bacteria? Round your answer to the nearest tenth of an hour. Answer: 18.3 Solution: P(t) = 1200 e^(0.05 t) We want to find t when P(t) = 3000. Set up the equation. 3000 = 1200 e^(0.05 t) Divide both sides by 1200 to isolate the exponential term.
   Full step-by-step solution

   We are given the population model: P(t) = 1200 e^(0.05 t) We want to find t when P(t) = 3000. Step 1: Set up the equation. 3000 = 1200 e^(0.05 t) Step 2: Divide both sides by 1200 to isolate the exponential term. 3000 / 1200 = e^(0.05 t) Simplify the fraction: 3000 ÷ 1200 = 30/12 = 5/2 = 2.5 So: 2.5 = e^(0.05 t) Step 3: Take the natural logarithm of both sides to solve for t. ln(2.5) = ln(e^(0.05 t)) Using the property ln(e^x) = x: ln(2.5) = 0.05 t Step 4: Solve for t. t = ln(2.5) / 0.05 Step 5: Compute ln(2.5). ln(2.5) ≈ 0.9162907319 (using a calculator) Step 6: Divide by 0.05. t ≈ 0.9162907319 / 0.05 Dividing by 0.05 is the same as multiplying by 20: t ≈ 0.9162907319 × 20 t ≈ 18.325814638 Step 7: Round to the nearest tenth of an hour. t ≈ 18.3 Final answer: 18.3 hours

5. Hana records the following data: (1, 12), (2, 36), (3, 108), (4, 324), (5, 972). Determine which function type (linear, quadratic, or exponential) best fits the data and write the approximate equation of the model. Answer: Exponential; y = 4 * 3^x Solution: Check first differences: 36-12 = 24, 108-36 = 72, 324-108 = 216, 972-324 = 648. Not constant, so not linear. Check second differences: 72-24 = 48, 216-72 = 144, 648-216 = 432.
   Full step-by-step solution

   Step 1: Check first differences: 36-12 = 24, 108-36 = 72, 324-108 = 216, 972-324 = 648. Not constant, so not linear. Step 2: Check second differences: 72-24 = 48, 216-72 = 144, 648-216 = 432. Not constant, so not quadratic. Step 3: Check ratios: 36/12 = 3, 108/36 = 3, 324/108 = 3, 972/324 = 3. Constant ratio of 3, so exponential. Step 4: Exponential form is y = a * b^x. b = 3. Use point (1, 12): 12 = a * 3^1 => a = 4. So y = 4 * 3^x. The answer is exponential; y = 4 * 3^x.

6. Ava records the following data: (1, 6), (2, 11), (3, 16), (4, 21), (5, 26). Which type of function best models this data: linear, quadratic, or exponential? Justify your answer and write the equation of the best-fit function. Answer: Linear; y = 5x + 1 Solution: Calculate first differences: 11-6 = 5, 16-11 = 5, 21-16 = 5, 26-21 = 5. The first differences are constant (all equal to 5). This indicates a linear function.
   Full step-by-step solution

   Step 1: Calculate first differences: 11-6 = 5, 16-11 = 5, 21-16 = 5, 26-21 = 5. The first differences are constant (all equal to 5). This indicates a linear function. Step 2: The slope m = 5. Using point (1, 6): 6 = 5(1) + b => b = 1. So equation is y = 5x + 1. Step 3: Check: x=2 gives 5(2)+1=11, x=3 gives 16, etc. The answer is linear; y = 5x + 1.

7. Aroha records the following data: (1, 3), (2, 9), (3, 27), (4, 81), (5, 243). Determine which type of function (linear, quadratic, or exponential) best fits this data and write the equation of the function. Answer: Exponential; y = 3^x Solution: Check for linear: First differences: 9-3=6, 27-9=18, 81-27=54, 243-81=162. Not constant, so not linear. Check for quadratic: Second differences: 18-6=12, 54-18=36, 162-54=108.
   Full step-by-step solution

   Step 1: Check for linear: First differences: 9-3=6, 27-9=18, 81-27=54, 243-81=162. Not constant, so not linear. Step 2: Check for quadratic: Second differences: 18-6=12, 54-18=36, 162-54=108. Not constant, so not quadratic. Step 3: Check for exponential: Ratios: 9/3=3, 27/9=3, 81/27=3, 243/81=3. Constant ratio of 3, so exponential. Step 4: The ratio is the base: y = a * 3^x. Using (1,3): 3 = a * 3^1 = 3a, so a = 1. Equation: y = 3^x. The answer is exponential; y = 3^x.

8. Liam records the following data: (1, 7), (2, 21), (3, 63), (4, 189). Which type of function (linear, quadratic, or exponential) best fits this data? Write the estimated equation of the function. Answer: exponential, y = 7 * 3^(x-1) or y = (7/3) * 3^x Solution: Check first differences: 21 - 7 = 14, 63 - 21 = 42, 189 - 63 = 126. Not constant, so not linear. Check second differences: 42 - 14 = 28, 126 - 42 = 84.
   Full step-by-step solution

   Step 1: Check first differences: 21 - 7 = 14, 63 - 21 = 42, 189 - 63 = 126. Not constant, so not linear. Step 2: Check second differences: 42 - 14 = 28, 126 - 42 = 84. Not constant, so not quadratic. Step 3: Check ratios: 21/7 = 3, 63/21 = 3, 189/63 = 3. Constant ratio of 3, so the data is exponential. Step 4: The general form is y = a * b^x. Using point (1, 7): 7 = a * b^1. Using point (2, 21): 21 = a * b^2. Divide: 21/7 = (a*b^2)/(a*b) => 3 = b. So b = 3. Step 5: Substitute b = 3 into 7 = a * 3^1 => 7 = 3a => a = 7/3. Step 6: Equation: y = (7/3) * 3^x. Alternatively, y = 7 * 3^(x-1). The answer is exponential, y = 7 * 3^(x-1) or y = (7/3) * 3^x.