Function Fitting

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Answer Key & Explanations

Function Fitting · Grade 11 · Worksheet 3

1. Ava is studying the cooling of a liquid in a chemistry lab. She records the temperature of the liquid every 10 minutes and obtains the following data: at t = 0 minutes, temperature = 96°C; at t = 10 minutes, temperature = 76°C; at t = 20 minutes, temperature = 61°C; at t = 30 minutes, temperature = 51°C; at t = 40 minutes, temperature = 46°C; at t = 50 minutes, temperature = 43°C. Which type of function (linear, quadratic, or exponential) best models this cooling data, and what is the approximate temperature at t = 60 minutes based on that model? Answer: Exponential; approximately 41°C Solution: Calculate first differences (temperature change each 10 minutes): 96-76 = 20, 76-61 = 15, 61-51 = 10, 51-46 = 5, 46-43 = 3. These differences (20, 15, 10, 5, 3) are not constant, so it is not linear.
   Full step-by-step solution

   Step 1: Calculate first differences (temperature change each 10 minutes): 96-76 = 20, 76-61 = 15, 61-51 = 10, 51-46 = 5, 46-43 = 3. These differences (20, 15, 10, 5, 3) are not constant, so it is not linear. Step 2: Calculate second differences: 15-20 = -5, 10-15 = -5, 5-10 = -5, 3-5 = -2. These are not constant (the last is -2 instead of -5), so it is not exactly quadratic. Step 3: Calculate ratios of consecutive temperatures: 76/96 = 0.7917, 61/76 = 0.8026, 51/61 = 0.8361, 46/51 = 0.9020, 43/46 = 0.9348. The ratios are not exactly constant, but the pattern of decreasing differences suggests an exponential decay approaching room temperature. The data closely fits an exponential model with a horizontal asymptote around 41°C. Step 4: Using the exponential decay pattern, the temperature decreases by a factor of roughly 0.8 each interval initially, then slows as it approaches 41°C. Extrapolating, at t = 60 minutes the temperature is approximately 41°C. The answer is exponential; approximately 41°C.

2. Tane records the following data: (1, 12), (2, 36), (3, 108), (4, 324), (5, 972). Which type of function (linear, quadratic, or exponential) best fits this data? Estimate the value of the function at x = 6 using the identified model. Answer: 2916 Solution: Check for linear pattern. Differences: 36-12=24, 108-36=72, 324-108=216, 972-324=648. Not constant, so not linear.
   Full step-by-step solution

   Step 1: Check for linear pattern. Differences: 36-12=24, 108-36=72, 324-108=216, 972-324=648. Not constant, so not linear. Step 2: Check for quadratic pattern. Second differences: 72-24=48, 216-72=144, 648-216=432. Not constant, so not quadratic. Step 3: Check for exponential pattern. Ratios: 36/12=3, 108/36=3, 324/108=3, 972/324=3. Constant ratio of 3, so exponential. Step 4: The function is of the form y = a * 3^x. Using (1,12): 12 = a * 3^1 = 3a, so a = 4. Thus y = 4 * 3^x. Step 5: Estimate at x = 6: y = 4 * 3^6 = 4 * 729 = 2916. The answer is 2916.

3. Emma records the following data: (1, 7), (2, 21), (3, 63), (4, 189), (5, 567). Determine which function type (linear, quadratic, or exponential) best fits the data, and estimate the value when x = 6. Answer: 1701 Solution: Check for linear: differences between y-values: 21-7=14, 63-21=42, 189-63=126, 567-189=378. Not constant, so not linear. Check for quadratic: second differences: 42-14=28, 126-42=84, 378-126=252.
   Full step-by-step solution

   Step 1: Check for linear: differences between y-values: 21-7=14, 63-21=42, 189-63=126, 567-189=378. Not constant, so not linear. Step 2: Check for quadratic: second differences: 42-14=28, 126-42=84, 378-126=252. Not constant, so not quadratic. Step 3: Check for exponential: ratios of consecutive y-values: 21/7=3, 63/21=3, 189/63=3, 567/189=3. Constant ratio of 3, so exponential. Step 4: The function is y = 7 * 3^(x-1). For x=6: y = 7 * 3^(5) = 7 * 243 = 1701. The answer is 1701.

4. Ava records the following data: (1, 6), (2, 11), (3, 16), (4, 21), (5, 26). Determine which function type (linear, quadratic, or exponential) best fits this data, and write the equation of the best-fitting 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 (5). Since first differences are constant, the data is linear.
   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 (5). Step 2: Since first differences are constant, the data is linear. Step 3: The slope m = 5. Use point (1, 6): y = mx + b => 6 = 5(1) + b => b = 1. Step 4: Equation: y = 5x + 1. The answer is linear; y = 5x + 1.

5. Noah is a marine biologist studying the population of a certain species of fish in a lake. He collects data over several years and records the estimated population (in hundreds) at the beginning of each year. The data is as follows: Year 0: 11 fish (hundreds), Year 1: 16 fish (hundreds), Year 2: 21 fish (hundreds), Year 3: 26 fish (hundreds), Year 4: 31 fish (hundreds). Determine which type of function (linear, quadratic, or exponential) best models the population growth, and write the equation of that function in the form P(t) = mt + b, P(t) = at^2 + bt + c, or P(t) = a * b^t, where t is the time in years. Answer: Linear, P(t) = 5t + 11 Solution: Calculate the first differences: 16 - 11 = 5, 21 - 16 = 5, 26 - 21 = 5, 31 - 26 = 5. The first differences are all equal to 5, so the data is linear. Since the function is linear, use the form P(t) = mt + b.
   Full step-by-step solution

   Step 1: Calculate the first differences: 16 - 11 = 5, 21 - 16 = 5, 26 - 21 = 5, 31 - 26 = 5. The first differences are all equal to 5, so the data is linear. Step 2: Since the function is linear, use the form P(t) = mt + b. The slope m is the constant first difference, so m = 5. Step 3: Use the point (0, 11) to find b. When t = 0, P = 11, so 11 = 5 * 0 + b, thus b = 11. Step 4: The equation is P(t) = 5t + 11. The answer is Linear, P(t) = 5t + 11.

6. Ava records the following data: (1, 6), (2, 11), (3, 16), (4, 21), (5, 26). Which type of function (linear, quadratic, or exponential) best fits this data? Estimate the value of the function at x = 6. Answer: Linear; 31 Solution: Calculate the first differences: 11 - 6 = 5, 16 - 11 = 5, 21 - 16 = 5, 26 - 21 = 5. The first differences are constant (5), so the data is linear. The linear function has slope m = 5.
   Full step-by-step solution

   Step 1: Calculate the first differences: 11 - 6 = 5, 16 - 11 = 5, 21 - 16 = 5, 26 - 21 = 5. The first differences are constant (5), so the data is linear. Step 2: The linear function has slope m = 5. Using point (1, 6), the equation is y = 5x + 1 (since 6 = 5*1 + 1). Step 3: Estimate at x = 6: y = 5*6 + 1 = 30 + 1 = 31. The answer is linear and 31.

7. Mere records the following data: (1, 12), (2, 36), (3, 108), (4, 324). Which type of function (linear, quadratic, or exponential) best models this data? Estimate the value when x = 5. Answer: exponential; 972 Solution: Check first differences: 36 - 12 = 24, 108 - 36 = 72, 324 - 108 = 216. Not constant, so not linear. Check second differences: 72 - 24 = 48, 216 - 72 = 144.
   Full step-by-step solution

   Step 1: Check first differences: 36 - 12 = 24, 108 - 36 = 72, 324 - 108 = 216. Not constant, so not linear. Step 2: Check second differences: 72 - 24 = 48, 216 - 72 = 144. Not constant, so not quadratic. Step 3: Check ratios: 36/12 = 3, 108/36 = 3, 324/108 = 3. Constant ratio of 3, so exponential. Step 4: The function is y = 12 * 3^(x-1). For x = 5: y = 12 * 3^4 = 12 * 81 = 972. The answer is exponential; 972.