Function Fitting

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Answer Key & Explanations

Function Fitting · Grade 11 · Worksheet 2

1. Emma is studying the cooling of a cup of coffee in her physics class. She measures the temperature of the coffee every 10 minutes and records the following data: at t = 0 min, temperature = 95°C; at t = 10 min, temperature = 71°C; at t = 20 min, temperature = 55°C; at t = 30 min, temperature = 44°C. The room temperature is constant at 21°C. Determine which type of function (linear, quadratic, or exponential) best models the temperature difference (coffee temperature minus room temperature) over time, and write the specific function that fits the data for the temperature difference as a function of time in minutes. Answer: Exponential; T_diff(t) = 74 * (0.67)^(t/10) Solution: Calculate the temperature difference (coffee temp minus room temp) for each time: At t=0: 95-21=74; at t=10: 71-21=50; at t=20: 55-21=34; at t=30: 44-21=23.
   Full step-by-step solution

   Step 1: Calculate the temperature difference (coffee temp minus room temp) for each time: At t=0: 95-21=74; at t=10: 71-21=50; at t=20: 55-21=34; at t=30: 44-21=23. Step 2: Check if the model is linear: First differences: 50-74=-24, 34-50=-16, 23-34=-11. These are not constant, so not linear. Step 3: Check if the model is quadratic: Second differences: (-16)-(-24)=8, (-11)-(-16)=5. These are not constant, so not quadratic. Step 4: Check if the model is exponential: Ratios of successive temperature differences: 50/74 ≈ 0.6757, 34/50 = 0.68, 23/34 ≈ 0.6765. These are approximately constant (about 0.68), so exponential model fits. Step 5: Write the exponential function in the form T_diff(t) = A * r^(t/10). At t=0, T_diff=74, so A=74. The common ratio over each 10-minute interval is approximately 0.68. Using the data from t=0 to t=10: 74 * r = 50, so r = 50/74 ≈ 0.6757. Using t=10 to t=20: 50 * r = 34, so r = 34/50 = 0.68. Taking average: r ≈ 0.676. So the function is T_diff(t) = 74 * (0.676)^(t/10). For simplicity, using r = 0.67 (rounded) gives T_diff(t) = 74 * (0.67)^(t/10). The answer is: Exponential; T_diff(t) = 74 * (0.67)^(t/10).

2. Noah 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 an approximate equation for the function. Answer: Exponential; y = 4 * 3^x Solution: Calculate first differences: 36-12=24, 108-36=72, 324-108=216, 972-324=648. These are not constant, so not linear. Calculate second differences: 72-24=48, 216-72=144, 648-216=432.
   Full step-by-step solution

   Step 1: Calculate first differences: 36-12=24, 108-36=72, 324-108=216, 972-324=648. These are not constant, so not linear. Step 2: Calculate second differences: 72-24=48, 216-72=144, 648-216=432. These are not constant, so not quadratic. Step 3: Calculate ratios of consecutive y-values: 36/12=3, 108/36=3, 324/108=3, 972/324=3. The ratio is constant at 3, so the data is exponential. Step 4: An exponential function has the form y = a * b^x. The common ratio b = 3. When x=1, y=12, so 12 = a * 3^1 = 3a, giving a = 4. Step 5: The equation is y = 4 * 3^x. The answer is exponential; y = 4 * 3^x.

3. Tane records the following data: (1, 5), (2, 15), (3, 45), (4, 135), (5, 405). Which type of function (linear, quadratic, or exponential) best models this data? Sketch a curve through the points and estimate the value when x = 6. Answer: exponential; y ≈ 1215 when x = 6 Solution: Examine the y-values: 5, 15, 45, 135, 405. Calculate first differences: 15-5=10, 45-15=30, 135-45=90, 405-135=270. Not constant, so not linear.
   Full step-by-step solution

   Step 1: Examine the y-values: 5, 15, 45, 135, 405. Step 2: Calculate first differences: 15-5=10, 45-15=30, 135-45=90, 405-135=270. Not constant, so not linear. Step 3: Calculate second differences: 30-10=20, 90-30=60, 270-90=180. Not constant, so not quadratic. Step 4: Calculate ratios: 15/5=3, 45/15=3, 135/45=3, 405/135=3. The ratio is constant (3), so the data is exponential. Step 5: The function is of the form y = a * 3^x. Using (1,5): 5 = a * 3^1 => a = 5/3. So y = (5/3) * 3^x. Step 6: Estimate for x=6: y = (5/3) * 3^6 = (5/3) * 729 = 5 * 243 = 1215. The answer is exponential; y ≈ 1215 when x = 6.

4. Aroha records the following data: (1, 5), (2, 11), (3, 19), (4, 29), (5, 41). Determine which type of function (linear, quadratic, or exponential) best fits this data and write the equation of the function that models it. Answer: Quadratic; y = x^2 + 3x + 1 Solution: Calculate first differences: 11-5=6, 19-11=8, 29-19=10, 41-29=12. These are not constant, so not linear. Calculate second differences: 8-6=2, 10-8=2, 12-10=2.
   Full step-by-step solution

   Step 1: Calculate first differences: 11-5=6, 19-11=8, 29-19=10, 41-29=12. These are not constant, so not linear. Step 2: Calculate second differences: 8-6=2, 10-8=2, 12-10=2. These are constant (2), so the data is quadratic. Step 3: Assume the quadratic is y = ax^2 + bx + c. Use the first three points: (1,5): a + b + c = 5; (2,11): 4a + 2b + c = 11; (3,19): 9a + 3b + c = 19. Step 4: Subtract the first equation from the second: (4a+2b+c) - (a+b+c) = 11-5 → 3a + b = 6. Step 5: Subtract the second from the third: (9a+3b+c) - (4a+2b+c) = 19-11 → 5a + b = 8. Step 6: Subtract the result of step 4 from step 5: (5a+b) - (3a+b) = 8-6 → 2a = 2 → a = 1. Step 7: Substitute a=1 into 3a+b=6: 3(1)+b=6 → b=3. Step 8: Substitute a=1, b=3 into a+b+c=5: 1+3+c=5 → c=1. Step 9: The quadratic function is y = x^2 + 3x + 1. Step 10: Check with (4,29): 4^2 + 3(4) + 1 = 16+12+1=29. Correct. Check with (5,41): 25+15+1=41. Correct. The answer is: Quadratic; y = x^2 + 3x + 1.

5. Noah is studying the growth of a certain type of algae in a pond. He measures the surface area covered by the algae (in square meters) over several days. His data is: Day 0: 5 m², Day 1: 15 m², Day 2: 45 m², Day 3: 135 m², Day 4: 405 m². Which type of function best models this data: linear, quadratic, or exponential? Explain your reasoning and write the equation that models the data. Answer: Exponential; y = 5 * 3^x Solution: Check for linear: First differences: 15-5=10, 45-15=30, 135-45=90, 405-135=270. These are not constant, so not linear. Check for quadratic: Second differences: 30-10=20, 90-30=60, 270-90=180.
   Full step-by-step solution

   Step 1: Check for linear: First differences: 15-5=10, 45-15=30, 135-45=90, 405-135=270. These are not constant, so not linear. Step 2: Check for quadratic: Second differences: 30-10=20, 90-30=60, 270-90=180. These are not constant, so not quadratic. Step 3: Check for exponential: Ratios: 15/5=3, 45/15=3, 135/45=3, 405/135=3. The ratio is constant at 3. Step 4: Write the exponential model: y = a * b^x. Initial value a = 5 (when x=0). Growth factor b = 3. So y = 5 * 3^x. The answer is: Exponential; y = 5 * 3^x.

6. Tane collects 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 equation of the best-fit function. Answer: Exponential; y = 4(3)^x Solution: Check for linear: first differences: 36-12=24, 108-36=72, 324-108=216, 972-324=648. Not constant, so not linear. Check for quadratic: second differences: 72-24=48, 216-72=144, 648-216=432.
   Full step-by-step solution

   Step 1: Check for linear: first differences: 36-12=24, 108-36=72, 324-108=216, 972-324=648. Not constant, so not linear. Step 2: Check for quadratic: second differences: 72-24=48, 216-72=144, 648-216=432. Not constant, so not quadratic. Step 3: Check for exponential: ratios: 36/12=3, 108/36=3, 324/108=3, 972/324=3. Constant ratio of 3, so exponential. Step 4: General form: y = a(b)^x. Using (1,12): 12 = a(b)^1 = ab. Using (2,36): 36 = a(b)^2 = ab^2. Divide: 36/12 = (ab^2)/(ab) = b, so b = 3. Then 12 = a(3) gives a = 4. Step 5: Equation: y = 4(3)^x. The answer is exponential; y = 4(3)^x.

7. Sophia records the following data: (1, 8), (2, 14), (3, 20), (4, 26), (5, 32). Determine which function type (linear, quadratic, or exponential) best fits the data and provide the equation of the best-fit function. Answer: Linear; y = 6x + 2 Solution: Calculate first differences: 14-8=6, 20-14=6, 26-20=6, 32-26=6. The first differences are constant (all equal to 6). This indicates a linear relationship.
   Full step-by-step solution

   Step 1: Calculate first differences: 14-8=6, 20-14=6, 26-20=6, 32-26=6. The first differences are constant (all equal to 6). This indicates a linear relationship. Step 2: Use the slope-intercept form y = mx + b. The slope m = 6 (the constant difference). Step 3: Substitute a point, say (1, 8): 8 = 6(1) + b => b = 2. Step 4: The equation is y = 6x + 2. The answer is linear; y = 6x + 2.