Function Fitting

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Answer Key & Explanations

Function Fitting · Grade 11 · Worksheet 3

1. 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.

2. 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.

3. Noah is monitoring the growth of a certain species of bamboo over several weeks. He records the height (in centimeters) of a stalk at the end of each week: Week 1: 12 cm, Week 2: 24 cm, Week 3: 48 cm, Week 4: 96 cm, Week 5: 192 cm. Based on this pattern, determine which type of function (linear, quadratic, or exponential) best models the data, and then use that model to estimate the height of the bamboo at the end of Week 7. Answer: 768 cm Solution: Calculate the first differences: Week 2 - Week 1 = 24 - 12 = 12; Week 3 - Week 2 = 48 - 24 = 24; Week 4 - Week 3 = 96 - 48 = 48; Week 5 - Week 4 = 192 - 96 = 96.
   Full step-by-step solution

   Step 1: Calculate the first differences: Week 2 - Week 1 = 24 - 12 = 12; Week 3 - Week 2 = 48 - 24 = 24; Week 4 - Week 3 = 96 - 48 = 48; Week 5 - Week 4 = 192 - 96 = 96. The first differences (12, 24, 48, 96) are not constant, so the data is not linear. Step 2: Calculate the second differences: 24 - 12 = 12; 48 - 24 = 24; 96 - 48 = 48. The second differences (12, 24, 48) are not constant, so the data is not quadratic. Step 3: Calculate the ratios of consecutive heights: 24/12 = 2; 48/24 = 2; 96/48 = 2; 192/96 = 2. The ratio is constant (2), so the data follows an exponential model with a growth factor of 2 per week. Step 4: The exponential function can be written as H(t) = 12 * 2^(t-1), where t is the week number starting from 1. For Week 7, t = 7: H(7) = 12 * 2^(7-1) = 12 * 2^6 = 12 * 64 = 768. The answer is 768 cm.

4. Sophia 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 provide the 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. The ratio is constant (3), so the data is exponential. Step 4: Exponential form is y = a * b^x. The common ratio b = 3. Using point (1,12): 12 = a * 3^1 => a = 4. So the equation is y = 4 * 3^x. The answer is exponential; y = 4 * 3^x.

5. Emma is designing a roller coaster and needs to model the track's shape. The track follows a cubic polynomial path described by f(x) = ax³ + bx² + cx + d. She knows the track passes through the points (-2, -15), (0, 1), (1, 0), and (2, 9). Determine the values of a, b, c, and d that define the track's equation. Answer: a = 2, b = -3, c = -4, d = 1 Solution: To find the equation of a polynomial that passes through given points, substitute each point into the general polynomial form to create a system of equations. For a cubic polynomial f(x) = ax³ + bx² + cx + d, you need four points to determine the four unknown coefficients.
   Full step-by-step solution

   To find the equation of a polynomial that passes through given points, substitute each point into the general polynomial form to create a system of equations. For a cubic polynomial f(x) = ax³ + bx² + cx + d, you need four points to determine the four unknown coefficients. Solving this system reveals the specific polynomial that fits all the given points exactly.

6. Emma is designing a roller coaster track that follows a cubic polynomial path. The track must pass through the points (0, 10), (2, 26), (4, 34), and (6, 50), where x represents the horizontal distance in meters and y represents the height in meters. Determine the equation of the cubic polynomial that models this roller coaster track. Answer: y = -0.5x^3 + 3x^2 - 1.5x + 10 Solution: To find a polynomial that passes through given points, we use the fact that each point must satisfy the polynomial equation.
   Full step-by-step solution

   To find a polynomial that passes through given points, we use the fact that each point must satisfy the polynomial equation. For a cubic polynomial, we have four unknown coefficients, and four points give us four equations. Solving this system of equations allows us to determine the exact polynomial that fits all the given data points.

7. Mere records the following data: (1, 15), (2, 45), (3, 135), (4, 405). Determine which function type (linear, quadratic, or exponential) best fits this data, and write the equation of the function that models it. Answer: Exponential; y = 5 * 3^x Solution: Check for linearity by finding first differences. 45 - 15 = 30, 135 - 45 = 90, 405 - 135 = 270. The first differences (30, 90, 270) are not constant, so it is not linear.
   Full step-by-step solution

   Step 1: Check for linearity by finding first differences. 45 - 15 = 30, 135 - 45 = 90, 405 - 135 = 270. The first differences (30, 90, 270) are not constant, so it is not linear. Step 2: Check for quadratic by finding second differences. 90 - 30 = 60, 270 - 90 = 180. The second differences (60, 180) are not constant, so it is not quadratic. Step 3: Check for exponential by finding ratios of consecutive y-values. 45/15 = 3, 135/45 = 3, 405/135 = 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 constant ratio b = 3. To find a, use the point (1, 15): 15 = a * 3^1, so 15 = 3a, thus a = 5. Step 5: The equation is y = 5 * 3^x. The answer is Exponential; y = 5 * 3^x.