Linear vs Exponential

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Answer Key & Explanations

Linear vs Exponential · Grade 9 · Worksheet 3

1. Aroha compares two functions: f(x) = 9x + 3 and g(x) = 9(3)^x. For x = 0, 1, 2, 3, which function grows faster and why? Answer: g(x) grows faster because it is exponential; f(x) is linear. Solution: Evaluate f(x) = 9x + 3 for x = 0, 1, 2, 3: f(0) = 3, f(1) = 12, f(2) = 21, f(3) = 30. The differences are +9 each time (constant rate of change).
   Full step-by-step solution

   Step 1: Evaluate f(x) = 9x + 3 for x = 0, 1, 2, 3: f(0) = 3, f(1) = 12, f(2) = 21, f(3) = 30. The differences are +9 each time (constant rate of change). Step 2: Evaluate g(x) = 9(3)^x for x = 0, 1, 2, 3: g(0) = 9, g(1) = 27, g(2) = 81, g(3) = 243. The ratios are ×3 each time (constant multiplicative factor). Step 3: Compare growth: from x=0 to x=3, f(x) increases by 27, while g(x) increases by 234. Exponential growth outpaces linear growth because the multiplicative factor causes increasingly larger jumps. The answer is g(x) grows faster because it is exponential.

2. Sophia is comparing two different investment accounts for her college fund. Account A starts with $6,000 and adds $360 each year. Account B starts with $6,000 and grows by 6% each year, meaning the balance is multiplied by 1.06 each year. Write functions A(t) and B(t) to model the balance in each account after t years. Determine which account represents linear growth and which represents exponential growth, and explain the difference in their growth patterns. Then calculate the balance in each account after 6 years, and state which account has a higher balance and by how much. Answer: Account B has $1,616.71 more than Account A after 6 years. Account A is linear; Account B is exponential. Solution: Write the function for Account A. It adds $360 each year starting from $6,000. This is linear: A(t) = 6000 + 360t.
   Full step-by-step solution

   Step 1: Write the function for Account A. It adds $360 each year starting from $6,000. This is linear: A(t) = 6000 + 360t. Step 2: Write the function for Account B. It multiplies by 1.06 each year starting from $6,000. This is exponential: B(t) = 6000 * (1.06)^t. Step 3: Identify growth types. Account A is linear because it has constant rate of change (+360 per year). Account B is exponential because it has constant multiplicative factor (1.06 per year). Step 4: Calculate Account A after 6 years: A(6) = 6000 + 360*6 = 6000 + 2160 = 8160 dollars. Step 5: Calculate Account B after 6 years: B(6) = 6000 * (1.06)^6. Compute (1.06)^6 = 1.06 * 1.06 * 1.06 * 1.06 * 1.06 * 1.06. Stepwise: 1.06^2 = 1.1236, 1.06^3 = 1.191016, 1.06^4 = 1.26247696, 1.06^5 = 1.3382255776, 1.06^6 = 1.418519112256. So B(6) = 6000 * 1.418519112256 = 8511.114673536 dollars. Round to nearest cent: $8,511.11. Step 6: Find the difference. Account B has more. Difference = 8511.11 - 8160 = 351.11 dollars. Step 7: Explain the difference. In linear growth (Account A), the same absolute amount ($360) is added each year, so the graph is a straight line. In exponential growth (Account B), the balance grows by a percentage each year, so the amount added increases each year (e.g., first year adds $360, second year adds about $381.60, etc.), causing the graph to curve upward and grow faster over time. Final answer: Account B has $351.11 more than Account A after 6 years. Account A is linear; Account B is exponential.

3. A research scientist is studying bacterial growth in a lab culture. The initial population is 500 bacteria, and the population doubles every 3 hours. Liam needs to determine the bacterial population after 9 hours. Write an exponential function to model this growth and calculate the population after 9 hours. Answer: 4000 Solution: We start with an initial population of 500 bacteria. The population doubles every 3 hours. We want the population after 9 hours.
   Full step-by-step solution

   Step 1: Understand the problem We start with an initial population of 500 bacteria. The population doubles every 3 hours. We want the population after 9 hours. Step 2: Write the exponential growth model General exponential growth formula: P(t) = P0 * (2)^(t / d) where: P0 = initial population = 500 d = doubling time = 3 hours t = time in hours P(t) = population at time t So the function is: P(t) = 500 * 2^(t / 3) Step 3: Substitute t = 9 hours into the function P(9) = 500 * 2^(9 / 3) P(9) = 500 * 2^3 Step 4: Simplify the exponent 2^3 = 8 So P(9) = 500 * 8 Step 5: Multiply 500 * 8 = 4000 Step 6: Conclusion The bacterial population after 9 hours is 4000.

4. A research scientist is studying bacterial growth in a lab culture. The initial population is 500 bacteria, and the population doubles every 3 hours. Write an exponential function in the form P(t) = a·b^t that models the bacterial population after t hours. Answer: P(t) = 500·2^(t/3) Solution: Identify the general form of the exponential function P(t) = a · b^t - t = time in hours - P(t) = population at time t - a = initial population - b = base of the exponential The problem says: initial population is 500 bacteria.
   Full step-by-step solution

   Let's go step-by-step. --- **Step 1: Identify the general form of the exponential function** We are told the function is in the form: P(t) = a · b^t where: - t = time in hours - P(t) = population at time t - a = initial population - b = base of the exponential --- **Step 2: Determine the initial population** The problem says: initial population is 500 bacteria. That means when t = 0, P(0) = 500. So: P(0) = a · b^0 = a · 1 = a Therefore a = 500. So far: P(t) = 500 · b^t --- **Step 3: Use the doubling time to find b** The population doubles every 3 hours. That means: P(3) = 2 × P(0) = 2 × 500 = 1000. Using our formula: P(3) = 500 · b^3 = 1000 Divide both sides by 500: b^3 = 2 So b = 2^(1/3) --- **Step 4: Write the function with b** P(t) = 500 · [2^(1/3)]^t Using exponent rules: [2^(1/3)]^t = 2^(t/3) So: P(t) = 500 · 2^(t/3) --- **Step 5: Check the doubling behavior** If t increases by 3, then: P(t+3) = 500 · 2^((t+3)/3) = 500 · 2^(t/3 + 1) = 500 · 2^(t/3) · 2 = 2 · P(t) Yes, it doubles every 3 hours. --- **Final Answer:** P(t) = 500 · 2^(t/3)

5. Emma is analyzing two different savings plans offered by her bank. Plan A offers a fixed annual increase of $150 to the account balance. Plan B offers a 5% annual interest rate, compounded yearly. If she deposits $1,500 into both plans initially, write functions A(t) and B(t) to represent the balance in each plan after t years. Identify which plan represents linear growth and which represents exponential growth, and explain the key difference in how the balances change over time. Answer: Plan A is linear (A(t) = 1500 + 150t). Plan B is exponential (B(t) = 1500(1.05)^t). In linear growth, a constant amount is added each year; in exponential growth, the balance multiplies by a constant factor each year. Solution: Analyze Plan A. It adds $150 every year. This means the balance increases by the same absolute amount each year.
   Full step-by-step solution

   Step 1: Analyze Plan A. It adds $150 every year. This means the balance increases by the same absolute amount each year. The function is A(t) = 1500 + 150t. This is a linear function because it has the form y = mx + b (here m = 150, b = 1500). Step 2: Analyze Plan B. It earns 5% interest compounded yearly. This means each year the balance is multiplied by 1 + 0.05 = 1.05. The function is B(t) = 1500(1.05)^t. This is an exponential function because it has the form y = a * b^t (here a = 1500, b = 1.05). Step 3: Identify the growth types. Plan A is linear. Plan B is exponential. Step 4: Explain the difference. In linear growth (Plan A), the change per year is constant: +$150 every year. The graph is a straight line. In exponential growth (Plan B), the change per year increases over time because you earn interest on previously earned interest. The graph curves upward, growing faster as t increases. The answer: Plan A is linear (A(t) = 1500 + 150t). Plan B is exponential (B(t) = 1500(1.05)^t). In linear growth, a constant amount is added each year; in exponential growth, the balance multiplies by a constant factor each year.

6. Kaia is comparing two functions graphed on a coordinate plane. The first function passes through the points (1, 3) and (3, 11). The second function passes through (1, 3) and (3, 27). Determine which function is linear and which is exponential. Explain your reasoning based on the patterns of change. Answer: The first function (points (1,3) and (3,11)) is linear. The second function (points (1,3) and (3,27)) is exponential. Solution: Analyze the first function with points (1,3) and (3,11). As x increases by 2 (from 1 to 3), the y-value increases from 3 to 11, a difference of 8. The rate of change is 8/2 = 4 per unit increase in x.
   Full step-by-step solution

   Step 1: Analyze the first function with points (1,3) and (3,11). As x increases by 2 (from 1 to 3), the y-value increases from 3 to 11, a difference of 8. The rate of change is 8/2 = 4 per unit increase in x. This constant rate of change indicates a linear function. Step 2: Analyze the second function with points (1,3) and (3,27). As x increases by 2 (from 1 to 3), the y-value increases from 3 to 27. The ratio is 27/3 = 9. Since the x-change is 2, the constant multiplier per unit x is sqrt(9) = 3. This means each time x increases by 1, y multiplies by 3, indicating an exponential function. Step 3: Conclusion: The first function is linear because it has a constant additive rate of change. The second function is exponential because it has a constant multiplicative rate of change. The answer is: The first function is linear. The second function is exponential.