Linear vs Exponential

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Answer Key & Explanations

Linear vs Exponential · Grade 9 · Worksheet 2

1. 2^(x+3) = 32 Answer: 2 Solution: Write 32 as a power of 2: 32 = 2^5 Substitute into the equation: 2^(x+3) = 2^5 Since the bases are equal, set the exponents equal: x + 3 = 5 Solve for x: x = 5 - 3 x = 2 The answer is 2.
   Full step-by-step solution

   Step 1: Write 32 as a power of 2: 32 = 2^5 Step 2: Substitute into the equation: 2^(x+3) = 2^5 Step 3: Since the bases are equal, set the exponents equal: x + 3 = 5 Step 4: Solve for x: x = 5 - 3 Step 5: x = 2 The answer is 2.

2. Emma plots two functions on the same coordinate grid. Function A passes through the points (0, 10) and (1, 15). Function B passes through the points (0, 10) and (1, 20). For x = 0 to x = 4, which function grows faster and why? Identify whether each function is linear or exponential, and explain how you can tell from the graph. Answer: Function A is linear; Function B is exponential. Function B grows faster because exponential growth multiplies by a constant factor, while linear growth adds a constant amount. Solution: Analyze Function A. Points: (0,10) and (1,15). As x increases by 1, y increases by 15 - 10 = 5.
   Full step-by-step solution

   Step 1: Analyze Function A. Points: (0,10) and (1,15). As x increases by 1, y increases by 15 - 10 = 5. This constant additive change (+5 each step) means Function A is linear. For x=0 to 4, y values: 10, 15, 20, 25, 30. Step 2: Analyze Function B. Points: (0,10) and (1,20). As x increases by 1, y multiplies by 20/10 = 2. This constant multiplicative change (×2 each step) means Function B is exponential. For x=0 to 4, y values: 10, 20, 40, 80, 160. Step 3: Compare growth. At x=4, Function A has y=30, Function B has y=160. Function B grows much faster because exponential growth (multiplying) outpaces linear growth (adding) over multiple steps. The graph of Function A is a straight line; the graph of Function B curves upward steeply.

3. Emma is comparing two different savings accounts for her summer job earnings. Account A earns simple interest, adding $75 each year to her initial deposit of $1,125. Account B earns compound interest at a rate of 5% per year, applied to the total balance each year, starting from the same initial deposit of $1,125. Write functions A(t) and B(t) to model the balance in each account after t years. Determine which account is linear and which is exponential, and calculate the balance in each account after 7 years to the nearest dollar. Which account has a higher balance after 7 years, and by how much? Answer: Account A is linear; Account B is exponential. After 7 years, Account A has $1,650, Account B has approximately $1,583. Account A has $67 more. Solution: Account A adds $75 each year. This is constant addition, so it is linear. The function is A(t) = 1125 + 75t, where t is years.
   Full step-by-step solution

   Step 1: Account A adds $75 each year. This is constant addition, so it is linear. The function is A(t) = 1125 + 75t, where t is years. Step 2: Account B multiplies by 1.05 each year (100% + 5% = 105% = 1.05). This is constant multiplication, so it is exponential. The function is B(t) = 1125 * (1.05)^t. Step 3: Calculate A(7): A(7) = 1125 + 75 * 7 = 1125 + 525 = 1650 dollars. Step 4: Calculate B(7): B(7) = 1125 * (1.05)^7. Compute step by step: (1.05)^1 = 1.05, (1.05)^2 = 1.1025, (1.05)^3 = 1.157625, (1.05)^4 = 1.21550625, (1.05)^5 = 1.2762815625, (1.05)^6 = 1.340095640625, (1.05)^7 = 1.40710042265625. Multiply by 1125: 1125 * 1.40710042265625 = 1582.98797548828125, which rounds to $1583. Step 5: Compare: A(7) = $1650, B(7) ≈ $1583. Account A has more by 1650 - 1583 = $67. The answer: Account A is linear, Account B is exponential. After 7 years, Account A has $1,650, Account B has approximately $1,583. Account A has $67 more.

4. Ava graphs two functions on the same coordinate plane. The first function passes through the points (0, 9) and (1, 15). The second function passes through (0, 9) and (1, 27). Both functions continue the same pattern. One is linear and the other is exponential. Based on the given points, which function is exponential and which is linear? Explain your reasoning using the pattern of change between the points. Answer: The first function (0,9) to (1,15) is linear; the second function (0,9) to (1,27) is exponential. Solution: Examine the first function's points: (0,9) and (1,15). As x increases by 1, y goes from 9 to 15. The change is 15 - 9 = 6.
   Full step-by-step solution

   Step 1: Examine the first function's points: (0,9) and (1,15). As x increases by 1, y goes from 9 to 15. The change is 15 - 9 = 6. This is a constant additive increase of 6. Therefore, the first function is linear (y = 6x + 9). Step 2: Examine the second function's points: (0,9) and (1,27). As x increases by 1, y goes from 9 to 27. The ratio is 27 / 9 = 3. This is a multiplicative factor of 3, not a constant addition. Therefore, the second function is exponential (y = 9 * 3^x). Step 3: Conclusion: The first function is linear because the y-values increase by a constant difference of 6. The second function is exponential because the y-values increase by a constant factor of 3. The answer is: The first function (0,9) to (1,15) is linear; the second function (0,9) to (1,27) is exponential.

5. Emma compares two functions: f(x) = 5x + 10 and g(x) = 5(2)^x. For x = 0, 1, 2, 3, 4, compute f(x) and g(x). Which function is linear and which is exponential? Explain the difference in how the y-values change as x increases by 1. Answer: f(x) is linear; g(x) is exponential. For f(x), y increases by a constant 5 each step. For g(x), y multiplies by 2 each step. Solution: Compute f(x) = 5x + 10 for x = 0, 1, 2, 3, 4. f(0) = 5(0) + 10 = 10 f(1) = 5(1) + 10 = 15 f(2) = 5(2) + 10 = 20 f(3) = 5(3) + 10 = 25 f(4) = 5(4) + 10 = 30 Compute g(x) = 5(2)^x for x = 0, 1, 2, 3, 4.
   Full step-by-step solution

   Step 1: Compute f(x) = 5x + 10 for x = 0, 1, 2, 3, 4. f(0) = 5(0) + 10 = 10 f(1) = 5(1) + 10 = 15 f(2) = 5(2) + 10 = 20 f(3) = 5(3) + 10 = 25 f(4) = 5(4) + 10 = 30 Step 2: Compute g(x) = 5(2)^x for x = 0, 1, 2, 3, 4. g(0) = 5(2)^0 = 5(1) = 10 g(1) = 5(2)^1 = 5(2) = 10 g(2) = 5(2)^2 = 5(4) = 20 g(3) = 5(2)^3 = 5(8) = 40 g(4) = 5(2)^4 = 5(16) = 80 Step 3: Analyze the changes. For f(x): Differences: 15-10=5, 20-15=5, 25-20=5, 30-25=5. Constant difference of 5 → linear. For g(x): Ratios: 10/10=1, 20/10=2, 40/20=2, 80/40=2. Constant ratio of 2 (except first step from 10 to 10) → exponential. Conclusion: f(x) is linear because y increases by a constant amount each step. g(x) is exponential because y multiplies by a constant factor each step.

6. Matiu is comparing two different savings plans for his summer job earnings. Plan A offers a fixed weekly allowance of $120, plus an additional $40 each week he works. Plan B offers a starting bonus of $100, and then the amount he receives doubles each week he works. Write functions A(t) and B(t) to model the total amount Matiu has earned after t weeks under each plan. Determine which plan represents linear growth and which represents exponential growth, and state how you can tell from the functions. Answer: Plan A is linear (A(t) = 120 + 40t); Plan B is exponential (B(t) = 100 × 2^t). Linear growth adds a constant amount each week, while exponential growth multiplies by a constant factor each week. Solution: Analyze Plan A. Matiu starts with $120 and earns $40 more each week. After 1 week: 120 + 40 = 160.
   Full step-by-step solution

   Step 1: Analyze Plan A. Matiu starts with $120 and earns $40 more each week. After 1 week: 120 + 40 = 160. After 2 weeks: 120 + 40 + 40 = 200. The pattern is adding 40 each week. This is a linear function: A(t) = 120 + 40t, where t is the number of weeks. Linear functions have a constant rate of change (slope). Step 2: Analyze Plan B. Matiu starts with $100 and the amount doubles each week. After 1 week: 100 × 2 = 200. After 2 weeks: 100 × 2 × 2 = 400. The pattern is multiplying by 2 each week. This is an exponential function: B(t) = 100 × 2^t. Exponential functions have a constant ratio between consecutive terms. Step 3: Distinguish the growth types. Plan A adds a constant amount ($40) each week, which is characteristic of linear growth. Plan B multiplies by a constant factor (2) each week, which is characteristic of exponential growth. Step 4: Conclusion. Plan A is linear (A(t) = 120 + 40t); Plan B is exponential (B(t) = 100 × 2^t).