Linear vs Exponential

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Answer Key & Explanations

Linear vs Exponential · Grade 9 · Worksheet 1

1. Noah is comparing two different investment accounts for his summer internship earnings. Account A starts with an initial deposit of $6,000 and adds a fixed amount of $360 each year. Account B starts with an initial deposit of $6,000 and grows by 6% each year, compounded annually. Write functions A(t) and B(t) to model the balance in each account after t years. Identify which account represents linear growth and which represents exponential growth, and explain the key difference in how the balances change over time. Answer: Account A is linear: A(t) = 6000 + 360t. Account B is exponential: B(t) = 6000(1.06)^t. In linear growth, a constant amount ($360) is added each year; in exponential growth, the balance multiplies by a constant factor (1.06) each year, causing the growth rate to increase over time. Solution: Analyze Account A. It adds $360 every year. This means the balance increases by the same absolute amount each year.
   Full step-by-step solution

   Step 1: Analyze Account A. It adds $360 every year. This means the balance increases by the same absolute amount each year. The function is A(t) = 6000 + 360t. This is a linear function because it has the form y = mx + b (here m = 360, b = 6000). Step 2: Analyze Account B. It earns 6% interest compounded annually. Each year, the balance is multiplied by 1 + 0.06 = 1.06. The function is B(t) = 6000(1.06)^t. This is an exponential function because it has the form y = a * b^t (here a = 6000, b = 1.06). Step 3: Identify the growth types. Account A is linear. Account B is exponential. Step 4: Explain the difference. In linear growth (Account A), the change per year is constant: +$360 every year. The graph is a straight line. In exponential growth (Account 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. Final answer: Account A is linear: A(t) = 6000 + 360t. Account B is exponential: B(t) = 6000(1.06)^t. In linear growth, a constant amount ($360) is added each year; in exponential growth, the balance multiplies by a constant factor (1.06) each year, causing the growth rate to increase over time.

2. A city planner is analyzing two different investment strategies for a municipal project. Strategy A grows according to the function A(t) = 5000 + 300t, where t is time in years. Strategy B grows according to the function B(t) = 4000 × (1.08)^t. The planner needs to determine which strategy represents exponential growth and which represents linear growth. Which statement correctly identifies the growth types? Answer: B. Strategy A is linear, Strategy B is exponential Solution: Analyze Strategy A: A(t) = 5000 + 300t This function adds a constant amount (300) each time period, which is characteristic of linear growth.
   Full step-by-step solution

   Step 1: Analyze Strategy A: A(t) = 5000 + 300t This function adds a constant amount (300) each time period, which is characteristic of linear growth. Step 2: Analyze Strategy B: B(t) = 4000 × (1.08)^t This function multiplies by a constant factor (1.08) each time period, which is characteristic of exponential growth. Step 3: Verify the patterns: - Linear functions have the form f(x) = mx + b - Exponential functions have the form f(x) = a × b^x Step 4: Strategy A matches the linear form with m = 300 and b = 5000 Strategy B matches the exponential form with a = 4000 and b = 1.08 Step 5: Therefore, Strategy A represents linear growth and Strategy B represents exponential growth. The correct answer is Strategy A is linear, Strategy B is exponential.

3. Aroha is analyzing two functions displayed on a coordinate grid. The first function, f(x), passes through the points (1, 9) and (3, 81). The second function, g(x), passes through the points (1, 9) and (3, 15). For each function, as x increases by 2 from x = 1 to x = 3, determine whether the function is linear or exponential. Justify your answer by describing the pattern of change in y. Answer: f(x) is exponential because y is multiplied by 9 when x increases by 2; g(x) is linear because y increases by a constant 6 when x increases by 2. Solution: Examine f(x) from (1, 9) to (3, 81). As x increases by 2, y goes from 9 to 81. Check the ratio: 81 divided by 9 equals 9.
   Full step-by-step solution

   Step 1: Examine f(x) from (1, 9) to (3, 81). As x increases by 2, y goes from 9 to 81. Check the ratio: 81 divided by 9 equals 9. This means y is multiplied by 9. This constant multiplication shows exponential growth. Step 2: Examine g(x) from (1, 9) to (3, 15). As x increases by 2, y goes from 9 to 15. Check the difference: 15 minus 9 equals 6. This means y increases by a constant 6. This constant addition shows a linear relationship. Step 3: Conclusion. f(x) is exponential because the y-values are multiplied by a constant factor (9) for a constant change in x. g(x) is linear because the y-values increase by a constant amount (6) for a constant change in x. The answer is: f(x) is exponential; g(x) is linear.

4. A right triangle is drawn on a coordinate plane with vertices at (0,0), (6,0), and (6,8). A circle is inscribed inside this triangle, tangent to all three sides. What is the radius of the inscribed circle? Answer: 2 Solution: A = (0,0) B = (6,0) C = (6,8) This is a right triangle with the right angle at B = (6,0) because AB is horizontal and BC is vertical.
   Full step-by-step solution

   Let's go step-by-step. --- **Step 1: Understand the triangle** Vertices: A = (0,0) B = (6,0) C = (6,8) This is a right triangle with the right angle at B = (6,0) because AB is horizontal and BC is vertical. Sides: AB: from (0,0) to (6,0) → length = 6 BC: from (6,0) to (6,8) → length = 8 AC: hypotenuse from (0,0) to (6,8) → length = sqrt((6-0)^2 + (8-0)^2) = sqrt(36 + 64) = sqrt(100) = 10 So sides: 6, 8, 10. --- **Step 2: Inradius formula for a right triangle** For any right triangle with legs a and b and hypotenuse c, the inradius r is given by: r = (a + b - c) / 2 --- **Step 3: Apply formula** Here a = 6, b = 8, c = 10. r = (6 + 8 - 10) / 2 r = (14 - 10) / 2 r = 4 / 2 r = 2 --- **Step 4: Conclusion** The radius of the inscribed circle is 2. --- **Final answer:** 2

5. Is f(x) = 7x + 2 linear or exponential? Is g(x) = 7(2)^x linear or exponential? Answer: f(x) is linear; g(x) is exponential Solution: Examine f(x) = 7x + 2. The variable x is raised to the first power (x^1) and is multiplied by a constant (7). The graph is a straight line with slope 7 and y-intercept 2.
   Full step-by-step solution

   Step 1: Examine f(x) = 7x + 2. The variable x is raised to the first power (x^1) and is multiplied by a constant (7). The graph is a straight line with slope 7 and y-intercept 2. This is a linear function. Step 2: Examine g(x) = 7(2)^x. The variable x is in the exponent. As x increases by 1, the output is multiplied by 2 (the base). The graph curves upward increasingly steeply. This is an exponential function. Step 3: Conclusion: f(x) is linear; g(x) is exponential.

6. 2^(x) = 32, x = ? Answer: 5 Solution: Step 1: Write the equation: 2^x = 32 Step 2: Express 32 as a power of 2: 32 = 2^5 Step 3: Substitute back into the equation: 2^x = 2^5 Step 4: Since the bases are equal, the exponents must be equal: x = 5 Step 5: Verify: 2^5 = 32, which matches the original equation The answer is 5.
   Full step-by-step solution

   Step 1: Write the equation: 2^x = 32 Step 2: Express 32 as a power of 2: 32 = 2^5 Step 3: Substitute back into the equation: 2^x = 2^5 Step 4: Since the bases are equal, the exponents must be equal: x = 5 Step 5: Verify: 2^5 = 32, which matches the original equation The answer is 5.