Exponential Form

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Answer Key & Explanations

Exponential Form · Grade 9 · Worksheet 2

1. 2^(3x+1) = 128 = ? Answer: 2 Solution: Write both sides with base 2: 2^(3x+1) = 2^7 Since the bases are equal, set the exponents equal: 3x + 1 = 7 Subtract 1 from both sides: 3x = 6 Divide both sides by 3: x = 2 The answer is 2.
   Full step-by-step solution

   Step 1: Write both sides with base 2: 2^(3x+1) = 2^7 Step 2: Since the bases are equal, set the exponents equal: 3x + 1 = 7 Step 3: Subtract 1 from both sides: 3x = 6 Step 4: Divide both sides by 3: x = 2 The answer is 2.

2. A population starts at 200 and doubles every 4 years. Write the exponential function in the form f(t) = a·b^t where t is measured in years. Answer: 200·2^(t/4) Solution: Identify the initial value a. The population starts at 200, so a = 200. Determine the growth factor b.
   Full step-by-step solution

   Step 1: Identify the initial value a. The population starts at 200, so a = 200. Step 2: Determine the growth factor b. The population doubles every 4 years, so the base should be 2. Step 3: Adjust the exponent for the time period. Since doubling occurs every 4 years, the exponent should be t/4. Step 4: Write the function: f(t) = 200·2^(t/4). The exponential function is f(t) = 200·2^(t/4).

3. A city's population is currently 80,000 people and is growing at a rate of 5% per year. Write an exponential function in the form P(t) = ab^t that models the population after t years. Answer: P(t) = 80000 * 1.05^t Solution: Identify the initial population value, which is 80,000. This becomes the coefficient 'a' in our function. Determine the growth factor 'b'.
   Full step-by-step solution

   Step 1: Identify the initial population value, which is 80,000. This becomes the coefficient 'a' in our function. Step 2: Determine the growth factor 'b'. Since the population grows by 5% each year, we convert 5% to decimal form: 5% = 0.05. Step 3: The growth factor is 1 plus the growth rate: 1 + 0.05 = 1.05. Step 4: Write the function using the initial population and growth factor: P(t) = 80000 * 1.05^t. The complete exponential function is P(t) = 80000 * 1.05^t.

4. A bacterial culture starts with 500 bacteria and doubles every 3 hours. The population can be modeled by the function P(t) = 500 × 2^(t/3), where t is time in hours. How many bacteria will there be after 9 hours? Answer: 4000 Solution: Initial bacteria count = 500 Doubling time = 3 hours Population model: P(t) = 500 × 2^(t/3) t = time in hours Identify the time given in the problem. t = 9 hours Substitute t = 9 into the formula.
   Full step-by-step solution

   Let's solve this step by step. We are given: Initial bacteria count = 500 Doubling time = 3 hours Population model: P(t) = 500 × 2^(t/3) t = time in hours Step 1: Identify the time given in the problem. t = 9 hours Step 2: Substitute t = 9 into the formula. P(9) = 500 × 2^(9/3) Step 3: Simplify the exponent. 9/3 = 3 So P(9) = 500 × 2^3 Step 4: Calculate 2^3. 2^3 = 2 × 2 × 2 = 8 Step 5: Multiply 500 by 8. 500 × 8 = 4000 Step 6: Interpret the result. After 9 hours, the bacteria population will be 4000. This makes sense because every 3 hours the population doubles: At t = 0: 500 At t = 3: 1000 At t = 6: 2000 At t = 9: 4000 Final answer: 4000

5. A right triangle is drawn on a coordinate plane with vertices at (0,0), (6,0), and (6,8). The triangle is then reflected across the line y = x. What is the area of the new triangle formed after reflection? Answer: 24 Solution: Identify the original triangle's vertices: A(0,0), B(6,0), C(6,8) Apply reflection across y = x by swapping x and y coordinates: A'(0,0), B'(0,6), C'(8,6) Calculate the area of the original triangle: base = 6, height = 8, area = (1/2) × 6 × 8 = 24 Reflection preserves area, so the new triangle…
   Full step-by-step solution

   Step 1: Identify the original triangle's vertices: A(0,0), B(6,0), C(6,8) Step 2: Apply reflection across y = x by swapping x and y coordinates: A'(0,0), B'(0,6), C'(8,6) Step 3: Calculate the area of the original triangle: base = 6, height = 8, area = (1/2) × 6 × 8 = 24 Step 4: Reflection preserves area, so the new triangle also has area = 24 The answer is 24.

6. A population starts at 1200 and decreases by 15% each year. Write the exponential function in the form f(t) = a·b^t. Answer: 1200·0.85^t Solution: The initial population is 1200, so a = 1200. A 15% decrease each year means the population retains 85% of its value from the previous year (100% - 15% = 85%). 85% as a decimal is 0.85, so b = 0.85.
   Full step-by-step solution

   Step 1: The initial population is 1200, so a = 1200. Step 2: A 15% decrease each year means the population retains 85% of its value from the previous year (100% - 15% = 85%). Step 3: 85% as a decimal is 0.85, so b = 0.85. Step 4: Write the function using the form f(t) = a·b^t. The exponential function is f(t) = 1200·0.85^t.

7. Sophia's investment of $1200 grows by 8% annually. Write the exponential function in the form f(t) = a·b^t. Answer: 1200·1.08^t Solution: The initial investment is $1200, so a = 1200. The investment grows by 8% annually, so the growth factor b = 1 + 0.08 = 1.08. Write the function in the form f(t) = a·b^t.
   Full step-by-step solution

   Step 1: The initial investment is $1200, so a = 1200. Step 2: The investment grows by 8% annually, so the growth factor b = 1 + 0.08 = 1.08. Step 3: Write the function in the form f(t) = a·b^t. Step 4: Substitute a = 1200 and b = 1.08 to get f(t) = 1200·1.08^t. The answer is 1200·1.08^t.

8. A pharmaceutical company is testing a new drug that decays exponentially in the bloodstream. A patient receives an initial dose of 800 mg, and after 4 hours, only 200 mg remain active. Write an exponential function in the form D(t) = ab^t that models the amount of active drug in milligrams after t hours. Answer: D(t) = 800 * (1/2)^(t/2) Solution: Exponential decay functions model situations where a quantity decreases by a constant factor over equal time intervals.
   Full step-by-step solution

   Exponential decay functions model situations where a quantity decreases by a constant factor over equal time intervals. To find the decay factor, you can determine what fraction of the original amount remains after the given time period, then find the hourly decay rate that would produce this result.

9. A right triangle is drawn on a coordinate plane with vertices at (0,0), (4,0), and (4,3). The triangle is then reflected across the y-axis. What is the area of the new triangle formed after reflection? Answer: 6 Solution: The vertices are (0,0), (4,0), and (4,3). - (0,0) is the origin. - (4,0) is 4 units to the right on the x-axis.
   Full step-by-step solution

   Step 1: Understand the original triangle The vertices are (0,0), (4,0), and (4,3). Plotting these: - (0,0) is the origin. - (4,0) is 4 units to the right on the x-axis. - (4,3) is directly above (4,0) by 3 units. So the triangle has a horizontal leg from (0,0) to (4,0) of length 4, and a vertical leg from (4,0) to (4,3) of length 3. This is a right triangle with legs 4 and 3. Step 2: Calculate the area of the original triangle Area = (1/2) * base * height = (1/2) * 4 * 3 = 6. Step 3: Reflect across the y-axis Reflecting across the y-axis means changing the sign of the x-coordinate of each point, while the y-coordinate stays the same. Original vertices: A = (0,0) → A' = (0,0) B = (4,0) → B' = (-4,0) C = (4,3) → C' = (-4,3) Step 4: Understand the new triangle New vertices: (0,0), (-4,0), (-4,3). This is still a right triangle: - Horizontal leg from (0,0) to (-4,0) has length 4 (absolute value). - Vertical leg from (-4,0) to (-4,3) has length 3. Step 5: Calculate the area of the new triangle Area = (1/2) * base * height = (1/2) * 4 * 3 = 6. Step 6: Conclusion Reflection does not change the size or shape of the triangle, only its position. Therefore, the area remains the same. Final answer: 6