Function Parameters

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Answer Key & Explanations

Function Parameters · Grade 9 · Worksheet 1

1. A research lab is studying bacterial growth in a petri dish. The initial bacterial population is 500. The population doubles every 3 hours. Write an exponential function P(t) that models the bacterial population after t hours, and determine how many bacteria will be present after 12 hours. Answer: 8000 Solution: Identify the initial population. The problem says the initial bacterial population is 500. So, initial value P0 = 500.
   Full step-by-step solution

   Step 1: Identify the initial population. The problem says the initial bacterial population is 500. So, initial value P0 = 500. Step 2: Determine the growth factor and period. The population doubles every 3 hours. That means the doubling time is 3 hours. Step 3: Write the general exponential growth formula. For exponential growth with doubling time T, the formula is: P(t) = P0 * (2)^(t / T) Here, P0 = 500, T = 3. Step 4: Substitute values into the formula. P(t) = 500 * 2^(t / 3) This is the exponential function modeling the population after t hours. Step 5: Use the function to find the population after 12 hours. Set t = 12: P(12) = 500 * 2^(12 / 3) P(12) = 500 * 2^4 Step 6: Calculate 2^4. 2^4 = 16. Step 7: Multiply by 500. 500 * 16 = 8000. Step 8: State the final answer. After 12 hours, the bacterial population will be 8000.

2. Noah is analyzing the depreciation of a piece of machinery. Its value, in dollars, after x years is modeled by the exponential function V(x) = 8400(0.85)^x. A graph of this function shows the value decreasing over time, starting from a point on the vertical axis at x=0 and curving downward to approach zero. What is the real-world meaning of the number 8400 in this model, and what does the number 0.85 represent? Answer: 8400 represents the initial value of the machinery in dollars; 0.85 represents the annual decay factor, meaning the machinery retains 85% of its value each year. Solution: The function is V(x) = 8400(0.85)^x. When x = 0, V(0) = 8400(0.85)^0 = 8400(1) = 8400. So 8400 is the initial value of the machinery at the time of purchase.
   Full step-by-step solution

   Step 1: The function is V(x) = 8400(0.85)^x. When x = 0, V(0) = 8400(0.85)^0 = 8400(1) = 8400. So 8400 is the initial value of the machinery at the time of purchase. Step 2: The base 0.85 is less than 1, indicating exponential decay. Since 0.85 = 85/100 = 85%, the machinery retains 85% of its value each year. This is called the decay factor. Step 3: Therefore, 8400 means the machinery was worth $8400 when new, and 0.85 means it loses 15% of its value each year (since 100% - 85% = 15%). Final answer: 8400 represents the initial value in dollars; 0.85 represents the annual decay factor (85% retained per year).

3. y = 27(1.02)^x models Isabella's savings account. What does 27 represent? What does 1.02 represent? Answer: 27 represents the initial amount in dollars, 1.02 represents the growth factor per time period Solution: The exponential function is in the form y = ab^x, where a is the initial value and b is the growth factor. In this context, y represents the total savings and x represents time periods.
   Full step-by-step solution

   Step 1: The exponential function is in the form y = ab^x, where a is the initial value and b is the growth factor. Step 2: In this context, y represents the total savings and x represents time periods. Step 3: The parameter 27 is the initial amount when x = 0: y = 27(1.02)^0 = 27(1) = 27 dollars. Step 4: The parameter 1.02 represents the growth factor per time period. Since 1.02 = 1 + 0.02, this means the savings grows by 2% each time period. Step 5: Therefore, 27 represents the initial amount in dollars, and 1.02 represents the growth factor per time period.

4. Hana is a conservation scientist tracking the population of a rare bird species on a remote island. The number of birds, N(t), can be modeled by the exponential function N(t) = 240(0.85)^t, where t is the number of years since the study began. Interpret the meaning of the parameters 240 and 0.85 in this context. Answer: 240 is the initial number of birds at the start of the study; 0.85 is the decay factor, meaning the population decreases by 15% each year. Solution: The function is N(t) = 240(0.85)^t, which is in the form y = a(b)^x, where a is the initial value and b is the growth or decay factor. The parameter 240 is the value when t = 0.
   Full step-by-step solution

   Step 1: The function is N(t) = 240(0.85)^t, which is in the form y = a(b)^x, where a is the initial value and b is the growth or decay factor. Step 2: The parameter 240 is the value when t = 0. This means at the start of the study (t = 0), there were 240 birds. Step 3: The parameter 0.85 is the base of the exponent. Since 0.85 is less than 1, this indicates exponential decay. Each year, the population is multiplied by 0.85, meaning it retains 85% of the previous year's population. Step 4: The decay rate is 1 - 0.85 = 0.15, or 15% per year. So, the population decreases by 15% each year. Final answer: 240 is the initial number of birds; 0.85 is the decay factor, meaning the population decreases by 15% per year.

5. Aroha is tracking the value of two different investments. Investment A is modeled by the linear function V_A(t) = 125t + 500, where V_A is the value in dollars after t months. Investment B is modeled by the exponential function V_B(t) = 500(1.07)^t, where V_B is the value in dollars after t months. Explain the real-world meaning of the numbers 125 and 500 in the linear model, and the numbers 500 and 1.07 in the exponential model. Answer: In the linear model, 500 represents the initial value of Investment A (in dollars), and 125 represents the monthly increase in value (in dollars per month). In the exponential model, 500 represents the initial value of Investment B (in dollars), and 1.07 represents the monthly growth factor (the value is multiplied by 1.07 each month, corresponding to a 7% monthly increase). Solution: Interpret the linear model V_A(t) = 125t + 500. The general form is y = mx + b, where b is the y-intercept and m is the slope. Here, b = 500 is the value when t = 0 (initial investment).
   Full step-by-step solution

   Step 1: Interpret the linear model V_A(t) = 125t + 500. The general form is y = mx + b, where b is the y-intercept and m is the slope. Here, b = 500 is the value when t = 0 (initial investment). The slope m = 125 means the value increases by 125 dollars each month. Step 2: Interpret the exponential model V_B(t) = 500(1.07)^t. The general form is y = ab^t, where a is the initial value and b is the growth factor. Here, a = 500 is the value when t = 0 (initial investment). The base b = 1.07 means each month the value is multiplied by 1.07, which is a 7% increase per month. Step 3: Final answer: In the linear model, 500 is the initial value in dollars, and 125 is the monthly increase in dollars per month. In the exponential model, 500 is the initial value in dollars, and 1.07 is the monthly growth factor (7% increase per month).

6. A right triangle is drawn on a coordinate plane with vertices at (0,0), (4,0), and (4,3). A circle is inscribed in this triangle, tangent to all three sides. What is the radius of the inscribed circle? Answer: 1 Solution: A = (0,0) B = (4,0) C = (4,3) This is a right triangle with the right angle at B = (4,0) 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 = (4,0) C = (4,3) This is a right triangle with the right angle at B = (4,0) because AB is horizontal and BC is vertical. Sides: AB = from (0,0) to (4,0) → length = 4 BC = from (4,0) to (4,3) → length = 3 AC = hypotenuse from (0,0) to (4,3) → length = sqrt( (4-0)^2 + (3-0)^2 ) = sqrt(16 + 9) = sqrt(25) = 5 So sides: 3, 4, 5. --- **Step 2: Incenter radius formula for a right triangle** For any triangle, the inradius r = Area / Semiperimeter. Area = (1/2) * base * height Here base = 4, height = 3 (legs of the right triangle). Area = (1/2) * 4 * 3 = 6. Semiperimeter s = (3 + 4 + 5) / 2 = 12 / 2 = 6. --- **Step 3: Compute r** r = Area / s = 6 / 6 = 1. --- **Step 4: Conclusion** The radius of the inscribed circle is 1. --- **Final answer:** 1

7. Liam is studying bacterial growth in his biology class. He observes that a certain bacterial culture starts with 200 bacteria and doubles every 3 hours. Write an exponential function B(t) that models the number of bacteria after t hours, then determine how many bacteria will be present after 9 hours. Answer: 1600 Solution: The culture starts with 200 bacteria. It doubles every 3 hours. So, after 3 hours → 400 bacteria, after 6 hours → 800 bacteria, etc.
   Full step-by-step solution

   Let's go step-by-step. --- **Step 1: Identify the initial amount and growth pattern** The culture starts with 200 bacteria. It doubles every 3 hours. So, after 3 hours → 400 bacteria, after 6 hours → 800 bacteria, etc. --- **Step 2: Write the general exponential growth formula** For exponential growth with doubling time: B(t) = initial × 2^(t / doubling time) Here: Initial = 200 Doubling time = 3 hours So: B(t) = 200 × 2^(t / 3) --- **Step 3: Use the formula to find B(9)** We want t = 9 hours. B(9) = 200 × 2^(9 / 3) B(9) = 200 × 2^3 B(9) = 200 × 8 --- **Step 4: Calculate** 200 × 8 = 1600 --- **Step 5: Conclusion** After 9 hours, there will be 1600 bacteria. --- **Final Answer:** 1600