Function Parameters

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Answer Key & Explanations

Function Parameters · Grade 9 · Worksheet 3

1. Matiu is studying the growth of a bacteria colony. The number of bacteria after t hours is modeled by the exponential function B(t) = 400(1.08)^t. The graph of this function is shown on a coordinate plane, with the horizontal axis labeled 'Time (hours)' and the vertical axis labeled 'Number of Bacteria'. Interpret the meaning of the parameters 400 and 1.08 in the context of the bacteria colony. Answer: 400 is the initial number of bacteria at t = 0 hours; 1.08 is the growth factor, meaning the number of bacteria increases by 8% each hour. Solution: Identify the general form of an exponential function: y = ab^x. Here, B(t) = 400(1.08)^t, so a = 400 and b = 1.08. Interpret the parameter a.
   Full step-by-step solution

   Step 1: Identify the general form of an exponential function: y = ab^x. Here, B(t) = 400(1.08)^t, so a = 400 and b = 1.08. Step 2: Interpret the parameter a. When t = 0, B(0) = 400(1.08)^0 = 400 * 1 = 400. This means at time 0 hours, there are 400 bacteria. So 400 represents the initial number of bacteria. Step 3: Interpret the parameter b. Since b = 1.08 > 1, the function models exponential growth. Each time t increases by 1, the bacteria count is multiplied by 1.08. A factor of 1.08 corresponds to an 8% increase (because 1.08 = 1 + 0.08, and 0.08 = 8/100 = 8%). So 1.08 means the bacteria population grows by 8% each hour. Step 4: Final answer: 400 is the initial number of bacteria at t = 0 hours; 1.08 is the growth factor, meaning the number of bacteria increases by 8% each hour.

2. Tane's investment grows according to y = 8200(1.07)^x, where y is the value after x years. What does 8200 represent? What does 1.07 represent? Answer: 8200 represents the initial investment amount, 1.07 represents the growth factor (7% annual growth rate) Solution: Identify the exponential function form: y = ab^x Compare Tane's function y = 8200(1.07)^x to the standard form The parameter 'a' = 8200 represents the initial value when x = 0 The parameter 'b' = 1.07 represents the growth factor per year Since 1.07 > 1, this indicates 7% annual growth (1.07 = 1…
   Full step-by-step solution

   Step 1: Identify the exponential function form: y = ab^x Step 2: Compare Tane's function y = 8200(1.07)^x to the standard form Step 3: The parameter 'a' = 8200 represents the initial value when x = 0 Step 4: The parameter 'b' = 1.07 represents the growth factor per year Step 5: Since 1.07 > 1, this indicates 7% annual growth (1.07 = 1 + 0.07) Step 6: Therefore, 8200 represents the initial investment amount, and 1.07 represents the growth factor showing 7% annual growth rate

3. Liam is studying bacterial growth in his biology class. He observes that a bacterial culture starts with 200 cells and doubles every 3 hours. Write an exponential function in the form f(t) = a * b^t that models the number of bacteria after t hours, then determine how many bacteria will be present after 12 hours. Answer: 3200 Solution: The problem says the culture starts with 200 cells. So at t = 0, f(0) = 200. The culture doubles every 3 hours.
   Full step-by-step solution

   Let's go step-by-step. --- **Step 1: Identify the initial value a** The problem says the culture starts with 200 cells. So at t = 0, f(0) = 200. Thus, in f(t) = a * b^t, we have a = 200. --- **Step 2: Determine the base b** The culture doubles every 3 hours. This means f(3) = 2 * f(0) = 400. So: 200 * b^3 = 400 Divide both sides by 200: b^3 = 2 So b = 2^(1/3). Thus the function is: f(t) = 200 * (2^(1/3))^t which simplifies to: f(t) = 200 * 2^(t/3). --- **Step 3: Verify the form f(t) = a * b^t** We have a = 200, b = 2^(1/3). So f(t) = 200 * (2^(1/3))^t is correct. --- **Step 4: Find the number of bacteria after 12 hours** We can use f(t) = 200 * 2^(t/3) for easier calculation. t = 12: f(12) = 200 * 2^(12/3) = 200 * 2^4 = 200 * 16 = 3200. --- **Step 5: Conclusion** The exponential function is f(t) = 200 * (2^(1/3))^t. After 12 hours, there are 3200 bacteria. --- **Final answer:** 3200

4. The graph below (described in words) shows the population of a rare bird species on an island over time. The population is modeled by an exponential function of the form y = a(b)^x, where y is the number of birds and x is the number of years since 2010. The graph passes through the points (0, 15) and (3, 120). What is the interpretation of the parameter 'a' in this context? What is the interpretation of the parameter 'b'? Answer: a = 15 represents the initial population of birds in 2010; b = 2 represents the growth factor per year (the population doubles each year). Solution: The function is y = a(b)^x. At x = 0 (year 2010), the graph passes through (0, 15), so y = 15 when x = 0. Substituting: 15 = a(b)^0 = a(1) = a.
   Full step-by-step solution

   Step 1: The function is y = a(b)^x. At x = 0 (year 2010), the graph passes through (0, 15), so y = 15 when x = 0. Substituting: 15 = a(b)^0 = a(1) = a. Thus, a = 15. Interpretation: a is the initial population in 2010, which is 15 birds. Step 2: At x = 3 (year 2013), the graph passes through (3, 120), so y = 120 when x = 3. Using a = 15: 120 = 15(b)^3. Divide both sides by 15: 8 = b^3. Take the cube root: b = 2. Interpretation: b is the growth factor per year. Since b = 2, the population doubles each year. Step 3: Final answer: a = 15 (initial population of 15 birds in 2010); b = 2 (the population doubles every year).

5. A city's population is currently 50,000 people and is growing at a rate of 4% per year. Meanwhile, a nearby town has a population of 80,000 people but is experiencing a decline of 3% per year due to economic changes. Write exponential functions for both populations, where P_c(t) represents the city's population and P_t(t) represents the town's population after t years. After how many years will the city's population first exceed the town's population? Answer: 7 Solution: City growth: P_c(t) = 50000 × (1.04)^t Town decline: P_t(t) = 80000 × (0.97)^t Set the functions equal to find when populations are equal 50000 × (1.04)^t = 80000 × (0.97)^t Divide both sides by 50000 (1.04)^t = 1.6 × (0.97)^t Divide both sides by (0.97)^t (1.04/0.97)^t = 1.6 1.04 ÷ 0.97 =…
   Full step-by-step solution

   Step 1: Write the exponential functions City growth: P_c(t) = 50000 × (1.04)^t Town decline: P_t(t) = 80000 × (0.97)^t Step 2: Set the functions equal to find when populations are equal 50000 × (1.04)^t = 80000 × (0.97)^t Step 3: Divide both sides by 50000 (1.04)^t = 1.6 × (0.97)^t Step 4: Divide both sides by (0.97)^t (1.04/0.97)^t = 1.6 Step 5: Calculate the ratio 1.04 ÷ 0.97 = 1.072164948 Step 6: Take natural log of both sides t × ln(1.072164948) = ln(1.6) Step 7: Calculate the logarithms ln(1.072164948) = 0.0696 ln(1.6) = 0.4700 Step 8: Solve for t t = 0.4700 ÷ 0.0696 = 6.75 Step 9: Since we need the first year when city exceeds town, round up to the next whole year t = 7 years The answer is 7 years.

6. The graph below (described in text) shows a line and an exponential curve on the same coordinate axes. The line passes through the points (0, 24) and (6, 0). The exponential curve passes through (0, 3) and (2, 12). (a) Write the equation of the line in the form y = mx + b. Then interpret the meaning of the slope m and the y-intercept b in the context of a scenario where the line models the decreasing height (in cm) of a burning candle over time x (in hours). (b) Write the equation of the exponential curve in the form y = a * b^x. Then interpret the meaning of the parameters a and b in the context of a scenario where the curve models the number of bacteria in a petri dish over time x (in hours). Answer: Line: y = -4x + 24. Slope -4 means the candle height decreases by 4 cm per hour; y-intercept 24 means initial height is 24 cm. Exponential: y = 3 * 2^x. Initial value a=3 means 3 bacteria initially; growth factor b=2 means the population doubles each hour. Solution: Find the equation of the line. Use points (0,24) and (6,0). Slope m = (0 - 24) / (6 - 0) = -24 / 6 = -4.
   Full step-by-step solution

   Step 1: Find the equation of the line. Use points (0,24) and (6,0). Slope m = (0 - 24) / (6 - 0) = -24 / 6 = -4. The y-intercept b is the y-coordinate when x=0, so b = 24. Equation: y = -4x + 24. Interpretation: The slope m = -4 means the candle height decreases by 4 cm each hour. The y-intercept b = 24 means the initial height of the candle is 24 cm. Step 2: Find the equation of the exponential curve. Use y = a * b^x. Point (0,3): 3 = a * b^0 = a * 1, so a = 3. Point (2,12): 12 = 3 * b^2. Divide both sides by 3: 4 = b^2. Take the positive square root: b = 2. Equation: y = 3 * 2^x. Interpretation: a = 3 means there were 3 bacteria initially at time 0. b = 2 means the bacteria population doubles every hour. Final answer: Line: y = -4x + 24. Slope -4 means the candle height decreases by 4 cm per hour; y-intercept 24 means initial height is 24 cm. Exponential: y = 3 * 2^x. Initial value a=3 means 3 bacteria initially; growth factor b=2 means the population doubles each hour.