Function Parameters

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Answer Key & Explanations

Function Parameters · Grade 9 · Worksheet 2

1. A graph (described below) shows the decay of a radioactive isotope over time. The y-axis represents the mass in grams, and the x-axis represents time in years. The curve passes through the points (0, 81) and (3, 27). The function is of the form y = a(b^x). What is the value of b, and what does it represent in the context of this decay? Answer: b = 1/3; it represents the decay factor, meaning the mass multiplies by 1/3 every year Solution: The function is y = a(b^x). At x = 0, y = 81, so a = 81. Step 2: At x = 3, y = 27, so 27 = 81(b^3).
   Full step-by-step solution

   Step 1: The function is y = a(b^x). At x = 0, y = 81, so a = 81. Step 2: At x = 3, y = 27, so 27 = 81(b^3). Step 3: Divide both sides by 81: 27/81 = b^3, which simplifies to 1/3 = b^3. Step 4: Take the cube root of both sides: b = (1/3)^(1/3) = 1/3. Step 5: Interpretation: b = 1/3 means the mass decays by a factor of 1/3 each year (i.e., each year the remaining mass is multiplied by 1/3). Final answer: b = 1/3, representing the annual decay factor.

2. Emma is studying the growth of a bacteria culture. She models the number of bacteria after x hours using the exponential function y = 150(1.25)^x. On a coordinate grid, she plots points for x = 0, 1, 2, 3, and 4. Describe what the number 150 and the number 1.25 represent in the context of the bacteria growth. Answer: 150 represents the initial number of bacteria at time 0 hours. 1.25 represents the growth factor: each hour, the number of bacteria multiplies by 1.25, meaning a 25% increase per hour. Solution: Identify the general form of an exponential function: y = a * b^x. Here, a = 150 and b = 1.25. Step 2: The parameter a is the initial value when x = 0.
   Full step-by-step solution

   Step 1: Identify the general form of an exponential function: y = a * b^x. Here, a = 150 and b = 1.25. Step 2: The parameter a is the initial value when x = 0. Substitute x = 0: y = 150 * (1.25)^0 = 150 * 1 = 150. So, at time 0 hours, there are 150 bacteria. Step 3: The parameter b is the growth factor. For x = 1, y = 150 * 1.25 = 187.5. For x = 2, y = 150 * (1.25)^2 = 150 * 1.5625 = 234.375. From x = 0 to x = 1, the population multiplies by 1.25. From x = 1 to x = 2, it multiplies by 1.25 again. So, each hour, the number of bacteria is multiplied by 1.25, which is a 25% increase per hour. Final answer: 150 is the initial number of bacteria; 1.25 is the hourly growth factor (25% increase per hour).

3. The graph below (described in text) shows the value of a rare comic book over time. The value V (in dollars) is modeled by the exponential function V(t) = 75(1.05)^t, where t is the number of years since 2010. A coordinate plane is drawn with the horizontal axis labeled 'Years since 2010 (t)' and the vertical axis labeled 'Value (V) in dollars'. The curve starts at (0, 75) and rises slowly, passing through points such as (10, approximately 122) and (20, approximately 199). Interpret the meaning of the parameters 75 and 1.05 in the context of the comic book's value. Answer: 75 is the initial value of the comic book in 2010, and 1.05 is the annual growth factor, meaning the value increases by 5% each year. Solution: Identify the general form of an exponential function: V(t) = a * b^t. Here, a = 75 and b = 1.05. Interpret a.
   Full step-by-step solution

   Step 1: Identify the general form of an exponential function: V(t) = a * b^t. Here, a = 75 and b = 1.05. Step 2: Interpret a. The parameter a represents the initial value when t = 0. Since t is the number of years since 2010, when t = 0, V(0) = 75 * (1.05)^0 = 75 * 1 = 75. So in the year 2010, the comic book was worth $75. Step 3: Interpret b. The parameter b is the growth factor. Since b = 1.05, the value is multiplied by 1.05 each year. A growth factor of 1.05 corresponds to a 5% increase each year (because 1.05 = 1 + 0.05, and 0.05 = 5%). Step 4: Conclusion. The number 75 means the comic book was initially worth $75 in 2010. The number 1.05 means the value grows by a factor of 1.05 each year, which is a 5% annual increase. Final answer: 75 is the initial value in 2010; 1.05 is the annual growth factor (5% increase per year).

4. Olivia is analyzing the value of a rare coin collection. The graph below (described in text) shows the value of the collection over time. The value in dollars, V, is modeled by the exponential function V(t) = 81(1.03)^t, where t is the number of years since the collection was first appraised in 2015. On the graph, the point (0, 81) is marked, and the curve passes through the point (25, 81(1.03)^25). Interpret the meaning of the number 81 and the number 1.03 in the context of the coin collection's value. Answer: The number 81 represents the initial value of the coin collection in dollars at t = 0 (the year 2015). The number 1.03 represents the annual growth factor, meaning the collection's value increases by 3% each year. Solution: Identify the general form of an exponential function. The given function is V(t) = 81(1.03)^t. This matches the form y = a(b)^x, where a is the initial value and b is the growth factor.
   Full step-by-step solution

   Step 1: Identify the general form of an exponential function. The given function is V(t) = 81(1.03)^t. This matches the form y = a(b)^x, where a is the initial value and b is the growth factor. Step 2: Interpret the parameter 81. When t = 0, V(0) = 81(1.03)^0 = 81 * 1 = 81. So, at the time of the first appraisal in 2015, the collection was worth $81. The number 81 represents the initial value of the coin collection. Step 3: Interpret the parameter 1.03. The base 1.03 is greater than 1, indicating exponential growth. Each year, the value is multiplied by 1.03. A multiplication by 1.03 is equivalent to a 3% increase because 1.03 = 1 + 0.03, and 0.03 as a percentage is 3%. Therefore, the number 1.03 means the collection's value increases by 3% each year. Final answer: The number 81 represents the initial value ($81) in 2015. The number 1.03 represents a 3% annual growth factor.

5. log₃(27) + log₂(16) = ? Answer: 7 Solution: Evaluate log₃(27). This asks '3 to what power equals 27?' Since 3³ = 27, log₃(27) = 3. Evaluate log₂(16).
   Full step-by-step solution

   Step 1: Evaluate log₃(27). This asks '3 to what power equals 27?' Since 3³ = 27, log₃(27) = 3. Step 2: Evaluate log₂(16). This asks '2 to what power equals 16?' Since 2⁴ = 16, log₂(16) = 4. Step 3: Add the results: 3 + 4 = 7. The answer is 7.

6. Sophia's investment grows according to y = 7200(1.15)^x. What does 7200 represent? What does 1.15 represent? Answer: 7200 represents the initial investment amount, 1.15 represents the growth factor per period Solution: Identify the function type: y = 7200(1.15)^x is an exponential function in the form y = ab^x. Interpret parameter a = 7200: In exponential growth models, 'a' represents the initial value when x = 0.
   Full step-by-step solution

   Step 1: Identify the function type: y = 7200(1.15)^x is an exponential function in the form y = ab^x. Step 2: Interpret parameter a = 7200: In exponential growth models, 'a' represents the initial value when x = 0. Therefore, 7200 represents Sophia's initial investment amount. Step 3: Interpret parameter b = 1.15: In exponential functions, 'b' represents the growth factor per time period. Since 1.15 is greater than 1, it indicates growth. The value 1.15 means the investment multiplies by 1.15 (or increases by 15%) each period. Step 4: Final interpretation: 7200 represents the initial investment amount, and 1.15 represents the growth factor per period.

7. 2^(x+1) = 32 Answer: 4 Solution: Write down the given equation. 2^(x+1) = 32 Recognize that both sides of the equation can be written as powers of 2. We know that 32 is the same as 2^5 because 2 * 2 * 2 * 2 * 2 = 32.
   Full step-by-step solution

   Step 1: Write down the given equation. 2^(x+1) = 32 Step 2: Recognize that both sides of the equation can be written as powers of 2. We know that 32 is the same as 2^5 because 2 * 2 * 2 * 2 * 2 = 32. So we rewrite the equation as: 2^(x+1) = 2^5 Step 3: Since the bases are the same (both are base 2), we can set the exponents equal to each other. This gives us the new equation: x + 1 = 5 Step 4: Solve for x by subtracting 1 from both sides of the equation. x + 1 - 1 = 5 - 1 x = 4 Step 5: State the final answer. The solution is x = 4.