Function Parameters

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Answer Key & Explanations

Function Parameters · Grade 9 · Worksheet 2

1. 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.

2. Emma is analyzing two different investment options. Option A is modeled by the linear function A(t) = 375t + 1500, where A(t) represents the total value in dollars after t years. Option B is modeled by the exponential function B(t) = 1500(1.07)^t, where B(t) represents the total value in dollars after t years. For each function, interpret the meaning of the numerical parameters (the coefficients and constants) in the context of Emma's investments. Specifically, what does the 1500 represent in both functions, what does the 375 represent in Option A, and what does the 1.07 represent in Option B? Answer: In both functions, the 1500 represents the initial investment amount of $1500. In Option A, the 375 represents the constant annual increase of $375 per year (linear growth). In Option B, the 1.07 represents a 7% annual growth factor, meaning the investment grows by 7% each year. Solution: Identify the initial value in both functions. For Option A: A(0) = 375(0) + 1500 = 1500. For Option B: B(0) = 1500(1.07)^0 = 1500(1) = 1500.
   Full step-by-step solution

   Step 1: Identify the initial value in both functions. For Option A: A(0) = 375(0) + 1500 = 1500. For Option B: B(0) = 1500(1.07)^0 = 1500(1) = 1500. So the 1500 in both functions represents the initial investment of $1500. Step 2: Interpret the 375 in Option A. The linear function is A(t) = 375t + 1500. This is in slope-intercept form y = mx + b, where m = 375 is the slope. The slope represents the rate of change: each year (t increases by 1), A(t) increases by 375. So the 375 means Emma's investment grows by a constant $375 each year. Step 3: Interpret the 1.07 in Option B. The exponential function is B(t) = 1500(1.07)^t. This is in the form y = ab^t, where b = 1.07 is the growth factor. Since 1.07 > 1, it represents growth. The growth rate is b - 1 = 1.07 - 1 = 0.07, which is 7% per year. So the 1.07 means Emma's investment grows by 7% each year. Final answer: In both functions, the 1500 represents the initial investment amount of $1500. In Option A, the 375 represents the constant annual increase of $375 per year (linear growth). In Option B, the 1.07 represents a 7% annual growth factor, meaning the investment grows by 7% each year.

3. Sophia's car depreciates according to the model y = 16000(0.91)^x, where y is the car's value after x years. What does 16000 represent? What does 0.91 represent? Answer: 16000 represents the initial value of the car, and 0.91 represents the annual decay factor (the car retains 91% of its value each year). Solution: Identify the parameters in the exponential model y = 16000(0.91)^x The parameter 16000 is multiplied by the exponential term, so it represents the initial value when x = 0 When x = 0, y = 16000(0.91)^0 = 16000(1) = 16000, confirming this is the starting value The parameter 0.91 is the base of…
   Full step-by-step solution

   Step 1: Identify the parameters in the exponential model y = 16000(0.91)^x Step 2: The parameter 16000 is multiplied by the exponential term, so it represents the initial value when x = 0 Step 3: When x = 0, y = 16000(0.91)^0 = 16000(1) = 16000, confirming this is the starting value Step 4: The parameter 0.91 is the base of the exponential term, which represents the multiplier each year Step 5: Since 0.91 < 1, this indicates decay, and specifically the car retains 91% of its value each year Step 6: Therefore, 16000 represents the initial car value, and 0.91 represents the annual decay factor

4. Noah invests money in a savings account that earns compound interest. The amount in the account after t years is modeled by the exponential function A(t) = 600(1.06)^t. Explain the meaning of the numbers 600 and 1.06 in the context of the investment. Answer: 600 represents the initial investment amount in dollars, and 1.06 represents a 6% annual growth factor. Solution: Identify the parts of the exponential function A(t) = 600(1.06)^t. The general form is y = ab^t, where a is the initial value and b is the growth factor. Interpret a = 600.
   Full step-by-step solution

   Step 1: Identify the parts of the exponential function A(t) = 600(1.06)^t. The general form is y = ab^t, where a is the initial value and b is the growth factor. Step 2: Interpret a = 600. When t = 0, A(0) = 600(1.06)^0 = 600 * 1 = 600. This means the initial amount invested is $600. Step 3: Interpret b = 1.06. Since b > 1, this represents exponential growth. The growth factor is 1.06, which corresponds to a growth rate of 1.06 - 1 = 0.06, or 6% per year. This means the account balance increases by 6% each year. The answer: 600 represents the initial investment of $600, and 1.06 represents a 6% annual growth factor.

5. Aroha is comparing two phone plans. Plan A charges a flat monthly fee of $35 plus $0.09 per text message. Plan B charges a flat monthly fee of $15 plus $0.15 per text message. For each plan, write a linear function in the form C(t) = mt + b, where t is the number of text messages sent in a month and C(t) is the total monthly cost. What does the parameter m represent in each function? Answer: The cost per text message Solution: For Plan A, the flat fee is $35 and the cost per text is $0.09. Here, m = 0.09 and b = 35. For Plan B, the flat fee is $15 and the cost per text is $0.15.
   Full step-by-step solution

   Step 1: For Plan A, the flat fee is $35 and the cost per text is $0.09. So the linear function is C(t) = 0.09t + 35. Here, m = 0.09 and b = 35. Step 2: For Plan B, the flat fee is $15 and the cost per text is $0.15. So the linear function is C(t) = 0.15t + 15. Here, m = 0.15 and b = 15. Step 3: In each function, m is the coefficient of t. Since t is the number of text messages, m tells us how much the total cost increases for each additional text message sent. Step 4: Therefore, m represents the cost per text message for each plan. Final answer: The cost per text message

6. A company's revenue is modeled by the function R(t) = 25,000(1.08)^t, where t is the number of years since 2020 and R(t) is the annual revenue in dollars. According to this model, what was the company's initial revenue in 2020? Answer: 25000 Solution: The function is R(t) = 25,000(1.08)^t, where t is years since 2020 The initial year 2020 corresponds to t = 0 Substitute t = 0 into the function: R(0) = 25,000(1.08)^0 Any number raised to the power of 0 equals 1, so (1.08)^0 = 1 Calculate R(0) = 25,000 × 1 = 25,000 The initial revenue in 2020…
   Full step-by-step solution

   Step 1: The function is R(t) = 25,000(1.08)^t, where t is years since 2020 Step 2: The initial year 2020 corresponds to t = 0 Step 3: Substitute t = 0 into the function: R(0) = 25,000(1.08)^0 Step 4: Any number raised to the power of 0 equals 1, so (1.08)^0 = 1 Step 5: Calculate R(0) = 25,000 × 1 = 25,000 Step 6: The initial revenue in 2020 was $25,000 The answer is 25000.