Function Parameters

lessonbunny.com

Answer Key & Explanations

Function Parameters · Grade 9 · Worksheet 1

1. 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).

2. Emma is analyzing the growth of two different investments. Investment A is modeled by the function A(t) = 125(1.07)^t, where A(t) represents the value in dollars after t years. Investment B is modeled by the function B(t) = 75t + 200, where B(t) also represents the value in dollars after t years. Interpret the meaning of the parameters 125 and 1.07 in Investment A, and the parameters 75 and 200 in Investment B. Which investment will have a higher value after 10 years? Answer: Investment A Solution: Interpret the parameters for Investment A: A(t) = 125(1.07)^t. The number 125 is the initial value of the investment in dollars at t = 0.
   Full step-by-step solution

   Step 1: Interpret the parameters for Investment A: A(t) = 125(1.07)^t. The number 125 is the initial value of the investment in dollars at t = 0. The number 1.07 is the growth factor, meaning the investment increases by 7% each year (since 1.07 = 1 + 0.07). Step 2: Interpret the parameters for Investment B: B(t) = 75t + 200. The number 75 is the constant rate of increase in dollars per year (slope). The number 200 is the initial value of the investment in dollars at t = 0 (y-intercept). Step 3: Calculate the value of Investment A after 10 years: A(10) = 125(1.07)^10. First, compute 1.07^10. 1.07^2 = 1.1449, 1.07^4 = (1.1449)^2 = 1.310796, 1.07^8 = (1.310796)^2 = 1.718186, then 1.07^10 = 1.07^8 * 1.07^2 = 1.718186 * 1.1449 = 1.967151 (rounded to six decimals). So A(10) = 125 * 1.967151 = 245.893875. Rounded to two decimals, A(10) = $245.89. Step 4: Calculate the value of Investment B after 10 years: B(10) = 75(10) + 200 = 750 + 200 = $950. Step 5: Compare: $950 > $245.89, so Investment B has a higher value after 10 years. The answer is Investment B.

3. A right circular cone has a height of 12 cm and a base radius of 5 cm. A smaller cone is formed by cutting parallel to the base, creating a cross-section at a height of 4 cm from the vertex. What is the radius of this smaller cross-section? Answer: 1.67 Solution: - Height \( H = 12 \) cm - Base radius \( R = 5 \) cm A smaller cone is formed by cutting parallel to the base at a height of 4 cm from the vertex. That means the smaller cone's height \( h = 4 \) cm.
   Full step-by-step solution

   Let's go step-by-step. --- **Step 1: Understand the problem** We have a large right circular cone with: - Height \( H = 12 \) cm - Base radius \( R = 5 \) cm A smaller cone is formed by cutting parallel to the base at a height of 4 cm from the vertex. That means the smaller cone's height \( h = 4 \) cm. We want the radius \( r \) of the cross-section at that height. --- **Step 2: Use similar triangles** When we slice a cone parallel to its base, the cross-section is a smaller cone similar to the original cone. In similar cones, the ratio of corresponding lengths is constant: \[ \frac{\text{radius of small cone}}{\text{radius of large cone}} = \frac{\text{height of small cone}}{\text{height of large cone}} \] That is: \[ \frac{r}{R} = \frac{h}{H} \] --- **Step 3: Substitute values** Here: - \( R = 5 \) - \( H = 12 \) - \( h = 4 \) \[ \frac{r}{5} = \frac{4}{12} \] --- **Step 4: Simplify** \[ \frac{r}{5} = \frac{1}{3} \] Multiply both sides by 5: \[ r = \frac{5}{3} \] --- **Step 5: Convert to decimal** \[ \frac{5}{3} = 1.666\ldots \approx 1.67 \] --- **Final Answer:** The radius of the cross-section at 4 cm from the vertex is \( 1.67 \) cm.

4. y = 2800(1.06)^x models Hana's investment growth. What does 2800 represent? What does 1.06 represent? Answer: 2800 represents the initial investment amount; 1.06 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. The parameter 2800 is the coefficient a, which represents the starting amount when x = 0.
   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: The parameter 2800 is the coefficient a, which represents the starting amount when x = 0. Step 3: The parameter 1.06 is the base b, which represents the multiplier for each time period. Since 1.06 = 1 + 0.06, this indicates 6% growth per period. Step 4: Therefore, 2800 represents Hana's initial investment amount, and 1.06 represents the growth factor showing her investment grows by 6% each time period.

5. Isabella's savings grow according to y = 72(1.07)^x. What does 72 represent? What does 1.07 represent? Answer: 72 is the initial savings amount, 1.07 is the growth factor (7% increase per period) Solution: The exponential function is in the form y = ab^x, where a is the initial value and b is the growth factor. The parameter 72 represents the starting amount of Isabella's savings when x = 0.
   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: The parameter 72 represents the starting amount of Isabella's savings when x = 0. Step 3: The parameter 1.07 represents the growth factor, meaning her savings multiply by 1.07 each period, which is a 7% increase. Step 4: Therefore, 72 is the initial savings and 1.07 indicates 7% growth per time period.

6. Aroha is tracking the value of a rare coin collection. She models the value V(t) in dollars after t years using the exponential function V(t) = 1250(1.07)^t. Interpret the meaning of the parameters 1250 and 1.07 in this context. Answer: 1250 is the initial value of the coin collection in dollars; 1.07 is the annual growth factor, meaning the value increases by 7% each year. Solution: The exponential function is given as V(t) = 1250(1.07)^t, where t is time in years. When t = 0, V(0) = 1250(1.07)^0 = 1250 * 1 = 1250. So 1250 represents the initial value of the coin collection in dollars.
   Full step-by-step solution

   Step 1: The exponential function is given as V(t) = 1250(1.07)^t, where t is time in years. Step 2: When t = 0, V(0) = 1250(1.07)^0 = 1250 * 1 = 1250. So 1250 represents the initial value of the coin collection in dollars. Step 3: The base 1.07 is the growth factor. Since 1.07 = 1 + 0.07, the value increases by 7% each year. This means after one year, the value is multiplied by 1.07. Step 4: Therefore, 1250 is the starting value (at t = 0), and 1.07 is the annual growth factor (7% increase per year).

7. log₂(64) = ? Answer: 6 Solution: We are solving log₂(64) = ? This means we need to find the exponent x such that 2^x = 64. Write 64 as a power of 2: 64 = 2 × 2 × 2 × 2 × 2 × 2 = 2^6.
   Full step-by-step solution

   Step 1: We are solving log₂(64) = ? Step 2: This means we need to find the exponent x such that 2^x = 64. Step 3: Write 64 as a power of 2: 64 = 2 × 2 × 2 × 2 × 2 × 2 = 2^6. Step 4: Therefore, 2^x = 2^6, so x = 6. Step 5: The answer is 6.