6. Given the following exponential functions, identify if it is an exponential growth or decay, identify the initial amount, and the rate of growth (as a percent) of the exponential function. Lastly, evaluate the function when \( t=4 \) \( \begin{array}{ll}\text { a) } y=32,000(0.75)^{t} & \text { b) } y=25(1.05)^{t} \\ \text { growth or decay? } & \text { growth or decay? } \\ \text { initial amount: } & \text { initial amount: } \\ \text { rate of growth: } & \text { rate of growth: } \\ \text { Evaluate for } t=4: & \text { Evaluate for }=4:\end{array} \)

Sure, let's analyze each exponential function step by step. --- ### **a) \( y = 32,000 \times (0.75)^{t} \)** 1. **Growth or Decay?** - **Decay** - **Reason:** The base of the exponential function is **0.75**, which is less than **1**. When the base is between **0** and **1**, the function represents exponential decay. 2. **Initial Amount:** - **\( 32,000 \)** - **Explanation:** The initial amount is the value of \( y \) when \( t = 0 \): \[ y(0) = 32,000 \times (0.75)^{0} = 32,000 \times 1 = 32,000 \] 3. **Rate of Growth (Decay):** - **25% Decay Rate** - **Calculation:** \[ \text{Decay Rate} = 1 - 0.75 = 0.25 \text{ or } 25\% \] - **Explanation:** Since the base is **0.75**, the quantity decreases by **25%** each time period. 4. **Evaluate for \( t = 4 \):** - **Calculation:** \[ y(4) = 32,000 \times (0.75)^{4} = 32,000 \times 0.31640625 = 10,156.25 \] - **Result:** \[ y(4) = 10,156.25 \] --- ### **b) \( y = 25 \times (1.05)^{t} \)** 1. **Growth or Decay?** - **Growth** - **Reason:** The base of the exponential function is **1.05**, which is greater than **1**. When the base is greater than **1**, the function represents exponential growth. 2. **Initial Amount:** - **\( 25 \)** - **Explanation:** The initial amount is the value of \( y \) when \( t = 0 \): \[ y(0) = 25 \times (1.05)^{0} = 25 \times 1 = 25 \] 3. **Rate of Growth:** - **5% Growth Rate** - **Explanation:** The base is **1.05**, indicating a **5%** increase each time period. 4. **Evaluate for \( t = 4 \):** - **Calculation:** \[ y(4) = 25 \times (1.05)^{4} = 25 \times 1.21550625 = 30.38765625 \] - **Result:** \[ y(4) \approx 30.39 \] *(Rounded to two decimal places)* --- ### **Summary Table** | **Function** | **a) \( y = 32,000 \times (0.75)^t \)** | **b) \( y = 25 \times (1.05)^t \)** | |------------------------|-------------------------------------------|---------------------------------------| | **Growth or Decay** | Decay | Growth | | **Initial Amount** | 32,000 | 25 | | **Rate of Growth** | 25% Decay Rate | 5% Growth Rate | | **\( y(4) \)** | 10,156.25 | 30.39 | --- **Note:** - When dealing with exponential functions: - **Base \( > 1 \):** Exponential Growth - **Base \( < 1 \):** Exponential Decay - The **initial amount** is always the coefficient of the exponential term when \( t = 0 \). - The **rate of growth or decay** is determined by how much the base deviates from **1**: - For growth: \( \text{Rate} = (\text{Base} - 1) \times 100\% \) - For decay: \( \text{Decay Rate} = (1 - \text{Base}) \times 100\% \) - To **evaluate** the function at a specific time \( t \), substitute the value of \( t \) into the function and compute the result.