\begin{tabular}{l} 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 \( t=4 \) \\ \( \begin{array}{ll}\text { a) } y=32,000(0.75)^{t} & \text { b) } y=25(1.05)^{t} \\ \text { growth or decay? } \\ \text { growth or decay? } \\ \text { initial amount: } \\ \text { rate of growth: } & \text { initial amount: } \\ \text { Evaluate for }=4: & \text { rate of growth: }\end{array} \) \\ \hline\end{tabular}

For the first function \( y = 32,000(0.75)^{t} \), this represents exponential decay because the base (0.75) is less than 1. The initial amount is 32,000, and the rate of decay is 25% (calculated as \( 1 - 0.75 = 0.25 \) or 25%). Evaluating the function for \( t = 4 \): \[ y = 32,000(0.75)^4 \approx 32,000 \times 0.3164 \approx 10,101.08 \] For the second function \( y = 25(1.05)^{t} \), this represents exponential growth since the base (1.05) is greater than 1. The initial amount is 25, and the rate of growth is 5%. Evaluating the function for \( t = 4 \): \[ y = 25(1.05)^4 \approx 25 \times 1.2155 \approx 30.39 \] So to summarize: a) Decay, initial amount: 32,000, rate of decay: 25%, evaluated at \( t=4 \): 10,101.08 b) Growth, initial amount: 25, rate of growth: 5%, evaluated at \( t=4 \): 30.39