Let's analyze each exponential expression and determine if it models growth or decay, and what the rate is. Here is a step-by-step breakdown for each expression:
1. For the expression [tex]\( 30(1.25)^x \)[/tex]:
- This is an exponential growth model because the base, 1.25, is greater than 1.
- The percentage rate of growth is calculated based on the factor 1.25. The rate is [tex]\( (1.25 - 1) \times 100\% = 0.25 \times 100\% = 25\% \)[/tex].
2. For the expression [tex]\( 500(0.75)^x \)[/tex]:
- This is an exponential decay model because the base, 0.75, is less than 1.
- The percentage rate of decay is calculated based on the factor 0.75. The rate is [tex]\( (1 - 0.75) \times 100\% = 0.25 \times 100\% = 25\% \)[/tex].
3. For the expression [tex]\( 2(2)^x \)[/tex]:
- This is an exponential growth model because the base, 2, is greater than 1.
- The percentage rate of growth is calculated based on the factor 2. The rate is [tex]\( (2 - 1) \times 100\% = 1 \times 100\% = 100\% \)[/tex].
4. For the expression [tex]\( 4,000(1.01)^x \)[/tex]:
- This is an exponential growth model because the base, 1.01, is greater than 1.
- The percentage rate of growth is calculated based on the factor 1.01. The rate is [tex]\( (1.01 - 1) \times 100\% = 0.01 \times 100\% = 1\% \)[/tex].
5. For the expression [tex]\( 7,000(0.99)^x \)[/tex]:
- This is an exponential decay model because the base, 0.99, is less than 1.
- The percentage rate of decay is calculated based on the factor 0.99. The rate is [tex]\( (1 - 0.99) \times 100\% = 0.01 \times 100\% = 1\% \)[/tex].
Now we compile this information into the table:
[tex]\[
\begin{array}{|c|c|c|}
\hline
\text{Expression} & \text{Growth or Decay} & \text{Rate} \\
\hline
30(1.25)^x & \text{growth} & 25\% \\
500(0.75)^x & \text{decay} & 25\% \\
2(2)^x & \text{growth} & 100\% \\
4,000(1.01)^x & \text{growth} & 1\% \\
7,000(0.99)^x & \text{decay} & 1\% \\
\hline
\end{array}
\][/tex]
This table provides the correct labels for each exponential expression, identifying whether it models growth or decay, along with the corresponding rates.
