- The given function is an exponential function \(y = 12^x\).
- By evaluating the expression at successive integer values \(x\) and \(x+1\), we find the ratio of consecutive \(y\)-values is \(\frac{12^{x+1}}{12^x} = 12\).
- This indicates that each \(y\)-value is \(12\) times the preceding one.
- The rule describing the growth is to \(\boxed{\text{multiply by 12}}\).
Explanation
- Analyzing the Exponential Growth Function
The function provided is \(y = 12^x\). We are asked to determine the growth rule for the \(y\)-values as the \(x\)-value increases by unit steps (from \(x = -1\) to \(3\)). Let's observe the provided table:
| \(x\) | \(y\) |
| :--- | :--- |
| \(-1\) | \(\approx 0.083\) (\(1/12\)) |
| \(0\) | \(1\) |
| \(1\) | \(12\) |
| \(2\) | \(144\) |
| \(3\) | \(1,728\) |
We can see that for any given \(x\), the value of \(y\) is \(12^x\). As we move to the next integer value of \(x\), which is \(x + 1\), the new value of \(y\) becomes \(y' = 12^{x+1}\).
- Verifying the Growth Rule
Using the exponent rules, we know that \(12^{x+1} = 12^x \cdot 12^1 = 12 \cdot 12^x\). Since \(y = 12^x\), we can substitute this to see that \(y' = 12 \cdot y\). This means that to get the next \(y\)-value, we must multiply the current \(y\)-value by \(12\). Let us verify this with the given data:
- From \(x = -1\) to \(x = 0\): \(0.083 \times 12 \approx 1\).
- From \(x = 0\) to \(x = 1\): \(1 \times 12 = 12\).
- From \(x = 1\) to \(x = 2\): \(12 \times 12 = 144\).
- From \(x = 2\) to \(x = 3\): \(144 \times 12 = 1,728\).
The pattern is consistent throughout the given values.
- Conclusion
Based on our analysis, the growth of the \(y\)-values over each unit interval of \(x\) is described by multiplying the previous \(y\)-value by \(12\). Among the given options, the correct rule is "multiply by \(12\)". Thus, the final answer is \(\boxed{\text{multiply by 12}}\).
Examples
Exponential growth is a common concept in real-world scenarios, such as the growth of a population or the accumulation of interest in a bank account. For instance, if a bacterial colony doubles every hour, the number of bacteria at any given time can be represented by an exponential function. Similarly, if you invest money at a constant annual interest rate, the total amount of money in your account grows exponentially over time. Understanding these patterns allows us to make accurate predictions about future values based on current trends.