- Observe that the investment return \(f(x)\) increases over time, narrowing options to an increasing quadratic or exponential function.
- Test for exponential growth by calculating ratios between consecutive intervals, finding a constant ratio of approximately 1.22019 per 5-year period.
- Confirm the exponential model fits the data perfectly, including the 20-year mark.
- Conclude that the function is an \(\boxed{\text{increasing exponential function}}\).
Explanation
- Data Analysis
The problem provides a table of investment returns over time: \((0, 10000)\), \((5, 12201.90)\), \((10, 14888.64)\), and \((20, 22167.15)\). By observing these values, it is clear that the function \(f(x)\) is strictly increasing as \(x\) increases, which immediately narrows the possible choices down to an "increasing quadratic function" or an "increasing exponential function."
- Testing for Exponential Growth
To determine if the growth is exponential, we verify if the function follows the form \(f(x) = C \cdot a^x\). Since \(f(0) = 10000\), we have \(C = 10000\). The growth factor for an interval of 5 years can be calculated by looking at the ratio of values:
\[\frac{f(5)}{f(0)} = \frac{12201.90}{10000} = 1.22019\]
\[\frac{f(10)}{f(5)} = \frac{14888.64}{12201.90} \approx 1.22019\]
Since the ratio over equal intervals of 5 years is constant, the function follows an exponential model. We can further verify this for the interval from 10 to 20 years (a span of 10 years, which is two 5-year intervals). The expected ratio for a 10-year period should be \((1.22019)^2 \approx 1.48886\). Indeed, \(\frac{22167.15}{14888.64} \approx 1.48886\). This confirms the function is exponential.
- Conclusion
Because the function is clearly increasing and follows the exponential growth model \(f(x) = 10000 \cdot (1.22019)^{x/5}\), we conclude that the function is an increasing exponential function. Therefore, the correct description is an increasing exponential function.
Examples
In finance, compound interest is a classic example of exponential growth. When an investment account earns interest not only on the initial principal but also on the accumulated interest from previous periods, the total return follows an exponential function. This explains why your savings grow faster over time, making it an essential concept for long-term financial planning and investment strategies.