The function provided is:

$f(n) = 8n^3 + 4n^2 + 15n + 20$

Step 1: Analyze the Growth Rate

To understand the growth rate of this function as $n$ increases, we can break it down by examining the asymptotic behavior. In Big-O notation, the growth rate is dominated by the term with the highest power of $n$ because, for large $n$, that term will grow much faster than the others.

In this case, the highest power of $n$ is $n^3$ (from the term $8n^3$). Therefore, as $n$ increases, the function behaves similarly to a cubic function. Thus, the growth rate of this function is O(n^3).

Step 2: Table Representation of $f(n)$ for Different $n$

Let’s compute the function $f(n)$ for several values of $n$ to see how it grows as $n$ increases.

$n$	$f(n) = 8n^3 + 4n^2 + 15n + 20$
1	$8(1^3) + 4(1^2) + 15(1) + 20 = 47$
2	$8(2^3) + 4(2^2) + 15(2) + 20 = 142$
3	$8(3^3) + 4(3^2) + 15(3) + 20 = 341$
4	$8(4^3) + 4(4^2) + 15(4) + 20 = 680$
5	$8(5^3) + 4(5^2) + 15(5) + 20 = 1197$

Step 3: Interpretation

The values in the table show that $f(n)$ grows rapidly as $n$ increases, consistent with the cubic nature of the function. For large $n$, the cubic term $8n^3$ becomes the dominant factor, and the lower-order terms ($4n^2$, $15n$, and 20) contribute less and less to the overall growth.

Thus, for large values of $n$, the function's growth is approximately proportional to $n^3$, confirming that the growth rate of this function is O(n^3).

Conclusion

The dominant term $8n^3$ leads to a cubic growth rate, and the function can be classified as having O(n^3) growth in terms of time complexity in the context of data structures and algorithms.

Do you want more details or have any questions?

Here are 5 related questions to further expand the discussion:

1. How does the growth rate change if we modify the coefficients of the polynomial?
2. What is the difference between $O(n^3)$ and $O(n^2)$ growth rates in terms of algorithm performance?
3. How can we estimate the time complexity of an algorithm given a different polynomial function?
4. Can we apply the same technique to analyze the growth rate of logarithmic or exponential functions?
5. What would the function's behavior be if one of the terms (e.g., $15n$) was removed?

Tip: Always focus on the highest-degree term when analyzing the growth rate of polynomial functions!