Discuss each of the formulas below, and determine a big-$O$, big-$Ω$, and big-Θ that are true for it. For right now, just use your intuition.

• $f_1(n)=3n^2+5n+3logn+25$
• $O(n^2) \times c$

Let $n=3$:

$(39)+(53)+(3*log_2(3))+25 = 71.754887502234$

Let $c = 4$:

$4*n^2, n=3 = 36$

Since $71.75 > 36,$ it is not a Big O relationship yet. We can plot these two equations in Desmos and find the intersection point. The first value of the intersection point should be the minimum value of n to make the relationship true.

Big O:

In this case, $n$ should be at least $9$ (or >8.332) to have a big O relationship (above the graph of the original equation)

We can find a $c$ that ensures that $c*n^2$ is always above the original equation graph for n ≥ 9.