which graph shows the cubic function ( f(x)=3x^{3}-5x^{2}+1 )?

Explanation:

Step1: Analyze the leading term

The leading term of (f(x)=3x^{3}-5x^{2}+1) is (3x^{3}). For a cubic function (y = ax^{3}+bx^{2}+cx + d) with (a>0), as (x\to+\infty), (y\to+\infty) and as (x\to-\infty), (y\to-\infty).

Step2: Check the (y -)intercept

Set (x = 0), then (f(0)=3\times0^{3}-5\times0^{2}+1=1). The (y -)intercept is ((0,1)).

Answer:

The graph with the following properties: as (x\to+\infty), (y\to+\infty); as (x\to-\infty), (y\to-\infty) and (y -)intercept at ((0,1)) (the bottom - left graph among the four options if we assume the standard orientation of the options).