Question

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).