Question

which is the graph of the function $f(x) = x^3 + 6x^2 + 11x + 6$?

Explanation:

Step1: Factor the cubic function

We factor \( f(x)=x^{3}+6x^{2}+11x + 6\). By trial and error, we find that \( f(-1)=(-1)^{3}+6(-1)^{2}+11(-1)+6=-1 + 6-11 + 6 = 0\), so \((x + 1)\) is a factor. Using polynomial division or synthetic division, we divide \(x^{3}+6x^{2}+11x + 6\) by \((x + 1)\). We get \(x^{2}+5x + 6\), which factors further into \((x + 2)(x+3)\). So \(f(x)=(x + 1)(x + 2)(x + 3)\).

Step2: Find the x - intercepts

The x - intercepts occur when \(f(x)=0\). So \((x + 1)(x + 2)(x + 3)=0\), which gives \(x=-1\), \(x=-2\), and \(x=-3\).

Step3: Analyze the end - behavior

For a cubic function of the form \(y = ax^{3}+bx^{2}+cx + d\) with \(a>0\) (here \(a = 1>0\)), as \(x
ightarrow-\infty\), \(y
ightarrow-\infty\) and as \(x
ightarrow\infty\), \(y
ightarrow\infty\).

Step4: Analyze the graph

The given graph has x - intercepts at \(x=-3\), \(x = - 2\), and \(x=-1\) (we can check the positions on the x - axis). Also, the end - behavior (as \(x
ightarrow-\infty\), the graph goes down and as \(x
ightarrow\infty\), the graph goes up) matches the end - behavior of a cubic function with \(a = 1>0\). The shape of the graph (with a local maximum and minimum between the roots) is consistent with the graph of a cubic function \(y=(x + 1)(x + 2)(x + 3)\).

Answer:

The graph shown (the one with x - intercepts at \(x=-3\), \(x=-2\), \(x = - 1\) and the correct end - behavior) is the graph of \(f(x)=x^{3}+6x^{2}+11x + 6\).