To graph the function $f(x) = x^3 + 3x^2 + 2$, let's go through the steps:

1. Identify the basic characteristics of the function:

• Degree: The function is a cubic polynomial, meaning it has degree 3.
• Leading Coefficient: The coefficient of $x^3$ is positive, so the graph will rise to the right and fall to the left.
• Intercepts: Let's find the $y$-intercept and the $x$-intercepts.

2. Calculate the intercepts:

• Y-intercept: Set $x = 0$. $f(0) = 0^3 + 3(0)^2 + 2 = 2$ So, the $y$-intercept is $(0, 2)$.

• X-intercepts: Solve $f(x) = 0$. $x^3 + 3x^2 + 2 = 0$ This can be solved either by factoring or using numerical methods.

3. Determine