To graph the polynomial function \( f(x) = 3x^3 + 12x^2 + 12x \), we start by analyzing its end behavior. The leading term is \( 3x^3 \), where the leading coefficient is positive (3) and the degree is odd (3). This indicates that as \( x \) approaches positive infinity, \( f(x) \) will also approach positive infinity, while as \( x \) approaches negative infinity, \( f(x) \) will approach negative infinity. Thus, the right side of the graph rises, and the left side falls.

Next, we find the x-intercepts by setting \( f(x) = 0 \). Factoring the function, we can extract the greatest common factor:

\[f(x) = 3x(x^2 + 4x + 4) = 3x(x + 2)^2\]

Setting this equal to zero gives us the factors \( 3x = 0 \) and \( (x + 2)^2 = 0 \). Solving these, we find the x-intercepts at \( x = 0 \) and \( x = -2 \). The multiplicity of the intercept at \( x = 0 \) is 1 (odd), indicating the graph will cross the x-axis at this point. The intercept at \( x = -2 \) has a multiplicity of 2 (even), meaning the graph will touch the x-axis and bounce back.

To find the y-intercept, we evaluate \( f(0) \), which yields:

\[f(0) = 3(0)(0 + 2)^2 = 0\]

Thus, the y-intercept is also at the origin (0, 0).

Next, we identify intervals of unknown behavior by examining the x-values around our known points. The intervals are:

• From \( -\infty \) to \( -2 \)
• From \( -2 \) to \( 0 \)
• From \( 0 \) to \( \infty \)

We select test points from each interval to evaluate \( f(x) \):

For \( x = -3 \):

\[f(-3) = 3(-3)(-3 + 2)^2 = 3(-3)(-1)^2 = -9\]

Thus, the point is \( (-3, -9) \).

For \( x = -1 \):

\[f(-1) = 3(-1)(-1 + 2)^2 = 3(-1)(1)^2 = -3\]

Thus, the point is \( (-1, -3) \).

For \( x = 1 \):

\[f(1) = 3(1)(1 + 2)^2 = 3(1)(3)^2 = 27\]

Thus, the point is \( (1, 27) \).

Now we can plot the points \( (-3, -9) \), \( (-1, -3) \), and \( (1, 27) \) on the graph. Connecting these points with a smooth curve, we observe the behavior at the x-intercepts: crossing at \( (0, 0) \) and touching at \( (-2, 0) \). The graph will extend downwards to negative infinity on the left and upwards to positive infinity on the right.

Finally, we check the number of turning points. The maximum number of turning points for a polynomial is given by \( n - 1 \), where \( n \) is the degree. Here, \( n = 3 \), so we can have a maximum of 2 turning points, which we confirm exist in our graph.

For the domain of polynomial functions, it is always all real numbers, expressed as \( (-\infty, \infty) \). The range, based on our graph, also extends from negative infinity to positive infinity, \( (-\infty, \infty) \), as the function continues to rise and fall without bound.

In summary, we have successfully graphed the polynomial function \( f(x) = 3x^3 + 12x^2 + 12x \), identified its x-intercepts at \( (0, 0) \) and \( (-2, 0) \), determined the y-intercept at \( (0, 0) \), and established that both the domain and range are \( (-\infty, \infty) \).