To graph the function $g(x) = (x + 4)^3 + 6$, let's follow the process of shifting, stretching, compressing, and reflecting starting with the graph of the basic cubic function $f(x) = x^3$.

Step-by-Step Graphing Process:

1. Start with the base function: The base function is $f(x) = x^3$, which is a standard cubic curve that passes through the origin $(0, 0)$ with the following key points:

   • $(-1, -1)$
   • $(0, 0)$
   • $(1, 1)$

   It has an "S" shape, with increasing behavior for $x > 0$ and decreasing for $x < 0$.

2. Shift left by 4 units: The term $(x + 4)$ means we shift the graph of $f(x) = x^3$ 4 units to the left. So, the key points are now:

   • $(-5, -1)$ (formerly $(-1, -1)$)
   • $(-4, 0)$ (formerly $(0, 0)$)
   • $(-3, 1)$ (formerly $(1, 1)$)

3. Shift up by 6 units: The $+6$ at the end means we shift the entire graph up by 6 units. The key points after this shift are:

   • $(-5, -1 + 6) = (-5, 5)$
   • $(-4, 0 + 6) = (-4, 6)$
   • $(-3, 1 + 6) = (-3, 7)$

Domain and Range:

• Domain: The domain of $g(x)$ is the set of all real numbers since a cubic function is defined for all $x \in \mathbb{R}$. Hence, the domain is $(-\infty, \infty)$.

• Range: The range of $g(x)$ is also all real numbers, as the cubic function takes on all values from negative infinity to positive infinity. Thus, the range is $(-\infty, \infty)$.

Key Points of the Transformed Function:

• $(-5, 5)$
• $(-4, 6)$
• $(-3, 7)$

Would you like more details or have any questions?

Here are five related questions for further practice:

1. How would the graph change if the function was $g(x) = -(x + 4)^3 + 6$?
2. What if the function was $g(x) = (x + 4)^3 - 6$? How would the range be affected?
3. What are the inflection points of the basic cubic function $f(x) = x^3$?
4. How do you find the derivative of $g(x) = (x + 4)^3 + 6$?
5. Can you generalize the transformation for a function $g(x) = (x + a)^3 + b$?

Tip: Always shift horizontally first before performing any vertical transformations when dealing with function transformations.