To graph the function $g(x) = (x + 4)^3 - 6$, we will begin by identifying the graph of the basic function, which is $f(x) = x^3$. Then, we apply the appropriate transformations (shifting, stretching, compressing, and/or reflecting).

1. Basic Function: $f(x) = x^3$

The graph of $f(x) = x^3$ is a cubic function with:

• An inflection point at the origin $(0, 0)$,
• The curve passing through $(1, 1)$, $(0, 0)$, and $(-1, -1)$,
• The graph increases to the right of the origin and decreases to the left of the origin.

2. Apply Transformations to $g(x) = (x + 4)^3 - 6$

We can break this down into two transformations:

• Horizontal Shift: The term $(x + 4)$ means the graph shifts 4 units to the left.
• Vertical Shift: The term $-6$ at the end indicates the graph shifts 6 units downward.

Thus, the graph of $f(x) = x^3$ is shifted left by 4 units and then down by 6 units to obtain the graph of $g(x) = (x + 4)^3 - 6$.

3. Domain and Range

• The domain of the function $g(x)$ is all real numbers because a cubic function is defined for all $x$: $\text{Domain}: (-\infty, \infty)$
• The range of the function $g(x)$ is also all real numbers because the cubic function tends towards $\infty$ as $x \to \infty$ and tends towards $-\infty$ as $x \to -\infty$: $\text{Range}: (-\infty, \infty)$

Summary of Transformations:

• The graph of $f(x) = x^3$ shifts 4 units left and 6 units down to form the graph of $g(x) = (x + 4)^3 - 6$.

Would you like to see the graph, or do you have any questions?

Here are 5 related questions to expand on the topic:

1. How do horizontal and vertical reflections affect the graph of a cubic function?
2. How would compressing or stretching the function vertically affect its shape?
3. What happens to the domain and range if we add an absolute value to $g(x)$?
4. How would adding or subtracting different constants inside the cubic term affect the transformation?
5. How would the graph change if $g(x) = - (x + 4)^3 - 6$?

Tip: Always apply horizontal shifts first before vertical shifts when graphing transformations.