To graph the function $(y + 1)^3 - 2$ and determine its domain, let's first analyze the function step by step.

1. Equation Rearrangement:

We have the equation: $(x = (y + 1)^3 - 2)$ This can be viewed as an equation in terms of $y$, meaning $x$ is dependent on $y$. It's an inverted cubic function, and the variable $y$ is inside the transformation $(y + 1)$, while the entire expression is shifted down by 2.

2. Domain:

The domain refers to the set of all possible values of $y$ for which the function is defined. Since this is a cubic function, there are no restrictions on $y$ (i.e., it can take any real value). Thus, the domain is: $\text{Domain}: \mathbb{R} \text{ (all real numbers)}$

3. Points:

To find the key points, let's pick several $y$-values and calculate corresponding $x$-values:

• When $y = 0$: $x = (0 + 1)^3 - 2 = 1^3 - 2 = 1 - 2 = -1$ Point: $(x, y) = (-1, 0)$

• When $y = -1$: $x = (-1 + 1)^3 - 2 = 0^3 - 2 = -2$ Point: $(x, y) = (-2, -1)$

• When $y = 1$: $x = (1 + 1)^3 - 2 = 2^3 - 2 = 8 - 2 = 6$ Point: $(x, y) = (6, 1)$

• When $y = -2$: $x = (-2 + 1)^3 - 2 = (-1)^3 - 2 = -1 - 2 = -3$ Point: $(x, y) = (-3, -2)$

• When $y = 2$: $x = (2 + 1)^3 - 2 = 3^3 - 2 = 27 - 2 = 25$ Point: $(x, y) = (25, 2)$

4. Graph Shape:

The graph of $x = (y + 1)^3 - 2$ is a cubic curve, flipped so that it's horizontal. The shape of a cubic function has a smooth curve with one point of inflection, and in this case, it's shifted and stretched based on the transformations applied.

5. Summary:

• Domain: All real numbers, $y \in (-\infty, \infty)$
• Key points:
  • $(-1, 0)$
  • $(-2, -1)$
  • $(6, 1)$
  • $(-3, -2)$
  • $(25, 2)$

Let me know if you want me to generate the graph!

Would you like any further details? Here are 5 related questions to expand on this:

1. How would the graph look if you added another transformation, such as multiplying by a constant?
2. What would happen to the points if the cubic function was reflected?
3. Can you describe the behavior of cubic functions in general?
4. How can we use calculus to find the turning point or inflection of a cubic function?
5. What is the range of this cubic function?

Tip: When graphing functions, always note any transformations like shifts, reflections, and stretching/compressing to understand how they affect the shape of the graph.