Let's begin by analyzing the functions $f(x) = x^3$ and $g(x) = x^3 + 1$.

1. Graph of $f(x) = x^3$:

The graph of $f(x) = x^3$ is a cubic function that passes through the origin (0,0) and has the following behavior:

• As $x \to -\infty$, $f(x) \to -\infty$
• As $x \to +\infty$, $f(x) \to +\infty$

2. Graph of $g(x) = x^3 + 1$:

The function $g(x) = x^3 + 1$ is similar to $f(x)$, but it is vertically shifted up by 1 unit. For every value of $x$, the output of $g(x)$ is 1 greater than that of $f(x)$.

Select integer values for $x$ from $-2$ to $2$:

Let’s calculate the corresponding values for both $f(x)$ and $g(x)$ when $x = -2, -1, 0, 1, 2$:

$x$	$f(x) = x^3$	$g(x) = x^3 + 1$
-2	$(-2)^3 = -8$	$(-2)^3 + 1 = -8 + 1 = -7$
-1	$(-1)^3 = -1$	$(-1)^3 + 1 = -1 + 1 = 0$
0	$0^3 = 0$	$0^3 + 1 = 0 + 1 = 1$
1	$1^3 = 1$	$1^3 + 1 = 1 + 1 = 2$
2	$2^3 = 8$	$2^3 + 1 = 8 + 1 = 9$

Description of the relationship between the graphs:

• The graph of $g(x)$ is a vertical translation of the graph of $f(x)$ by 1 unit upward. This means that for each $x$, the corresponding value of $g(x)$ is always 1 more than $f(x)$.

I can now graph these functions for you if you'd like! Would you like me to do that, or do you have any further questions about this?

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Here are a few related questions for you:

1. How does the transformation affect the domain and range of the function?
2. What happens if we modify the function to $g(x) = x^3 - 1$?
3. How do other cubic transformations, like $x^3 + 2$ or $x^3 - 3$, affect the graph?
4. What would happen if we compared the graphs of $f(x) = x^3$ and $g(x) = -x^3$?
5. What effect does adding a constant to the exponent have on the graph of a cubic function?

Tip: When analyzing transformations of graphs, remember that adding a constant to the function (like $f(x) + c$) results in a vertical shift of the graph, where the sign of $c$ determines the direction (up if $c > 0$, down if $c < 0$).