To graph the function $g(x) = x^3 - 6$, we can apply transformations to the standard cubic function $f(x) = x^3$.

Step 1: Start with the basic graph of $f(x) = x^3$

The graph of $f(x) = x^3$ is a symmetric curve passing through the origin $(0, 0)$, with inflection points at $(0, 0)$. It increases rapidly as $x$ becomes positive and decreases rapidly as $x$ becomes negative. It looks like a smooth S-shape.

Step 2: Apply the transformation

The given function is $g(x) = x^3 - 6$. This is a vertical translation of the standard cubic function.

• The "-6" means that the entire graph of $f(x) = x^3$ is shifted 6 units downward.
• The shape of the graph remains unchanged (it still has the same "S" shape), but every point on the graph is moved 6 units lower on the $y$-axis.

Step 3: Graphing $g(x) = x^3 - 6$

To graph this, you can take key points from $f(x) = x^3$ and subtract 6 from the $y$-coordinates:

• For $x = -2$, $f(x) = (-2)^3 = -8$. So, $g(-2) = -8 - 6 = -14$.
• For $x = -1$, $f(x) = (-1)^3 = -1$. So, $g(-1) = -1 - 6 = -7$.
• For $x = 0$, $f(x) = (0)^3 = 0$. So, $g(0) = 0 - 6 = -6$.
• For $x = 1$, $f(x) = (1)^3 = 1$. So, $g(1) = 1 - 6 = -5$.
• For $x = 2$, $f(x) = (2)^3 = 8$. So, $g(2) = 8 - 6 = 2$.

Plotting these points and connecting them with the characteristic "S" shape of a cubic graph will give you the graph of $g(x) = x^3 - 6$.

Summary of Transformation:

• Translation: 6 units downward.
• Shape: The graph remains the same in terms of its curve (it's still cubic with an inflection point at $(0, -6)$).

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Would you like more details or a visual representation? Feel free to ask!

Here are some related questions you might find interesting:

1. What happens if we shift the cubic function $f(x) = x^3$ to the right or left instead of up or down?
2. How does the graph of $g(x) = x^3 + 4$ compare to the graph of $g(x) = x^3 - 6$?
3. What is the effect of scaling the cubic function, such as multiplying by a constant like $2x^3$?
4. How would you graph the function $g(x) = -x^3 - 6$? What changes occur compared to $g(x) = x^3 - 6$?
5. What are some real-world applications where cubic functions and their transformations are used?

Tip: When shifting graphs, always remember that the transformation depends on the sign and magnitude of the constant added to or subtracted from the function!