We are tasked with analyzing the graph transformation from the parent function $f(x) = x^3$ to the translated function $g(x) = (x-2)^3 - 3$. Let's break this down:

Step 1: Understand the transformation

The function $g(x) = (x-2)^3 - 3$ involves two transformations applied to the parent function $f(x) = x^3$:

1. Horizontal Shift: The term $(x - 2)$ means the graph is shifted 2 units to the right.
2. Vertical Shift: The term $-3$ means the graph is shifted 3 units downward.

Thus, the graph of $f(x) = x^3$ is moved right by 2 and down by 3.

Step 2: Identify the new key point

For the parent function $f(x) = x^3$, the key point is the origin $(0, 0)$. After the transformation:

• The point $(0, 0)$ shifts to $(2, -3)$.

Step 3: Analyze the graph's shape

The cubic function $x^3$ has the same general shape across transformations. It is symmetric about its inflection point (the key point), with the left side decreasing and the right side increasing.

Step 4: Match with the correct graph

From the graphs provided, look for the one that has:

• The inflection point at $(2, -3)$,
• The same cubic shape as $f(x) = x^3$.

The correct graph will reflect the horizontal and vertical shifts described above.

Would you like me to provide further details or clarification?
Here are some follow-up questions and a tip to deepen understanding:

5 Questions to Expand Understanding

1. How does the general formula for transformations of functions help analyze shifts and stretches?
2. What happens to the graph of $g(x)$ if the $-3$ is changed to $+3$?
3. How would the transformation change if $g(x) = (x + 2)^3 - 3$?
4. How can you determine the inflection point of any cubic function?
5. Why does the function retain the same overall shape despite the transformations?

Tip

When analyzing transformations, always look for changes inside the parentheses (horizontal shifts) and outside (vertical shifts). This systematic approach ensures you won't miss any details!