We are given two functions:

• $f(x) = x^2$
• $g(x) = (x - 2)^2 - 3$

Describing the transformation:

1. Horizontal Translation: The function $g(x) = (x - 2)^2 - 3$ contains $(x - 2)$ inside the square. This indicates a horizontal translation. Specifically, the graph is shifted 2 units to the right compared to the graph of $f(x) = x^2$.

2. Vertical Translation: The term $-3$ outside the square means the graph is shifted 3 units downward.

Filling in the blanks:

The graph of $g$ is a translation:

• 2 units right and
• 3 units down

of the graph of $f$.

Would you like a further explanation or a graph of these transformations? Here are some related questions you could explore:

1. What other transformations exist beyond translations?
2. How does the transformation affect the vertex of a parabola?
3. How would the graph look if the transformation was $g(x) = (x + 2)^2 + 3$?
4. What is the general form for describing transformations of quadratic functions?
5. Can you describe a reflection transformation for a quadratic function?

Tip: When analyzing transformations, always look for changes to the input (horizontal shifts) and the output (vertical shifts).