The problem on your screen involves sketching the graph of the quadratic function $f(x) = (x - 2)^2 - 4$ by using translations (shifts) and comparing it to the graph of $f(x) = x^2$.

Let’s break down the solution step by step:

1. Standard Quadratic Function: The basic graph of $f(x) = x^2$ is a parabola that opens upwards with its vertex at $(0, 0)$.

2. Transformations: The given function is $f(x) = (x - 2)^2 - 4$. This involves two transformations:

   • The term $(x - 2)$ represents a horizontal shift 2 units to the right.
   • The term $-4$ represents a vertical shift 4 units downward.

3. Final Graph Description: So, the graph of $f(x) = (x - 2)^2 - 4$ is a parabola identical to the graph of $f(x) = x^2$, but it is shifted 2 units to the right and 4 units downward.

To fill in the blanks:

• The graph is shifted 2 units to the right.
• The graph is shifted 4 units down.

Would you like more details on graph transformations, or any specific questions on the concept?

Here are 5 follow-up questions that expand on this:

1. What is the general form of a quadratic function that involves horizontal and vertical shifts?
2. How would the graph change if the quadratic function included a coefficient, like $f(x) = 2(x - 2)^2 - 4$?
3. What is the vertex of the transformed parabola $f(x) = (x - 2)^2 - 4$?
4. Can you describe how to determine the direction in which a parabola opens?
5. What other types of transformations could be applied to quadratic functions?

Tip: When graphing functions, always pay attention to horizontal and vertical shifts as they dictate the new position of the vertex of a parabola.