https://file.aiquickdraw.com/mathbot/file/1728609063020zbc86gry.jpg

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?

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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.

Sketch the graph of the quadratic function f(x) = (x - 2)^2 - 4 by using translations (vertical or horizontal shifts). Compare the graph to the graph of f(x) = x^2.

Learn how to graph the quadratic function f(x) = (x - 2)^2 - 4 using translations and compare it with the basic graph of f(x) = x^2. Understand horizontal and vertical shifts in graph transformations. Suitable for Grades 9-11.

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