Use graphing technology to find the range of the function f, of, x, equals, left parenthesis, x, minus, 3, right parenthesis, squared, .f(x)=(x−3)^2

To find the range of the function \( f(x) = (x - 3)^2 \) using graphing technology, let's first analyze the function and then describe the graph.

### Analyzing the Function
The function \( f(x) = (x - 3)^2 \) is a quadratic function in standard form. It represents a parabola that opens upwards (since the coefficient of the squared term is positive). The vertex of the parabola is at \( (3, 0) \), which is the lowest point on the graph.

### Graphing the Function
When you graph \( f(x) = (x - 3)^2 \), you'll notice the following key features:

- **Vertex**: The vertex of the parabola is at \( (3, 0) \).
- **Axis of Symmetry**: The line \( x = 3 \) is the axis of symmetry for the parabola.
- **Direction**: The parabola opens upwards, meaning the function values increase as \( x \) moves away from the vertex on both sides.

### Determining the Range
Since the parabola opens upwards and the lowest point is at \( y = 0 \) (when \( x = 3 \)), the range of \( f(x) \) is all values of \( y \) that are greater than or equal to 0.

### Conclusion
Thus, the range of the function \( f(x) = (x - 3)^2 \) is:
\[
\text{Range} = [0, \infty)
\]

Would you like more details on the graph or have any questions?

Here are some related questions:
1. What is the domain of the function \( f(x) = (x - 3)^2 \)?
2. How would the graph of \( f(x) = (x - 3)^2 \) change if we added a constant term?
3. What is the range of the function \( f(x) = - (x - 3)^2 \)?
4. How can we determine the vertex of a quadratic function in general form?
5. How does the coefficient of the squared term affect the direction of the parabola?

**Tip:** The vertex form of a quadratic function, \( f(x) = a(x-h)^2 + k \), directly shows the vertex \( (h, k) \), making it easier to determine the range.

Discover how to find the range of the quadratic function f(x) = (x - 3)^2 using graphing technology. Understand the vertex form of a quadratic function and its impact on determining the range. Suitable for advanced high school students.

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