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.