Given a quadratic function: f(x) =x^2-2x-3. What is the range?
	all real numbers        B. {-4,-3,0,3,4}...       C. no range

To find the range of the quadratic function \( f(x) = x^2 - 2x - 3 \), we first complete the square to rewrite it in vertex form.

### Step 1: Complete the square
Start with:
\[
f(x) = x^2 - 2x - 3
\]

To complete the square, take the coefficient of \(x\) (which is \(-2\)), divide it by 2, and square it:
\[
\left(\frac{-2}{2}\right)^2 = 1
\]

Now, rewrite the function:
\[
f(x) = (x^2 - 2x + 1) - 1 - 3 = (x - 1)^2 - 4
\]

So, the function in vertex form is:
\[
f(x) = (x - 1)^2 - 4
\]

### Step 2: Analyze the vertex form
The vertex form \( f(x) = (x - 1)^2 - 4 \) reveals that the vertex is at \( (1, -4) \). Since the coefficient of the squared term \( (x - 1)^2 \) is positive, the parabola opens upwards. 

The minimum value of the function is \(-4\), and since the parabola opens upwards, \( f(x) \) can take all values greater than or equal to \(-4\).

### Conclusion:
The range of the function is:
\[
\boxed{[-4, \infty)}
\]

However, none of the options provided match this exact result. If you must choose from the given options, it's likely that "All real numbers" was meant to suggest an infinite range, but technically this is incorrect since the range is bounded below by \(-4\).

Would you like more details or further clarification on this? Here are 5 related questions you can consider:

1. How do you determine the vertex of a quadratic function?
2. What is the significance of completing the square in analyzing quadratics?
3. How do you find the domain and range of any quadratic function?
4. How does the sign of the leading coefficient affect the range of a quadratic?
5. What other methods can you use to find the minimum or maximum of a quadratic function?

**Tip:** The vertex form of a quadratic function is very useful for identifying both the vertex and the range quickly.

Given a quadratic function: f(x) =x^2 - 2x - 3. What is the range?

Explore how to find the range of the quadratic function f(x) = x^2 - 2x - 3 by completing the square and rewriting it in vertex form. Understand key concepts such as vertex and range of parabolas. This problem is suitable for students in Grades 9-12.

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