To find the range of the function defined by the equation \( y^2 = x^2 + 2x - 3 \), we'll proceed as follows:

1. **Rewrite the equation**: \( y^2 = x^2 + 2x - 3 \)
2. **Complete the square** for the expression on the right side to make it easier to analyze.

### Completing the Square

Given:
\[ y^2 = x^2 + 2x - 3 \]

Complete the square for the expression in \( x \):
\[ x^2 + 2x = (x + 1)^2 - 1 \]

Thus, the equation becomes:
\[ y^2 = (x + 1)^2 - 1 - 3 \]
\[ y^2 = (x + 1)^2 - 4 \]

### Analyze the Expression

We can rewrite this as:
\[ y^2 + 4 = (x + 1)^2 \]

Since the left side represents \( y^2 + 4 \) and is always non-negative and greater than or equal to 4 (because \( y^2 \geq 0 \)), the right side, \( (x + 1)^2 \), must also be greater than or equal to 4.

Thus, the expression \( (x + 1)^2 \) reaches its minimum value of 4 when \( x + 1 = \pm 2 \), i.e., \( x = 1 \) or \( x = -3 \).

### Finding the Range of \( y \)

From the equation \( y^2 + 4 = (x + 1)^2 \), we can solve for \( y^2 \):
\[ y^2 = (x + 1)^2 - 4 \]

Since \( (x + 1)^2 \geq 4 \), we have:
\[ (x + 1)^2 - 4 \geq 0 \]

Thus:
\[ y^2 \geq 0 \]

The smallest value \( y^2 \) can take is 0, which happens when \( (x + 1)^2 = 4 \). There is no upper bound for \( y^2 \), as \( (x + 1)^2 \) can increase indefinitely.

Therefore, the range of \( y^2 \) is \( [0, \infty) \). Since \( y^2 \geq 0 \), the value of \( y \) can be any real number, both positive and negative.

### Range of the Function

The range of the function \( y \) is:
\[ y \in (-\infty, \infty) \]

Do you have any questions, or would you like a more detailed explanation?

Here are some related questions you might explore:

1. What is the domain of the function \( y^2 = x^2 + 2x - 3 \)?
2. How would the range change if the equation was \( y = x^2 + 2x - 3 \)?
3. What is the range of the function \( y = \sqrt{x^2 + 2x - 3} \)?
4. How would the function change if we add a constant to both sides?
5. Can \( y \) be a complex number in this equation?
6. What transformations are applied to the parent function \( y = x^2 \) in this case?
7. What is the significance of completing the square in this context?
8. How would the equation change if the term \( -3 \) was instead \( +3 \)?

**Tip:** Completing the square is a valuable technique for transforming quadratic expressions, making it easier to analyze their properties such as vertex, range, and roots.

Discover how to find the range of the quadratic function y² = x² + 2x - 3. Learn about completing the square and analyzing quadratic expressions. Suitable for high school students studying quadratic equations.

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