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To find the critical points and domain endpoints for the function \( y = x^2 - 6x + 7 \), let’s proceed with the following steps:

### 1. **Domain**
Since \( y = x^2 - 6x + 7 \) is a polynomial, it is defined for all real values of \( x \). Therefore, the domain is:
\[
(-\infty, \infty)
\]

### 2. **Finding Critical Points**
Critical points occur where the derivative is zero or undefined. Let’s calculate the derivative of \( y \) with respect to \( x \):

\[
y' = \frac{d}{dx}(x^2 - 6x + 7) = 2x - 6
\]

Set the derivative equal to zero to find critical points:
\[
2x - 6 = 0
\]
\[
x = 3
\]

Now, substitute \( x = 3 \) back into the original function to find the corresponding \( y \)-value:
\[
y = 3^2 - 6 \cdot 3 + 7 = 9 - 18 + 7 = -2
\]

So, the critical point is at \( (3, -2) \).

### 3. **Analyzing the Critical Point**
Since the second derivative \( y'' = 2 \) is positive, the function is concave up at \( x = 3 \), indicating that this critical point is a minimum.

### 4. **Domain Endpoints**
Since the function has an infinite domain \((-\infty, \infty)\), there are no finite domain endpoints.

### Summary
- **Critical Point**: \( (3, -2) \) (local minimum)
- **Domain Endpoints**: None (due to the infinite domain)

Would you like further details or have any questions?

### Follow-up Questions
1. How would the critical points change if the function was \( y = -x^2 - 6x + 7 \)?
2. Can we verify the nature of the critical point using the First Derivative Test?
3. What would the range of the function be, given that it’s a parabola opening upwards?
4. How do we determine points of inflection for polynomial functions?
5. What are the implications of an infinite domain for finding domain endpoints?

### Tip
In a quadratic function, the vertex represents the only critical point if the parabola opens upwards or downwards, making it either a minimum or maximum, respectively.

Determine all critical points and all domain endpoints for the function y = x^2 - 6x + 7.

This problem involves finding critical points and domain endpoints for the quadratic function y = x^2 - 6x + 7. Learn how to compute derivatives and apply the second derivative test to analyze the function's behavior. Suitable for students in Grades 10-12.

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