can you determine if the relation $$ \left(x+4\right)^2+\left(y+6\right)^2=25\text{ defines }y\text{ as a function of }x. $$

To determine whether the relation $ (x+4)^2 + (y+6)^2 = 25 $ defines $ y $ as a function of $ x $, we can analyze the equation geometrically and algebraically.

### Step 1: Identify the type of equation
The given equation is in the standard form of a circle:

\[
(x - h)^2 + (y - k)^2 = r^2
\]

Here, $ (h, k) $ is the center of the circle, and $ r $ is the radius. Comparing with the given equation:

\[
(x + 4)^2 + (y + 6)^2 = 25
\]

We can rewrite it as:

\[
(x - (-4))^2 + (y - (-6))^2 = 5^2
\]

This represents a circle with:
- **Center**: $ (-4, -6) $
- **Radius**: $ 5 $

### Step 2: Determine if the relation defines $ y $ as a function of $ x $
For a relation to define $ y $ as a function of $ x $, each value of $ x $ must correspond to exactly one value of $ y $. 

However, a circle does not meet this criterion. A vertical line (which corresponds to a constant $ x $) generally intersects a circle at two points, meaning for a single $ x $ value, there could be two different $ y $ values. This violates the definition of a function.

### Conclusion
The relation $ (x+4)^2 + (y+6)^2 = 25 $ **does not** define $ y $ as a function of $ x $ because a circle is not a function. Specifically, there can be multiple $ y $-values for a given $ x $-value.

Would you like more details or have any questions?

Here are some related questions:

1. What is the vertical line test and how does it apply to this situation?
2. How can you determine if a given relation is a function algebraically?
3. How does the equation of a circle differ from the equation of a function?
4. Can you rewrite this relation to express $ y $ explicitly in terms of $ x $?
5. What other types of conic sections also fail to define $ y $ as a function of $ x $?

**Tip:** When analyzing whether a relation defines a function, remember that each input (x-value) must map to exactly one output (y-value).

Can you determine if the relation (x+4)^2+(y+6)^2=25 defines y as a function of x?

Explore whether the circle equation (x+4)^2 + (y+6)^2 = 25 defines y as a function of x. This problem covers key geometry concepts such as the equation of a circle and the vertical line test. Suitable for students in Grades 10-12.

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