Equation: x² + y 2 = 1.
Solve the given equation using the “assumption method” let x = 4.  Sketch it after solving.Then determine whether y is a function of x.

Let's go through the steps to solve and analyze the equation:

### Step 1: Given Equation
The equation provided is:

\[
x^2 + y^2 = 1
\]

### Step 2: Apply the Assumption Method (Let \( x = 4 \))
We are asked to assume \( x = 4 \) and then solve for \( y \).

Substituting \( x = 4 \) into the equation:

\[
4^2 + y^2 = 1
\]

Simplifying further:

\[
16 + y^2 = 1
\]

### Step 3: Solve for \( y \)
Now, solve for \( y \):

\[
y^2 = 1 - 16
\]
\[
y^2 = -15
\]

Taking the square root of both sides:

\[
y = \pm\sqrt{-15}
\]

Since the square root of a negative number is not real, \( y \) would be an imaginary number in this case:

\[
y = \pm i\sqrt{15}
\]

### Step 4: Sketch the Graph
The original equation \( x^2 + y^2 = 1 \) represents a circle with a radius of 1 centered at the origin (0, 0).

However, substituting \( x = 4 \) into the equation leads to a value of \( y \) that is not real, indicating that there are no real points on this circle corresponding to \( x = 4 \).

### Step 5: Determine if \( y \) is a Function of \( x \)
For \( y \) to be a function of \( x \), each \( x \) value must correspond to exactly one \( y \) value. 

In this case, since \( x = 4 \) gives no real solution for \( y \), \( y \) is not a function of \( x \) over this domain. 

Additionally, the equation \( x^2 + y^2 = 1 \) generally yields two values of \( y \) for most \( x \) values within the domain \([-1, 1]\), meaning it fails the vertical line test and is not a function of \( x \).

### Summary
- **No real solution** exists for \( y \) when \( x = 4 \).
- The graph of \( x^2 + y^2 = 1 \) is a circle centered at the origin with a radius of 1.
- \( y \) is not a function of \( x \) because each \( x \) does not map to a single \( y \) value.

Would you like to see a detailed graph of the circle or have any other questions?

Here are 5 related questions to explore the topic further:
1. What is the maximum value of \( x \) for which \( y \) can have real solutions in the equation \( x^2 + y^2 = 1 \)?
2. How would the graph change if the equation was \( x^2 + y^2 = 4 \)?
3. What happens to the solutions for \( y \) when \( x \) is negative?
4. Can you find the points where the circle intersects the x-axis and y-axis?
5. How does the concept of imaginary numbers come into play when solving equations like this?

**Tip:** When working with the equation of a circle, remember that \( x \) and \( y \) values must satisfy \( x^2 + y^2 \leq r^2 \) for real number solutions, where \( r \) is the radius of the circle.

Explore how to solve the equation x^2 + y^2 = 1 using the assumption method and determine if y is a function of x. This problem involves understanding equations of circles, imaginary numbers, and functions. Suitable for high school students.

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