The given graph shows a semicircle centered at the origin with a radius of 1. This type of graph corresponds to the equation of a circle.

For a circle centered at the origin $(0, 0)$ with radius $r$, the equation is: $x^2 + y^2 = r^2$

Given the graph, the radius $r = 1$, thus the equation of the full circle is: $x^2 + y^2 = 1$

Since the graph shows only the upper semicircle, we need to consider only the positive values of $y$. Therefore, the equation representing the upper semicircle is: $y = \sqrt{1 - x^2}$

Determining the Domain

The domain of this function includes all $x$ values for which the expression under the square root is non-negative: $1 - x^2 \geq 0$ $x^2 \leq 1$ $-1 \leq x \leq 1$

Thus, the domain of the function $y = \sqrt{1 - x^2}$ is: $[-1, 1]$

Summary

• The function $y = \sqrt{1 - x^2}$ describes the upper semicircle.
• The domain of this function is $[-1, 1]$.

Would you like more details or have any questions?

Here are some related questions you might find interesting:

1. What is the range of the function $y = \sqrt{1 - x^2}$?
2. How would the graph change if the equation were $y = -\sqrt{1 - x^2}$?
3. What is the area of the semicircle described by $y = \sqrt{1 - x^2}$?
4. How do you derive the equation of a circle from its geometric properties?
5. What would be the equation and graph of a semicircle centered at $(h, k)$ with radius $r$?

Tip

When analyzing functions, always start by identifying their type, which helps in determining their key properties such as domain, range, and behavior.