Question

Find an expression for the function whose graph is the given curve.

The top half of the circle x^2 + (y − 3)^2 = 4

Answer 1

The expression for the function whose graph is the top half of the given circle is y = 3 ± √(4 - x^2).

Step-by-step explanation:

The student is asking for the expression of a function that represents the top half of a circle. The equation of the circle given is x^2 + (y − 3)^2 = 4. To find the function for the top half of the circle, we need to solve for y taking into account that we only want the positive square root since we are interested in the top part of the circle.

Rewrite the given equation as:

(y - 3)^2 = 4 - x^2

Then, take the square root of both sides, remember to add the constraint that y must be greater than or equal to 3 because this is the top half:

y - 3 = √(4 - x^2)

Finally, solve for y:

y = 3 + √(4 - x^2)

This is the function that represents the top half of the circle. Note that this function is not defined for x values where 4 - x^2 is negative, since the square root of a negative number is not a real number.

Answer 2

We can accomplish this by:

(y-3)^2=4-x^2

Now, lets take the square root of both sides:

y-3=±sqrt(4-x^2) y=±sqrt(4-x^2)+3

Now, we only need the top half or the positive half so this means that our answer is simply:

y=sqrt(4-x^2)+3