Question

Find an expression for the function whose graph is the given curve. The top half of the circle\({{\bf{x}}^{\bf{2}}}{\bf{ + }}{\left( {{\bf{y - 2}}} \right)^{\bf{2}}}{\bf{ = 4}}\)

Short answer

The expression for the top half of the circle is\({\bf{y = 2 + }}\sqrt {{\bf{4 - }}{{\bf{x}}^{\bf{2}}}} \).

Step-by-step solution

Determine the combined expression for the bottom and top half of the circle

The given equation can be written as:

\(\begin{aligned}{c}{x^2} + {\left( {y - 2} \right)^2} = 4\\y - 2 = \pm \sqrt {4 - {x^2}} \\y = 2 \pm \sqrt {4 - {x^2}} \end{aligned}\)

Determine the expression of the top half of the circle

The plus sign of the above expression gives the top half of the circle and the negative sign gives the bottom half of the circle so the expression for the bottom half of the given parabola is given as:

\(y = 2 + \sqrt {4 - {x^2}} \)

Therefore, the expression for the top half of the circle is\(y = 2 + \sqrt {4 - {x^2}} \).