Question

Find an expression for the function whose graph is the given curve. The bottom half of the parabola\({\bf{x + }}{\left( {{\bf{y - 1}}} \right)^{\bf{2}}}{\bf{ = 0}}\)

Short answer

The expression for the bottom half of the parabola is\({\bf{y = 1 - }}\sqrt {{\bf{ - x}}} \).

Step-by-step solution

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

The equation of the parabola is\({\bf{x + }}{\left( {{\bf{y - 1}}} \right)^{\bf{2}}}{\bf{ = 0}}.\)

The given equation can be written as:

\(\begin{array}{c}x + {\left( {y - 1} \right)^2} = 0\\y - 1 = \pm \sqrt { - x} \\y = 1 \pm \sqrt { - x} \end{array}\)

Determine the expression of the bottom half of the parabola

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

\(y = 1 - \sqrt { - x} \)

Therefore, the expression for the bottom half of the parabola is\(y = 1 - \sqrt { - x} \).