Question

find an expression for the function whose graph is the given curve. the bottom half of the parabola ( x + (y - 6)^2 = 0 ) ( y = )

Explanation:

Step1: Rearrange the equation

We start with the given equation of the parabola: \( x + (y - 6)^2 = 0 \). We want to solve for \( y \) in terms of \( x \). First, we isolate the squared term:
\( (y - 6)^2 = -x \)

Step2: Take the square root

To solve for \( y \), we take the square root of both sides. Remember that when we take the square root, we have two solutions: one with the positive square root and one with the negative square root. However, we are interested in the bottom half of the parabola, which corresponds to the negative square root (since the bottom half will have \( y \) values less than the vertex's \( y \)-coordinate, and taking the negative square root will give the lower half). So we have:
\( y - 6 = -\sqrt{-x} \)

Step3: Solve for \( y \)

Now we solve for \( y \) by adding 6 to both sides of the equation:
\( y = 6 - \sqrt{-x} \)

Answer:

\( 6 - \sqrt{-x} \)