To find a formula for the function whose graph represents the bottom half of the parabola given by the equation [tex]\( x + (y-5)^2 = 0 \)[/tex], follow these steps:
1. Rewrite the Equation:
The original equation is:
[tex]\[
x + (y - 5)^2 = 0
\][/tex]
2. Isolate the Quadratic Term:
To solve for [tex]\( y \)[/tex], start by isolating the quadratic term on one side of the equation:
[tex]\[
(y - 5)^2 = -x
\][/tex]
3. Take the Square Root:
To solve for [tex]\( y \)[/tex], take the square root of both sides. Remember that when taking the square root, you should consider both the positive and negative roots. However, since we are interested in the bottom half of the parabola, we will consider only the negative root:
[tex]\[
y - 5 = -\sqrt{-x}
\][/tex]
4. Solve for [tex]\( y \)[/tex]:
Finally, solve for [tex]\( y \)[/tex] by adding 5 to both sides of the equation:
[tex]\[
y = 5 - \sqrt{-x}
\][/tex]
Thus, the formula for the function representing the bottom half of the parabola defined by the equation [tex]\( x + (y-5)^2 = 0 \)[/tex] is:
[tex]\[
y = 5 - \sqrt{-x}
\][/tex]
This function [tex]\( y = 5 - \sqrt{-x} \)[/tex] will trace out the bottom half of the given parabola.
