Find an expression for the function whose graph is the bottom half of the parabola x + (y - 9)2 = 0.
y=?
some help would be appreciated! thanks
y = √-x + 6, x ≤ 0
To find an expression for the function whose graph is the bottom half of the parabola x + (y - 9)² = 0, we need to isolate the variable y.
Step 1: Rewrite the equation in standard form
Start by expanding the equation x + (y - 9)² = 0:
x + y² - 18y + 81 = 0
Step 2: Rearrange the equation
Move the x and constant term to the right side of the equation:
y² - 18y = -x - 81
Step 3: Complete the square
To complete the square, we add and subtract half of the coefficient of y (which in this case is -18) squared.
y² - 18y + (-18/2)² = -x - 81 + (-18/2)²
This simplifies to:
y² - 18y + 81 = -x - 81 + 81
y² - 18y + 81 = -x
Step 4: Factor the perfect square trinomial
The left side of the equation is a perfect square trinomial that can be factored as:
(y - 9)² = -x
Step 5: Take the square root of both sides
Taking the square root of both sides gives us:
y - 9 = ±√(-x)
Step 6: Solve for y
Solve for y by adding 9 to both sides:
y = 9 ± √(-x)
Therefore, the expression for the function whose graph is the bottom half of the parabola x + (y - 9)² = 0 is:
y = 9 - √(-x)
Note: It's important to note that the square root of a negative number (√(-x)) involves imaginary numbers. So, the expression y = 9 - √(-x) represents the bottom half of the parabola on the complex plane.