blake rotated the parabola y equals x/2 by 180 ° around a point the new vertex is (6, - 8) what is the equation of the new parabola

Answer 1

To rotate the parabola y = x/2 by 180° around a point, we can use the rotation formula:

x' = x₀ + (x - x₀)cosθ - (y - y₀)sinθ
y' = y₀ + (x - x₀)sinθ + (y - y₀)cosθ

Since we are rotating 180° around the point (6, -8), we have x₀ = 6 and y₀ = -8.

Substituting these values, we get the following equations for the new coordinates (x', y'):

x' = 6 + (x - 6)cos(180°) - (y - (-8))sin(180°)
y' = -8 + (x - 6)sin(180°) + (y - (-8))cos(180°)

Simplifying these equations:

x' = 6 + 6 - x + 8sin(180°) - ycos(180°)
y' = -8 - 8sin(180°) + y - 8cos(180°)

Since sin(180°) = 0 and cos(180°) = -1, the equations become:

x' = 12 - x - y
y' = -16 + y

Therefore, the equation of the new rotated parabola is:

y' = (x' - 12)² - 16

Simplifying this equation further:

y' = x'² - 24x' + 144 - 16

y' = x'² - 24x' + 128

Hence, the equation of the new parabola is y = x² - 24x + 128.