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
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.