3Daet.com | Transformations

Graph transformations.

Mathematics.

Date Created:Monday January 22nd, 2007 02:36 AM
Date Modified:Saturday August 02nd, 2008 12:13 PM

ƒ(x) = 3cos(1/2x - π/2)

find vertical asymptotes

VA: 1/2x - π/2 = nπ/2; n odd integer
^{because tanθ = sinθ/cosθ; VA => cosθ = 0}

VA: x = nπ + π/2; n odd integer
^{multiplied by 2 and added π/2}

find x-intercepts

x-int: 1/2x - π/2 = nπ; n any integer
^{because tanθ = sinθ/cosθ; x-int => sinθ = 0}

x-int: x = 2nπ + π/2; n any integer
^{multiplied by 2 and added π/2}

find period
P/ω

period: ωx = 2π => 01/2x = 2π
^{because Period T = 2π/ω => ωx = 2π}

period: x = 4π
^{multiplied by 2}

find beginning (phase shift)
Φ/ω

beginning: ωx - Φ = 0 => 1/2x - π/2 = 0

beginning: x = Φ/ω => x = π
^{added Φ and multipled by 2}

find ending
Φ/ω + P/ω

ending: ωx - Φ = 2π => 1/2x - π/2 = 2π

ending: ωx = 2π + Φ => 1/2x = 2π + π/2
^{added Φ}

ending: x = Φ/ω + 2π/ω => x = π + 4π => 5π
^{multiplied by 2}