5.2 transformations of sinusoidal functions
TRANSCRIPT
Pre-Calc 12
5.2 Transformations of Sinusoidal
Functions
Big Idea:
Understanding the characteristics of families of functions allows us to model and understand
relationships and to build connections between classes of functions.
Curricular Competencies:
Explain and justify math ideas and decisions
Visualize to explore math
Vertical Displacement and Phase Shift
For periodic functions, a vertical translation is called a vertical displacement, while a horizontal
translation is called a phase shift.
Example 1: Sketch the graph of π¦ = sin(π₯ β 30Β°) + 1 for at least one cycle.
Vertical displacement: Period: Amplitude:
Phase shift: Domain: Range:
Pre-Calc 12
Example 2: Sketch the graph of π¦ = βcos(π₯ + π) β 1 for at least one cycle.
Vertical displacement: Period: Amplitude:
Phase shift: Domain: Range:
Transformation connection β¦.
π = ππ(π(π β π)) + π π = ππππ(π(π β π)) + π
Pre-Calc 12
Example 3: Sketch the graph of π¦ = 3sin (2π₯ β2π
3) + 2 for at least one cycle.
Vertical displacement: Period: Amplitude:
Phase shift: Domain: Range:
Example 4: Sketch the graph of π¦ = β2cosπ
6(π₯ + 3) β 1 for at least one cycle.
Vertical displacement: Period: Amplitude:
Phase shift: Domain: Range:
Pre-Calc 12
Equation of π¦ = ππ πππ(π₯ β π) + π or π¦ = ππππ π(π₯ β π) + π
π =ππππβππππ
π π =
ππ
π· π = starting point π = midline
Example 5: Write sinusoidal equations of the form π¦ = ππ πππ(π₯ β π) + π f and π¦ = ππππ π(π₯ β π) + π
to represent the function shown in the graph.
Example 6: Prince Rupert, British Columbia, has the deepest natural harbor in North America. The
depth, d, in meters, of the berths for the ships can be approximated by the equation π(π‘) = 8πππ π
6π‘ +
12, where π‘ is the time, in hours, after the first high tide.
a) Using your graphing calculator, graph the function over 2 cycles.
b) What is the period of the tide?
c) An ocean liner requires a minimum of 13 m
of water to dock safely. Determine the
number of hours per cycle the ocean liner
can safely dock.
d) If the minimum depth of the berth occurs
at 6 h, determine the depth of the water.
At what other times is the water level at a
Minimum?
Assignment: p 250 1 acef, 2 acef, 3-5, 6ac, 7ac, 9, 10, 13, 14-16, 20, 26
Hint β¦ 5280 ft = 1 mi