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PRE-CALCULUS GRAPHS OF PERIODIC FUNCTIONS Sinusoidal Graphs
Periodic Functions:
The Graph of Sine:
Domain: Range: Period: The Graph of the Cosine:
Domain: Range: Period: Transforming Graphs: A point rotates around a circle of radius 3. Sketch a graph of the y coordinate of the point.
Amplitude: If π = πππππ½, then the value in front of the sine function is called the The letter A is designated as the amplitude factor: y = A circle with radius 3 feet is mounted with its center 4 feet off the ground. The point closest to the ground is labeled P. Sketch a graph of the height above ground of the point P as the circle is rotated, then find a function that gives the height in terms of the angle of rotation.
Equation: The Midline of a Periodic Function: The center value of a sinusoidal function is the value that the function oscillates above and below is called the midline. It corresponds to a ________________________ shift. π(π½) = πππ(π½) + π π = ____ ππ πππ πππ ππππ The Period of a Function: The value at which the sinusoidal function will _____________ its cycle.
π(π½) = πππ(π©π½) ππ
π© π πππππππππ πππ _______________
Find the Amplitudes, Midlines and Periods of the functions below:
π = βππππ (π
ππ½) + π π =
π
ππππ (
π
ππ) β π
Write a formula for the periodic function graphed below.
Homework for Sinusoidal Graphs Determine the Amplitude, Midline and Period for the Sinusoidal Functions. Sketch a
graph.
π = ππππ (π
ππ½) π = βπππ(ππ) β π
π = βππππ (π
ππ½) + π π = πππ(ππ) β π
Determine the Amplitude, Midline and Period of the functions below. Write the function.
PRE-CALCULUS GRAPHS OF PERIODIC FUNCTIONS More on Sinusoidal Graphs
What we have learned so far: Given an equation in the form: f(t) = Asin(Bt) + k or f(t) = cos(Bt) + k A is the ________________ stretch and is the ___________________. B is the ________________ stretch/compression is determines the ____________ P, by the formula π· =
k is the _____________ shift and determines the _________________
A point completes 1 revolution every 2 minutes around a circle of radius 5. Graph, then state its amplitude, midline and period if it starts at (5, 0). Write an equation for the function.
Determine the midline, amplitude and period of the function: f(t) = 3 sin(2t) + 1 and then graph.
Find the formula for the sinusoidal function graphed here.
Graph the following after stating the amplitude, midline and period.
π(π) = βπ πππ (π
ππ½) β π π = π πππ (ππ½) + π
Homework for More on Sinusoidal Graphs If a sinusoidal function starts on the midline at point (0, 3), has an amplitude of 2, and a period of 4, write the formula for the function. Outside temperature over the course of a day can be modeled as a sinusoidal function. Suppose you know the temperature is 50 degrees at midnight and the high and low temperature during the day are 57 and 43 degrees, respectively. Assuming t is the number of hours since midnight, find a function for the temperature, D, in terms of t. A Ferris wheel is 25 meters in diameter and boarded from a platform that is 1 meter above the ground. The six o'clock position on the Ferris wheel is level with the loading platform. The wheel completes 1 full revolution in 10 minutes. The function h(t) gives your height in meters above the ground t minutes after the wheel begins to turn.
a. Find the amplitude, midline, and period of h(t).
b. Find a formula for the height function h(t)
c. How high are you off the ground after 5 minutes?
Find the amplitude, midline and period of the following functions. Then graph each.
π = ππππ (π
ππ½) π = βπππ(ππ) β π
π = βππππ (π
ππ½) + π π = πππ(ππ) β π
PRE-CALCULUS GRAPHS OF PERIODIC FUNCTIONS Even More on Sinusoidal Graphs
Horizontal Shifts: Given an equation in the form:
π(π) = π¨πππ(π©(π β π)) + π ππ π(π) = π¨πππ(π©(π β π)) + π
h is the horizontal shift of the function. Give the period, shift and direction of the following:
π = ππππ((ππ½ β π)) π = βππππ ((π
ππ + π ))
Sketch a graph of π(π) = ππππ (π
ππ β
π
π)
Find a formula for the function graphed here.
When it comes to finding the shift, there are 4 different combinations using sine and cosine and shifts. Cosine shifted ___ to the ________ or a negative cosine shifted ___ to the ________. Sine shifted ___ to the _________ or a negative sine shifted _____ to the _________.
Write an equation of a sine function with a midline of -6, that is reflected about the x-axis,
with an amplitude of 3, a period of 6 and a shift of π
π to the left.
π = πππ( ( π½ ))
Write an equation of a cosine function with a midline of 2, with an amplitude of 5, a period
of ππ
π and a shift of 4 to the right.
π = πππ( ( π½ ))
Homework for Still More on Sinusoidal Graph
Write a formula for the graphs below.
For each of the following equations, find the amplitude, period, horizontal shift and midline.
π = ππππ(π(π + π)) + π π = ππππ(ππ β ππ) + π π = πππ (π
ππ + π ) β π
Find the formula for the graphs below.
Graph the following functions:
π = ππππ(ππ β π) + π π = βπππ (π
ππ½ +
ππ
π) β π
PRE-CALCULUS GRAPHS OF PERIODIC FUNCTIONS Graphs of the Other Trig Functions
The Tangent Graph: π = πππ π½
The Period of the tangent graph is _____. Domain: _____________ Range: __________ Find the formula for the graph below.
Sketch the graph of: π(π½) = ππππ (
π
ππ½)
The Graph of Cotangent: π = πππ π½
The Period of the Cotangent graph is _____. Domain: _____________ Range: __________
The Graph of the Secant: π = πππ π½
The Period of the Secant graph is _____. Domain: _____________ Range: ____________ The Graph of the Cosecant: π = πππ π½
The Period of the Cosecant graph is _____. Domain: _____________ Range: ____________
Sketch the graph of π = ππππ (π
ππ½) + π (Hint: First sketch π = ππππ (
π
ππ½) + π)
Homework for Graphs of the Other Trig Functions
Given the graph of π = ππππ (π
ππ½) + π, sketch the graph of π = ππππ (
π
ππ½) + π
Label each graph as tangent, secant, cotangent or cosecant.
Find the period and horizontal shift of each and then sketch:
π = ππππ(ππ β π) π = ππππ (π
π(π + π))
Write the formula for each graph: